This is asymptote.info, produced by makeinfo version 7.2 from
asymptote.texi.

This file documents ‘Asymptote’, version .

   <https://asymptote.sourceforge.io>

   Copyright © 2004-2026 Andy Hammerlindl, John Bowman, and Tom Prince.

     Permission is granted to copy, distribute and/or modify this
     document under the terms of the GNU Lesser General Public License
     (see the file LICENSE in the top-level source directory).

INFO-DIR-SECTION Languages
START-INFO-DIR-ENTRY
* asymptote: (asymptote/asymptote). Vector graphics language.
END-INFO-DIR-ENTRY


File: asymptote.info,  Node: Top,  Next: Description,  Up: (dir)

Asymptote
*********

This file documents ‘Asymptote’, version .

   <https://asymptote.sourceforge.io>

   Copyright © 2004-2026 Andy Hammerlindl, John Bowman, and Tom Prince.

     Permission is granted to copy, distribute and/or modify this
     document under the terms of the GNU Lesser General Public License
     (see the file LICENSE in the top-level source directory).

* Menu:

* Description::                 What is ‘Asymptote’?
* Installation::                Downloading and installing
* Tutorial::                    Getting started
* Drawing commands::            Four primitive drawing commands
* Bezier curves::               Path connectors and direction specifiers
* Programming::                 The ‘Asymptote’ vector graphics language
* LaTeX usage::                 Embedding ‘Asymptote’ commands within LaTeX
* User modules::                Modules commonly used by users
* Specialty modules::           Modules for specialized applications
* Developer modules::           Modules for ‘Asymptote’ developers
* Options::                     Command-line options
* Interactive mode::            Typing ‘Asymptote’ commands interactively
* GUI::                         Graphical user interface
* Command-Line Interface::      Remote HTTP API to Asymptote server
* Language server protocol::    Help when writing code
* PostScript to Asymptote::     ‘Asymptote’ backend to ‘pstoedit’
* Help::                        Where to get help and submit bug reports
* Debugger::                    Squish those bugs!
* Credits::                     Contributions and acknowledgments
* General Index::               General index

 -- The Detailed Node Listing --

Installation

* UNIX binary distributions::   Prebuilt ‘UNIX’ binaries
* macOS X binary distributions::  Prebuilt ‘macOS X’ binaries
* Microsoft Windows::           Prebuilt ‘Microsoft Windows’ binary
* Configuring::                 Configuring ‘Asymptote’ for your system
* Search paths::                Where ‘Asymptote’ looks for your files
* Compiling from UNIX source::  Building ‘Asymptote’ from scratch
* Editing modes::               Convenient ‘emacs’ and ‘vim’ modes
* Git::                         Getting the latest development source
* Building the documentation::  Building the documentation
* Uninstall::                   Goodbye, ‘Asymptote’!

Tutorial

* Drawing in batch mode::       Run ‘Asymptote’ on a text file
* Drawing in interactive mode::  Running ‘Asymptote’ interactively
* Figure size::                 Specifying the figure size
* Labels::                      Adding LaTeX labels
* Paths::                       Drawing lines and curves

Drawing commands

* draw::                        Draw a path on a picture or frame
* fill::                        Fill a cyclic path on a picture or frame
* clip::                        Clip a picture or frame to a cyclic path
* label::                       Label a point on a picture

Programming

* Data types::                  void, bool, int, real, pair, triple, string
* Paths and guides::            Bezier curves
* Pens::                        Colors, line types, line widths, font sizes
* Transforms::                  Affine transforms
* Frames and pictures::         Canvases for immediate and deferred drawing
* Deferred drawing::            Withholding drawing until all data is available
* Files::                       Reading and writing your data
* Variable initializers::       Initialize your variables
* Structures::                  Organize your data
* Operators::                   Arithmetic and logical operators
* Implicit scaling::            Avoiding those ugly *s
* Functions::                   Traditional and higher-order functions
* Arrays::                      Dynamic vectors
* Casts::                       Implicit and explicit casts
* Import::                      Importing external ‘Asymptote’ modules
* Iterators::                   Iterating over structures
* Static::                      Where to allocate your variable?
* Autounravel::                 Adding associated libraries to structures
* Hash functions::              Hashing your data
* Collections (containers)::    Maps, sets, queues, and iterator tools

Operators

* Arithmetic & logical::        Basic mathematical operators
* Self & prefix operators::     Increment and decrement
* User-defined operators::      Overloading operators
* Bracket operators::           Bracket operators
* Cast operators::              Type conversion

Functions

* Default arguments::           Default values can appear anywhere
* Named arguments::             Assigning function arguments by keyword
* Rest arguments::              Functions with a variable number of arguments
* Mathematical functions::      Standard libm functions

Arrays

* Slices::                      Python-style array slices

Import

* Templated imports::           Imports parametrized by generic types

Iterators

* range::                       Iterating over integers

Autounravel

* Where fields are autounraveled::  Where autounraveling happens
* Where autounravel is legal::  Valid autounravel locations

Hash functions

* Hashing user-defined structures::  Implementing hash methods for custom types

Collections (containers)

* Pairs::                       Generic pair of two types with hash support
* Iterators and utilities::     Iterable structures and manipulation tools
* Maps::                        Hashmaps and b-tree maps (dictionaries/associative arrays)
* Using Maps as Sets::          ‘Map_K_bool’ for set operations
* Sets::                        Hash sets and sorted b-tree sets
* Using Sets as Maps::          ‘Set_Pair_K_V’ to map keys to values
* Queues::                      First-in-first-out (FIFO) queue
* Wrapping arrays::             Arrays that can be used as keys or values

Iterators and utilities

* collections.iter(T)::  Iterable structures
* Combining (zipping) iterators::  Iterating over multiple containers simultaneously
* Attaching a counter to an iterator::  Enumerating elements with indices

Maps

* Initializing a Map::   Creating hashmaps and b-tree maps
* The Map API::          Methods for map operations

Sets

* Initializing a Set::   Creating hash sets and b-tree sets
* The Set API::          Methods for set operations
* The SortedSet API::    Additional methods for sorted sets

User modules

* graph::                       2D linear & logarithmic graphs
* three::                       3D vector graphics
* graph3::                      3D linear & logarithmic graphs
* palette::                     Color density images and palettes
* colormap::                    Colormaps from matplotlib
* texcolors::                   CMYK TeX colors
* x11colors::                   RGB ‘X11’ colors
* fontsize::                    Nonstandard font sizes
* contour::                     Contour lines
* geometry::                    Geometry routines
* grid3::                       3D grids
* interpolate::                 Interpolation routines
* markers::                     Custom path marker routines
* math::                        Extend ‘Asymptote’'s math capabilities
* patterns::                    Custom fill and draw patterns
* roundedpath::                 Round the sharp corners of paths
* slopefield::                  Slope fields
* smoothcontour3::              Smooth implicit surfaces
* solids::                      3D solid geometry
* stats::                       Statistics routines and histograms
* tube::                        3D rotation minimizing tubes

Specialty modules

* animation::                   Embedded PDF and MPEG movies
* animate::                     Portable clickable PDF movies with controls
* annotate::                    Annotate your PDF files
* babel::                       Interface to LaTeX ‘babel’ package
* binarytree::                  Binary tree drawing module
* bsp::                         Binary space partition for hidden surface removal
* CAD::                         2D CAD pen and measurement functions (DIN 15)
* contour3::                    Contour surfaces
* drawtree::                    Tree drawing module
* embed::                       Embedding movies, sounds, and 3D objects
* external::                    Portable movie embedding
* feynman::                     Feynman diagrams
* flowchart::                   Flowchart drawing routines
* labelpath::                   Drawing curved labels
* labelpath3::                  Drawing curved labels in 3D
* lmfit::                       Least-squares fitting
* MetaPost::                    ‘MetaPost’ compatibility routines
* obj::                         3D obj files
* ode::                         Ordinary differential equations
* pstoedit::                    ‘PostScript’ to ‘Asymptote’ conversion
* rational::                    Rational number arithmetic
* rationalSimplex::             Simplex method for rational numbers
* simplex::                     Linear programming: simplex method
* size10::                      Named font sizes for 10pt LaTeX documents
* size11::                      Named font sizes for 11pt LaTeX documents
* slide::                       Making presentations with ‘Asymptote’
* syzygy::                      Syzygy and braid drawing module
* tree::                        Dynamic binary search tree
* trembling::                   Wavy lines

Developer modules

* plain::                       Default ‘Asymptote’ base file
* bezulate::                    Subdivision surface module
* simplex2::                    Two-variable simplex method
* v3d::                         V3D 3D graphics format

Graphical User Interface

* GUI installation::            Installing ‘xasy’
* GUI usage::                   Using ‘xasy’ to edit objects


File: asymptote.info,  Node: Description,  Next: Installation,  Prev: Top,  Up: Top

1 Description
*************

‘Asymptote’ is a powerful descriptive vector graphics language that
provides a mathematical coordinate-based framework for technical
drawing.  Labels and equations are typeset with LaTeX, for overall
document consistency, yielding the same high level of typesetting
quality that LaTeX provides for scientific text.  By default it produces
‘PostScript’ output, but it can also generate ‘OpenGL’, ‘PDF’, ‘SVG’,
‘WebGL’, ‘V3D’, and legacy ‘PRC’ vector graphics, along with any format
that the ‘ImageMagick’ package can produce.  You can even try it out in
your Web browser without installing it, using the ‘Asymptote Web
Application’:

   <https://asymptote.ualberta.ca>

   It is also possible to send remote commands to this server via the
curl utility (*note Command-Line Interface::).

   A major advantage of ‘Asymptote’ over other graphics packages is that
it is a high-level programming language, as opposed to just a graphics
application: it can therefore exploit the best features of the script
(command-driven) and graphical-user-interface (GUI) methods for
producing figures.  The rudimentary GUI ‘xasy’ included with the package
allows one to move script-generated objects around.  To make ‘Asymptote’
accessible to the average user, this GUI is currently being developed
into a full-fledged interface that can generate objects directly.
However, the script portion of the language is now ready for general use
by users who are willing to learn a few simple ‘Asymptote’ graphics
commands (*note Drawing commands::).

   ‘Asymptote’ is mathematically oriented (one can use complex
multiplication to rotate a vector) and uses LaTeX to do the typesetting
of labels.  This is an important feature for scientific applications.
It was inspired by an earlier drawing program (with a weaker syntax and
capabilities) called ‘MetaPost’.

   The ‘Asymptote’ vector graphics language provides:

   • a standard for typesetting mathematical figures, just as TeX/LaTeX
     is the de-facto standard for typesetting equations.

   • LaTeX typesetting of labels, for overall document consistency;

   • the ability to generate and embed 3D vector WebGL graphics within
     HTML files;

   • the ability to generate and embed 3D vector PRC graphics within PDF
     files;

   • a natural coordinate-based framework for technical drawing,
     inspired by ‘MetaPost’, with a much cleaner, powerful C++-like
     programming syntax;

   • compilation of figures into virtual machine code for speed, without
     sacrificing portability;

   • the power of a script-based language coupled to the convenience of
     a GUI;

   • customization using its own C++-like graphics programming language;

   • sensible defaults for graphical features, with the ability to
     override;

   • a high-level mathematically oriented interface to the ‘PostScript’
     language for vector graphics, including affine transforms and
     complex variables;

   • functions that can create new (anonymous) functions;

   • deferred drawing that uses the simplex method to solve overall size
     constraint issues between fixed-sized objects (labels and
     arrowheads) and objects that should scale with figure size;

   Many of the features of ‘Asymptote’ are written in the ‘Asymptote’
language itself.  While ‘Asymptote’ is designed for mathematical
typesetting needs, one can write ‘Asymptote’ modules that tailor it to
specific applications.  For example, a scientific graphing module is
available (*note graph::).  Examples of ‘Asymptote’ code and output,
including animations, are available at
     <https://asymptote.sourceforge.io/gallery/>
Clicking on an example file name in this manual, like ‘Pythagoras’, will
display the PDF output, whereas clicking on its ‘.asy’ extension will
show the corresponding ‘Asymptote’ code in a separate window.

   Links to many external resources, including an excellent user-written
‘Asymptote’ tutorial can be found at
     <https://asymptote.sourceforge.io/links.html>
   A quick reference card for ‘Asymptote’ is available at
     <https://asymptote.sourceforge.io/asyRefCard.pdf>


File: asymptote.info,  Node: Installation,  Next: Tutorial,  Prev: Description,  Up: Top

2 Installation
**************

* Menu:

* UNIX binary distributions::   Prebuilt ‘UNIX’ binaries
* macOS X binary distributions::  Prebuilt ‘macOS X’ binaries
* Microsoft Windows::           Prebuilt ‘Microsoft Windows’ binary
* Configuring::                 Configuring ‘Asymptote’ for your system
* Search paths::                Where ‘Asymptote’ looks for your files
* Compiling from UNIX source::  Building ‘Asymptote’ from scratch
* Editing modes::               Convenient ‘emacs’ and ‘vim’ modes
* Git::                         Getting the latest development source
* Building the documentation::  Building the documentation
* Uninstall::                   Goodbye, ‘Asymptote’!

After following the instructions for your specific distribution, please
see also *note Configuring::.

We recommend subscribing to new release announcements at
     <https://sourceforge.net/projects/asymptote>
Users may also wish to monitor the ‘Asymptote’ forum:
     <https://sourceforge.net/p/asymptote/discussion/409349>


File: asymptote.info,  Node: UNIX binary distributions,  Next: macOS X binary distributions,  Up: Installation

2.1 UNIX binary distributions
=============================

We release both ‘tgz’ and RPM binary distributions of ‘Asymptote’.  The
root user can install the ‘Linux x86_64’ ‘tgz’ distribution of version
‘x.xx’ of ‘Asymptote’ with the commands:
tar -C / -zxf asymptote-x.xx.x86_64.tgz
texhash
The ‘texhash’ command, which installs LaTeX style files, is optional.
The executable file will be ‘/usr/local/bin/asy’ and example code will
be installed by default in ‘/usr/local/share/doc/asymptote/examples’.

Fedora users can easily install a recent version of ‘Asymptote’ with the
command
dnf --enablerepo=rawhide install asymptote

To install the latest version of ‘Asymptote’ on a Debian-based
distribution (such as Ubuntu) follow the instructions for compiling from
‘UNIX’ source (*note Compiling from UNIX source::).  Alternatively,
Debian users can install one of Hubert Chathi's prebuilt ‘Asymptote’
binaries from
     <https://ftp.debian.org/debian/pool/main/a/asymptote>


File: asymptote.info,  Node: macOS X binary distributions,  Next: Microsoft Windows,  Prev: UNIX binary distributions,  Up: Installation

2.2 macOS X binary distributions
================================

‘macOS X’ users can either compile the ‘UNIX’ source code (*note
Compiling from UNIX source::) or install the ‘Asymptote’ binary
available at

   <https://www.macports.org/>

   or at

   <https://brew.sh/>

Note that many ‘macOS X’ (and FreeBSD) systems lack the GNU ‘readline’
library.  For full interactive functionality, GNU ‘readline’ version 4.3
or later must be installed.


File: asymptote.info,  Node: Microsoft Windows,  Next: Configuring,  Prev: macOS X binary distributions,  Up: Installation

2.3 Microsoft Windows
=====================

Users of the ‘Microsoft Windows’ operating system can install the
self-extracting ‘Asymptote’ executable ‘asymptote-x.xx-setup.exe’, where
‘x.xx’ denotes the latest version.

   A working TeX implementation (we recommend
<https://www.tug.org/texlive> or <https://miktex.org>) will be required
to typeset labels.  You will also need to install ‘GPL Ghostscript’
version 9.56 or later from <https://www.ghostscript.com/>.

   To view ‘PostScript’ output, you can install the program ‘Sumatra
PDF’ available from <https://www.sumatrapdfreader.org/>.

   The ‘ImageMagick’ package from
<https://www.imagemagick.org/script/binary-releases.php>

is required to support output formats other than HTML, PDF, SVG, and PNG
(*note magick::).  The ‘Python 3’ interpreter from
<https://www.python.org> is only required if you wish to try out the
graphical user interface (*note GUI::).

Example code will be installed by default in the ‘examples’ subdirectory
of the installation directory (by default, ‘C:\Program
Files\Asymptote’).


File: asymptote.info,  Node: Configuring,  Next: Search paths,  Prev: Microsoft Windows,  Up: Installation

2.4 Configuring
===============

In interactive mode, or when given the ‘-V’ option (the default when
running ‘Asymptote’ on a single file under ‘MSDOS’), ‘Asymptote’ will
automatically invoke your ‘PostScript’ viewer (‘evince’ under ‘UNIX’) to
display graphical output.  The ‘PostScript’ viewer should be capable of
automatically redrawing whenever the output file is updated.  The ‘UNIX’
‘PostScript’ viewer ‘gv’ supports this (via a ‘SIGHUP’ signal).  Users
of ‘ggv’ will need to enable ‘Watch file’ under ‘Edit/PostScript Viewer
Preferences’.

   Configuration variables are most easily set as ‘Asymptote’ variables
in an optional configuration file ‘config.asy’ (*note configuration
file::).  For example, the setting ‘pdfviewer’ specifies the location of
the PDF viewer.  Here are the default values of several important
configuration variables under ‘UNIX’:

import settings;
pdfviewer="acroread";
htmlviewer="google-chrome";
psviewer="evince";
display="display";
animate="animate";
gs="gs";
libgs="";

Under ‘MSDOS’, the viewer settings ‘htmlviewer’, ‘pdfviewer’,
‘psviewer’, ‘display’, and ‘animate’ default to the string ‘cmd’,
requesting the application normally associated with each file type.  The
(installation-dependent) default values of ‘gs’ and ‘libgs’ are
determined automatically from the ‘Microsoft Windows’ registry.  The
‘gs’ setting specifies the location of the ‘PostScript’ processor
‘Ghostscript’, available from <https://www.ghostscript.com/>.

The configuration variable ‘htmlviewer’ specifies the browser to use to
display 3D ‘WebGL’ output.  The default setting is ‘google-chrome’ under
‘UNIX’ and ‘cmd’ under ‘Microsoft Windows’.  Note that ‘Internet
Explorer’ does not support ‘WebGL’; ‘Microsoft Windows’ users should set
their default html browser to ‘chrome’ or ‘microsoft-edge’.  By default,
2D and 3D ‘HTML’ images expand to the enclosing canvas; this can be
disabled by setting the configuration variable ‘absolute’ to ‘true’.

   On ‘UNIX’ systems, to support automatic document reloading of ‘PDF’
files in ‘Adobe Reader’, we recommend copying the file ‘reload.js’ from
the ‘Asymptote’ system directory (by default,
‘/usr/local/share/asymptote’ under ‘UNIX’ to
‘~/.adobe/Acrobat/x.x/JavaScripts/’, where ‘x.x’ represents the
appropriate ‘Adobe Reader’ version number.  The automatic document
reload feature must then be explicitly enabled by putting
import settings;
pdfreload=true;
pdfreloadOptions="-tempFile";
in the ‘Asymptote’ configuration file.  This reload feature is not
useful under ‘MSDOS’ since the document cannot be updated anyway on that
operating system until it is first closed by ‘Adobe Reader’.

   The configuration variable ‘dir’ can be used to adjust the search
path (*note Search paths::).

   By default, ‘Asymptote’ attempts to center the figure on the page,
assuming that the paper type is ‘letter’.  The default paper type may be
changed to ‘a4’ with the configuration variable ‘papertype’.  Alignment
to other paper sizes can be obtained by setting the configuration
variables ‘paperwidth’ and ‘paperheight’.

   These additional configuration variables normally do not require
adjustment:
config
texpath
texcommand
dvips
dvisvgm
convert
asygl

   Warnings (such as "unbounded" and "offaxis") may be enabled or
disabled with the functions
warn(string s);
nowarn(string s);
or by directly modifying the string array ‘settings.suppress’, which
lists all disabled warnings.

   Configuration variables may also be set or overwritten with a
command-line option:
asy -psviewer=evince -V venn

   Alternatively, system environment versions of the above configuration
variables may be set in the conventional way.  The corresponding
environment variable name is obtained by converting the configuration
variable name to upper case and prepending ‘ASYMPTOTE_’: for example, to
set the environment variable
ASYMPTOTE_PAPERTYPE="a4";
under ‘Microsoft Windows XP’:
  1. Click on the ‘Start’ button;
  2. Right-click on ‘My Computer’;
  3. Choose ‘View system information’;
  4. Click the ‘Advanced’ tab;
  5. Click the ‘Environment Variables’ button.


File: asymptote.info,  Node: Search paths,  Next: Compiling from UNIX source,  Prev: Configuring,  Up: Installation

2.5 Search paths
================

In looking for ‘Asymptote’ files, ‘asy’ will search the following paths,
in the order listed:
  1. The current directory;
  2. A list of one or more directories specified by the configuration
     variable ‘dir’ or environment variable ‘ASYMPTOTE_DIR’ (separated
     by ‘:’ under UNIX and ‘;’ under ‘MSDOS’);
  3. The directory specified by the environment variable
     ‘ASYMPTOTE_HOME’; if this variable is not set, the directory ‘.asy’
     in the user's home directory (‘%USERPROFILE%\.asy’ under ‘MSDOS’)
     is used;
  4. The ‘Asymptote’ system directory (by default,
     ‘/usr/local/share/asymptote’ under ‘UNIX’ and ‘C:\Program
     Files\Asymptote’ under ‘MSDOS’).
  5. The ‘Asymptote’ examples directory (by default,
     ‘/usr/local/share/doc/asymptote/examples’ under ‘UNIX’ and
     ‘C:\Program Files\Asymptote\examples’ under ‘MSDOS’).


File: asymptote.info,  Node: Compiling from UNIX source,  Next: Editing modes,  Prev: Search paths,  Up: Installation

2.6 Compiling from UNIX source
==============================

To compile and install a ‘UNIX’ executable from the source release
‘asymptote-x.xx.src.tgz’ in the subdirectory ‘x.xx’ under

   <https://sourceforge.net/projects/asymptote/files/>

   execute the commands:
gunzip asymptote-x.xx.src.tgz
tar -xf asymptote-x.xx.src.tar
cd asymptote-x.xx

   Then compile ‘Asymptote’ with the commands
./configure
make all
make install
You may need to replace the last line with ‘sudo make install’ after
first running ‘make all’ without root privileges (to avoid generating
files with incorrect permissions).  Be sure to use GNU ‘make’ (on
non-GNU systems this command may be called ‘gmake’).  To build the
documentation, you may need to install the ‘texinfo-tex’ package.  If
you get errors from a broken ‘texinfo’ or ‘pdftex’ installation, simply
put
     <https://asymptote.sourceforge.io/asymptote.pdf>
in the directory ‘doc’ and repeat the command ‘make all’.

For a (default) system-wide installation, the last command should be
done as the root user.  To install without root privileges, change the
‘./configure’ command to
./configure --prefix=$HOME/asymptote
   One can disable use of the Boehm garbage collector by configuring
with ‘./configure --disable-gc’.  For a list of other configuration
options, say ‘./configure --help’.  For example, under ‘macOS X’, one
can tell configure to use the ‘clang’ compilers and look for header
files and libraries in nonstandard locations:
./configure CC=clang CXX=clang++ CPPFLAGS=-I/opt/local/include LDFLAGS=-L/opt/local/lib

   If you are compiling ‘Asymptote’ with ‘gcc’, you will need a version
that supports C++17.  For full interactive functionality, you will need
version 4.3 or later of the GNU ‘readline’ library.  The file
‘gcc3.3.2curses.patch’ in the ‘patches’ directory can be used to patch
the broken curses.h header file (or a local copy thereof in the current
directory) on some ‘AIX’ and ‘IRIX’ systems.

   The ‘FFTW’ library is only required if you want ‘Asymptote’ to be
able to take Fourier transforms of data (say, to compute an audio power
spectrum).  The ‘GSL’ library is only required if you require the
special functions that it supports.

   If you don't want to install ‘Asymptote’ system wide, just make sure
the compiled binary ‘asy’ and GUI script ‘xasy’ are in your path and set
the configuration variable ‘dir’ to point to the directory ‘base’ (in
the top level directory of the ‘Asymptote’ source code).


File: asymptote.info,  Node: Editing modes,  Next: Git,  Prev: Compiling from UNIX source,  Up: Installation

2.7 Editing modes
=================

Users of ‘emacs’ can edit ‘Asymptote’ code with the mode ‘asy-mode’,
after enabling it by putting the following lines in their ‘.emacs’
initialization file, replacing ‘ASYDIR’ with the location of the
‘Asymptote’ system directory (by default, ‘/usr/local/share/asymptote’
or ‘C:\Program Files\Asymptote’ under ‘MSDOS’):
(add-to-list 'load-path "ASYDIR")
(autoload 'asy-mode "asy-mode.el" "Asymptote major mode." t)
(autoload 'lasy-mode "asy-mode.el" "hybrid Asymptote/Latex major mode." t)
(autoload 'asy-insinuate-latex "asy-mode.el" "Asymptote insinuate LaTeX." t)
(add-to-list 'auto-mode-alist '("\\.asy$" . asy-mode))

Particularly useful key bindings in this mode are ‘C-c C-c’, which
compiles and displays the current buffer, and the key binding ‘C-c ?’,
which shows the available function prototypes for the command at the
cursor.  For full functionality you should also install the Apache
Software Foundation package ‘two-mode-mode’:
     <https://github.com/erlyaws/yaws/blob/master/two-mode-mode.el>
The hybrid mode ‘lasy-mode’ can be used to edit a LaTeX file containing
embedded ‘Asymptote’ code (*note LaTeX usage::).  This mode can be
enabled within ‘latex-mode’ with the key sequence ‘M-x lasy-mode <RET>’.
On ‘UNIX’ systems, additional keywords will be generated from all ‘asy’
files in the space-separated list of directories specified by the
environment variable ‘ASYMPTOTE_SITEDIR’.  Further documentation of
‘asy-mode’ is available within ‘emacs’ by pressing the sequence keys
‘C-h f asy-mode <RET>’.

   Fans of ‘vim’ can customize ‘vim’ for ‘Asymptote’ with

‘cp /usr/local/share/asymptote/asy.vim ~/.vim/syntax/asy.vim’

and add the following to their ‘~/.vimrc’ file:
augroup filetypedetect
au BufNewFile,BufRead *.asy     setf asy
augroup END
filetype plugin on

   If any of these directories or files don't exist, just create them.
To set ‘vim’ up to run the current asymptote script using ‘:make’ just
add to ‘~/.vim/ftplugin/asy.vim’:
setlocal makeprg=asy\ %
setlocal errorformat=%f:\ %l.%c:\ %m

   Syntax highlighting support for the KDE editor ‘Kate’ can be enabled
by running ‘asy-kate.sh’ in the ‘/usr/local/share/asymptote’ directory
and putting the generated ‘asymptote.xml’ file in
‘~/.local/share/org.kde.syntax-highlighting/syntax/’.


File: asymptote.info,  Node: Git,  Next: Building the documentation,  Prev: Editing modes,  Up: Installation

2.8 Git
=======

The following commands are needed to install the latest development
version of ‘Asymptote’ using ‘git’:
git clone https://github.com/vectorgraphics/asymptote

cd asymptote
./autogen.sh
./configure
make all
make install

To compile without optimization On ‘Ubuntu’ systems, you may need to
first install the required dependencies:
apt-get build-dep asymptote


File: asymptote.info,  Node: Building the documentation,  Next: Uninstall,  Prev: Git,  Up: Installation

2.9 Building the documentation
==============================

To build the PDF documentation doc/asymptote.pdf:
make man
   To build the HTML documentation doc/png/index.html:
make html

   The ‘asy’ executable is required for compiling the diagrams in the
documentation.


File: asymptote.info,  Node: Uninstall,  Prev: Building the documentation,  Up: Installation

2.10 Uninstall
==============

To uninstall a ‘Linux x86_64’ binary distribution
tar -zxvf asymptote-x.xx.x86_64.tgz | xargs --replace=% rm /%
texhash

To uninstall all ‘Asymptote’ files installed from a source distribution,
use the command
make uninstall


File: asymptote.info,  Node: Tutorial,  Next: Drawing commands,  Prev: Installation,  Up: Top

3 Tutorial
**********

* Menu:

* Drawing in batch mode::       Run ‘Asymptote’ on a text file
* Drawing in interactive mode::  Running ‘Asymptote’ interactively
* Figure size::                 Specifying the figure size
* Labels::                      Adding LaTeX labels
* Paths::                       Drawing lines and curves

A concise introduction to ‘Asymptote’ is given here.  For a more
thorough introduction, see the excellent ‘Asymptote’ tutorial written by
Charles Staats:

   <https://asymptote.sourceforge.io/asymptote_tutorial.pdf>

   Another ‘Asymptote’ tutorial is available as a wiki, with images
rendered by an online Asymptote engine:

   <https://www.artofproblemsolving.com/wiki/?title=Asymptote_(Vector_Graphics_Language)>


File: asymptote.info,  Node: Drawing in batch mode,  Next: Drawing in interactive mode,  Up: Tutorial

3.1 Drawing in batch mode
=========================

To draw a line from coordinate (0,0) to coordinate (100,100), create a
text file ‘test.asy’ containing
draw((0,0)--(100,100));

Then execute the command
asy -V test
Alternatively, ‘MSDOS’ users can drag and drop ‘test.asy’ onto the
Desktop ‘asy’ icon (or make ‘Asymptote’ the default application for the
extension ‘asy’).

This method, known as _batch mode_, outputs a ‘PostScript’ file
‘test.eps’.  If you prefer PDF output, use the command line
asy -V -f pdf test
   In either case, the ‘-V’ option opens up a viewer window so you can
immediately view the result:

                             [./diagonal]
Here, the ‘--’ connector joins the two points ‘(0,0)’ and ‘(100,100)’
with a line segment.


File: asymptote.info,  Node: Drawing in interactive mode,  Next: Figure size,  Prev: Drawing in batch mode,  Up: Tutorial

3.2 Drawing in interactive mode
===============================

Another method is _interactive mode_, where ‘Asymptote’ reads individual
commands as they are entered by the user.  To try this out, enter
‘Asymptote’'s interactive mode by clicking on the ‘Asymptote’ icon or
typing the command ‘asy’.  Then type
draw((0,0)--(100,100));
followed by ‘Enter’, to obtain the above image.

   At this point you can type further ‘draw’ commands, which will be
added to the displayed figure, ‘erase’ to clear the canvas,
input test;
to execute all of the commands contained in the file ‘test.asy’, or
‘quit’ to exit interactive mode.  You can use the arrow keys in
interactive mode to edit previous lines.  The tab key will automatically
complete unambiguous words; otherwise, hitting tab again will show the
possible choices.  Further commands specific to interactive mode are
described in *note Interactive mode::.


File: asymptote.info,  Node: Figure size,  Next: Labels,  Prev: Drawing in interactive mode,  Up: Tutorial

3.3 Figure size
===============

In ‘Asymptote’, coordinates like ‘(0,0)’ and ‘(100,100)’, called
_pairs_, are expressed in ‘PostScript’ "big points" (1 ‘bp’ = 1/72
‘inch’) and the default line width is ‘0.5bp’.  However, it is often
inconvenient to work directly in ‘PostScript’ coordinates.  The next
example produces identical output to the previous example, by scaling
the line ‘(0,0)--(1,1)’ to fit a rectangle of width ‘100.5bp’ and height
‘100.5bp’ (the extra ‘0.5bp’ accounts for the line width):
size(100.5,100.5);
draw((0,0)--(1,1));

                             [./diagonal]

   One can also specify the size in ‘pt’ (1 ‘pt’ = 1/72.27 ‘inch’),
‘cm’, ‘mm’, or ‘inches’.  Two nonzero size arguments (or a single size
argument) restrict the size in both directions, preserving the aspect
ratio.  If 0 is given as a size argument, no restriction is made in that
direction; the overall scaling will be determined by the other direction
(*note size::):
size(0,100.5);
draw((0,0)--(2,1),Arrow);

                            [./bigdiagonal]

   To connect several points and create a cyclic path, use the ‘cycle’
keyword:
size(3cm);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);

                              [./square]
For convenience, the path ‘(0,0)--(1,0)--(1,1)--(0,1)--cycle’ may be
replaced with the predefined variable ‘unitsquare’, or equivalently,
‘box((0,0),(1,1))’.

   To make the user coordinates represent multiples of exactly ‘1cm’:
unitsize(1cm);
draw(unitsquare);


File: asymptote.info,  Node: Labels,  Next: Paths,  Prev: Figure size,  Up: Tutorial

3.4 Labels
==========

Adding labels is easy in ‘Asymptote’; one specifies the label as a
double-quoted LaTeX string, a coordinate, and an optional alignment
direction:
size(3cm);
draw(unitsquare);
label("$A$",(0,0),SW);
label("$B$",(1,0),SE);
label("$C$",(1,1),NE);
label("$D$",(0,1),NW);

                            [./labelsquare]

   ‘Asymptote’ uses the standard compass directions ‘E=(1,0)’,
‘N=(0,1)’, ‘NE=unit(N+E)’, and ‘ENE=unit(E+NE)’, etc., which along with
the directions ‘up’, ‘down’, ‘right’, and ‘left’ are defined as pairs in
the ‘Asymptote’ base module ‘plain’ (a user who has a local variable
named ‘E’ may access the compass direction ‘E’ by prefixing it with the
name of the module where it is defined: ‘plain.E’).


File: asymptote.info,  Node: Paths,  Prev: Labels,  Up: Tutorial

3.5 Paths
=========

This example draws a path that approximates a quarter circle, terminated
with an arrowhead:
size(100,0);
draw((1,0){up}..{left}(0,1),Arrow);

                           [./quartercircle]
Here the directions ‘up’ and ‘left’ in braces specify the outgoing and
incoming directions at the points ‘(1,0)’ and ‘(0,1)’, respectively.

   In general, a path is specified as a list of points (or other paths)
interconnected with ‘--’, which denotes a straight line segment, or
‘..’, which denotes a cubic spline (*note Bezier curves::).  Specifying
a final ‘..cycle’ creates a cyclic path that connects smoothly back to
the initial node, as in this approximation (accurate to within 0.06%) of
a unit circle:
path unitcircle=E..N..W..S..cycle;

An ‘Asymptote’ path, being connected, is equivalent to a ‘PostScript
subpath’.  The ‘^^’ binary operator, which requests that the pen be
moved (without drawing or affecting endpoint curvatures) from the final
point of the left-hand path to the initial point of the right-hand path,
may be used to group several ‘Asymptote’ paths into a ‘path[]’ array
(equivalent to a ‘PostScript’ path):
size(0,100);
path unitcircle=E..N..W..S..cycle;
path g=scale(2)*unitcircle;
filldraw(unitcircle^^g,evenodd+yellow,black);


                             [./superpath]

The ‘PostScript’ even-odd fill rule here specifies that only the region
bounded between the two unit circles is filled (*note fillrule::).  In
this example, the same effect can be achieved by using the default zero
winding number fill rule, if one is careful to alternate the orientation
of the paths:
filldraw(unitcircle^^reverse(g),yellow,black);

   The ‘^^’ operator is used by the ‘box(triple, triple)’ function in
the module ‘three’ to construct the edges of a cube ‘unitbox’ without
retracing steps (*note three::):
import three;

currentprojection=orthographic(5,4,2,center=true);

size(5cm);
size3(3cm,5cm,8cm);

draw(unitbox);

dot(unitbox,red);

label("$O$",(0,0,0),NW);
label("(1,0,0)",(1,0,0),S);
label("(0,1,0)",(0,1,0),E);
label("(0,0,1)",(0,0,1),Z);

                               [./cube]

   See section *note graph:: (or the online ‘Asymptote’ gallery and
external links posted at <https://asymptote.sourceforge.io>) for further
examples, including two-dimensional and interactive three-dimensional
scientific graphs.  Additional examples have been posted by Philippe
Ivaldi at <https://blog.piprime.fr/asymptote/>.


File: asymptote.info,  Node: Drawing commands,  Next: Bezier curves,  Prev: Tutorial,  Up: Top

4 Drawing commands
******************

All of ‘Asymptote’'s graphical capabilities are based on four primitive
commands.  The three ‘PostScript’ drawing commands ‘draw’, ‘fill’, and
‘clip’ add objects to a picture in the order in which they are executed,
with the most recently drawn object appearing on top.  The labeling
command ‘label’ can be used to add text labels and external EPS images,
which will appear on top of the ‘PostScript’ objects (since this is
normally what one wants), but again in the relative order in which they
were executed.  After drawing objects on a picture, the picture can be
output with the ‘shipout’ function (*note shipout::).

   If you wish to draw ‘PostScript’ objects on top of labels (or
verbatim ‘tex’ commands; *note tex::), the ‘layer’ command may be used
to start a new ‘PostScript/LaTeX’ layer:
void layer(picture pic=currentpicture);

   The ‘layer’ function gives one full control over the order in which
objects are drawn.  Layers are drawn sequentially, with the most recent
layer appearing on top.  Within each layer, labels, images, and verbatim
‘tex’ commands are always drawn after the ‘PostScript’ objects in that
layer.

   A page break can be generated with the command
void newpage(picture pic=currentpicture);

   While some of these drawing commands take many options, they all have
sensible default values (for example, the picture argument defaults to
currentpicture).

* Menu:

* draw::                        Draw a path on a picture or frame
* fill::                        Fill a cyclic path on a picture or frame
* clip::                        Clip a picture or frame to a cyclic path
* label::                       Label a point on a picture


File: asymptote.info,  Node: draw,  Next: fill,  Up: Drawing commands

4.1 draw
========

void draw(picture pic=currentpicture, Label L="", path g,
          align align=NoAlign, pen p=currentpen,
          arrowbar arrow=None, arrowbar bar=None, margin margin=NoMargin,
          Label legend="", marker marker=nomarker);

   Draw the path ‘g’ on the picture ‘pic’ using pen ‘p’ for drawing,
with optional drawing attributes (Label ‘L’, explicit label alignment
‘align’, arrows and bars ‘arrow’ and ‘bar’, margins ‘margin’, legend,
and markers ‘marker’).  Only one parameter, the path, is required.  For
convenience, the arguments ‘arrow’ and ‘bar’ may be specified in either
order.  The argument ‘legend’ is a Label to use in constructing an
optional legend entry.

   Bars ‘bar’ are useful for indicating dimensions.  The possible values
of ‘bar’ are ‘None’, ‘BeginBar’, ‘EndBar’ (or equivalently ‘Bar’), and
‘Bars’ (which draws a bar at both ends of the path).  Each of these bar
specifiers (except for ‘None’) will accept an optional real argument
that denotes the length of the bar in ‘PostScript’ coordinates.  The
default bar length is ‘barsize(pen)’.

   The possible values of ‘arrow’ are ‘None’, ‘Blank’ (which draws no
arrows or path), ‘BeginArrow’, ‘MidArrow’, ‘EndArrow’ (or equivalently
‘Arrow’), and ‘Arrows’ (which draws an arrow at both ends of the path).

   There are also arrow versions with slightly modified default values
of ‘size’ and ‘angle’ suitable for curved arrows: ‘BeginArcArrow’,
‘EndArcArrow’ (or equivalently ‘ArcArrow’), ‘MidArcArrow’, and
‘ArcArrows’.

   For example:
draw((0,0)--(1,1),arrow=Arrows);

   All of the arrow specifiers except for ‘None’ and ‘Blank’ may be
given optional arguments, for example:
draw((0,0)--(1,1),arrow=Arrow(
     arrowhead=HookHead,size=3mm,angle=20,filltype=Draw,position=0.9));

   The function ‘Arrow’ has the signature
arrowbar Arrow(arrowhead arrowhead=DefaultHead,
               real size=0, real angle=arrowangle,
               filltype filltype=null, position position=EndPoint)
Calling ‘Arrow()’ returns ‘Arrow’, which is an ‘arrowbar’ object.  The
parameters are:

   • ‘arrowhead’ can be one of the predefined arrowhead styles
     ‘DefaultHead’, ‘SimpleHead’, ‘HookHead’, ‘TeXHead’.
   • real ‘size’ is the arrowhead size in ‘PostScript’ coordinates.

     The default arrowhead size when drawn with a pen ‘p’ is
     ‘arrowsize(p)’.
   • real ‘angle’ is the arrowhead angle in degrees.

   • filltype ‘filltype’ (*note filltype::),
   • (except for ‘MidArrow’ and ‘Arrows’) real ‘position’ (in the sense
     of ‘point(path p, real t)’) along the path where the tip of the
     arrow should be placed.

   Margins ‘margin’ can be used to shrink the visible portion of a path
by ‘labelmargin(p)’ to avoid overlap with other drawn objects.

   Typical values of ‘margin’ are:
‘NoMargin’
‘BeginMargin’
‘EndMargin’
     (equivalently ‘Margin’)
‘Margins’
     leaves a margin at both ends of the path.
‘Margin(real begin, real end=begin)’
     specify the size of the beginning and ending margin, respectively,
     in multiples of the units ‘labelmargin(p)’ used for aligning
     labels.
‘BeginPenMargin’
‘EndPenMargin’
     (equivalently ‘PenMargin’)
‘PenMargins’
‘PenMargin(real begin, real end=begin)’
     specify a margin in units of the pen line width, taking account of
     the pen line width when drawing the path or arrow.
‘DotMargin’
     an abbreviation for ‘PenMargin(-0.5*dotfactor,0.5*dotfactor)’, used
     to draw from the usual beginning point just up to the boundary of
     an end dot of width ‘dotfactor*linewidth(p)’.
‘BeginDotMargin’
‘DotMargins’
     work similarly.
‘TrueMargin(real begin, real end=begin)’
     specify a margin directly in ‘PostScript’ units, independent of the
     pen line width.

   The use of arrows, bars, and margins is illustrated by the examples
‘Pythagoras.asy’ and ‘sqrtx01.asy’.  Since bars are intended for
indicating precise dimensions, they should only be drawn with NoMargin
(the default).

   The legend for a picture ‘pic’ can be fit and aligned to a frame with
the routine:
frame legend(picture pic=currentpicture, int perline=1,
             real xmargin=legendmargin, real ymargin=xmargin,
             real linelength=legendlinelength,
             real hskip=legendhskip, real vskip=legendvskip,
             real maxwidth=0, real maxheight=0,
             bool hstretch=false, bool vstretch=false, pen p=currentpen);
Here ‘xmargin’ and ‘ymargin’ specify the surrounding x and y margins,
‘perline’ specifies the number of entries per line (default 1; 0 means
choose this number automatically), ‘linelength’ specifies the length of
the path lines, ‘hskip’ and ‘vskip’ specify the line skip (as a multiple
of the legend entry size), ‘maxwidth’ and ‘maxheight’ specify optional
upper limits on the width and height of the resulting legend (0 means
unlimited), ‘hstretch’ and ‘vstretch’ allow the legend to stretch
horizontally or vertically, and ‘p’ specifies the pen used to draw the
bounding box.  The legend frame can then be added and aligned about a
point on a picture ‘dest’ using ‘add’ or ‘attach’ (*note add about::).

   To draw a dot, simply draw a path containing a single point.  The
‘dot’ command defined in the module ‘plain’ draws a dot having a
diameter equal to an explicit pen line width or the default line width
magnified by ‘dotfactor’ (6 by default), using the specified filltype
(*note filltype::) or ‘dotfilltype’ (‘Fill’ by default):
void dot(frame f, pair z, pen p=currentpen, filltype filltype=dotfilltype);
void dot(picture pic=currentpicture, pair z, pen p=currentpen,
         filltype filltype=dotfilltype);
void dot(picture pic=currentpicture, Label L, pair z, align align=NoAlign,
         string format=defaultformat, pen p=currentpen, filltype filltype=dotfilltype);
void dot(picture pic=currentpicture, Label[] L=new Label[], pair[] z,
         align align=NoAlign, string format=defaultformat, pen p=currentpen,
         filltype filltype=dotfilltype);
void dot(picture pic=currentpicture, path[] g, pen p=currentpen,
         filltype filltype=dotfilltype);
void dot(picture pic=currentpicture, Label L, pen p=currentpen,
         filltype filltype=dotfilltype);

   If the variable ‘Label’ is given as the ‘Label’ argument to the third
routine, the ‘format’ argument will be used to format a string based on
the dot location (here ‘defaultformat’ is ‘"$%.4g$"’).  The fourth
routine draws a dot at every point of a pair array ‘z’.  One can also
draw a dot at every node of a path:
void dot(picture pic=currentpicture, Label[] L=new Label[],
         explicit path g, align align=RightSide, string format=defaultformat,
         pen p=currentpen, filltype filltype=dotfilltype);
   See *note pathmarkers:: and *note markers:: for more general methods
for marking path nodes.

   To draw a fixed-sized object (in ‘PostScript’ coordinates) about the
user coordinate ‘origin’, use the routine
void draw(pair origin, picture pic=currentpicture, Label L="", path g,
          align align=NoAlign, pen p=currentpen, arrowbar arrow=None,
          arrowbar bar=None, margin margin=NoMargin, Label legend="",
          marker marker=nomarker);


File: asymptote.info,  Node: fill,  Next: clip,  Prev: draw,  Up: Drawing commands

4.2 fill
========

void fill(picture pic=currentpicture, path g, pen p=currentpen);

   Fill the interior region bounded by the cyclic path ‘g’ on the
picture ‘pic’, using the pen ‘p’.

   There is also a convenient ‘filldraw’ command, which fills the path
and then draws in the boundary.  One can specify separate pens for each
operation:
void filldraw(picture pic=currentpicture, path g, pen fillpen=currentpen,
              pen drawpen=currentpen);

   This fixed-size version of ‘fill’ allows one to fill an object
described in ‘PostScript’ coordinates about the user coordinate
‘origin’:
void fill(pair origin, picture pic=currentpicture, path g, pen p=currentpen);

This is just a convenient abbreviation for the commands:
picture opic;
fill(opic,g,p);
add(pic,opic,origin);

   The routine
void filloutside(picture pic=currentpicture, path g, pen p=currentpen);
fills the region exterior to the path ‘g’, out to the current boundary
of picture ‘pic’.

   Lattice gradient shading varying smoothly over a two-dimensional
array of pens ‘p’, using fill rule ‘fillrule’, can be produced with
void latticeshade(picture pic=currentpicture, path g, bool stroke=false,
                  pen fillrule=currentpen, pen[][] p)
   If ‘stroke=true’, the region filled is the same as the region that
would be drawn by ‘draw(pic,g,zerowinding)’; in this case the path ‘g’
need not be cyclic.  The pens in ‘p’ must belong to the same color
space.  One can use the functions ‘rgb(pen)’ or ‘cmyk(pen)’ to promote
pens to a higher color space, as illustrated in the example file
‘latticeshading.asy’.

   Axial gradient shading varying smoothly from ‘pena’ to ‘penb’ in the
direction of the line segment ‘a--b’ can be achieved with
void axialshade(picture pic=currentpicture, path g, bool stroke=false,
                pen pena, pair a, bool extenda=true,
                pen penb, pair b, bool extendb=true);
The boolean parameters ‘extenda’ and ‘extendb’ indicate whether the
shading should extend beyond the axis endpoints ‘a’ and ‘b’.  An example
of axial shading is provided in the example file ‘axialshade.asy’.

   Radial gradient shading varying smoothly from ‘pena’ on the circle
with center ‘a’ and radius ‘ra’ to ‘penb’ on the circle with center ‘b’
and radius ‘rb’ is similar:
void radialshade(picture pic=currentpicture, path g, bool stroke=false,
                 pen pena, pair a, real ra, bool extenda=true,
                 pen penb, pair b, real rb, bool extendb=true);
The boolean parameters ‘extenda’ and ‘extendb’ indicate whether the
shading should extend beyond the radii ‘a’ and ‘b’.  Illustrations of
radial shading are provided in the example files ‘shade.asy’,
‘ring.asy’, and ‘shadestroke.asy’.

   Gouraud shading using fill rule ‘fillrule’ and the vertex colors in
the pen array ‘p’ on a triangular lattice defined by the vertices ‘z’
and edge flags ‘edges’ is implemented with
void gouraudshade(picture pic=currentpicture, path g, bool stroke=false,
                  pen fillrule=currentpen, pen[] p, pair[] z,
                  int[] edges);
void gouraudshade(picture pic=currentpicture, path g, bool stroke=false,
                  pen fillrule=currentpen, pen[] p, int[] edges);
In the second form, the elements of ‘z’ are taken to be successive nodes
of path ‘g’.  The pens in ‘p’ must belong to the same color space.
Illustrations of Gouraud shading are provided in the example file
‘Gouraud.asy’.  The edge flags used in Gouraud shading are documented on
pages 270-274 of the PostScript Language Reference (3rd edition):
     <https://www.adobe.com/jp/print/postscript/pdfs/PLRM.pdf>

   Tensor product shading using clipping path ‘g’, fill rule ‘fillrule’
on patches bounded by the n cyclic paths of length 4 in path array ‘b’,
using the vertex colors specified in the n \times 4 pen array ‘p’ and
internal control points in the n \times 4 array ‘z’, is implemented with
void tensorshade(picture pic=currentpicture, path[] g, bool stroke=false,
                 pen fillrule=currentpen, pen[][] p, path[] b=g,
                 pair[][] z=new pair[][]);
If the array ‘z’ is empty, Coons shading, in which the color control
points are calculated automatically, is used.  The pens in ‘p’ must
belong to the same color space.  A simpler interface for the case of a
single patch (n=1) is also available:
void tensorshade(picture pic=currentpicture, path g, bool stroke=false,
                 pen fillrule=currentpen, pen[] p, path b=g,
                 pair[] z=new pair[]);
   One can also smoothly shade the regions between consecutive paths of
a sequence using a given array of pens:
void draw(picture pic=currentpicture, pen fillrule=currentpen, path[] g,
          pen[] p);
Illustrations of tensor product and Coons shading are provided in the
example files ‘tensor.asy’, ‘Coons.asy’, ‘BezierPatch.asy’, and
‘rainbow.asy’.

   More general shading possibilities are available using TeX engines
that produce PDF output (*note texengines::): the routine
void functionshade(picture pic=currentpicture, path[] g, bool stroke=false,
                   pen fillrule=currentpen, string shader);
shades on picture ‘pic’ the interior of path ‘g’ according to fill rule
‘fillrule’ using the ‘PostScript’ calculator routine specified by the
string ‘shader’; this routine takes 2 arguments, each in [0,1], and
returns ‘colors(fillrule).length’ color components.  Function shading is
illustrated in the example ‘functionshading.asy’.

   The following routine uses ‘evenodd’ clipping together with the ‘^^’
operator to unfill a region:

void unfill(picture pic=currentpicture, path g);


File: asymptote.info,  Node: clip,  Next: label,  Prev: fill,  Up: Drawing commands

4.3 clip
========

void clip(picture pic=currentpicture, path g, stroke=false,
          pen fillrule=currentpen);

   Clip the current contents of picture ‘pic’ to the region bounded by
the path ‘g’, using fill rule ‘fillrule’ (*note fillrule::).  If
‘stroke=true’, the clipped portion is the same as the region that would
be drawn with ‘draw(pic,g,zerowinding)’; in this case the path ‘g’ need
not be cyclic.  While clipping has no notion of depth (it transcends
layers and even pages), one can localize clipping to a temporary
picture, which can then be added to ‘pic’.  For an illustration of
picture clipping, see the first example in *note LaTeX usage::.


File: asymptote.info,  Node: label,  Prev: clip,  Up: Drawing commands

4.4 label
=========

void label(picture pic=currentpicture, Label L, pair position,
           align align=NoAlign, pen p=currentpen, filltype filltype=NoFill)

   Draw Label ‘L’ on picture ‘pic’ using pen ‘p’.  If ‘align’ is
‘NoAlign’, the label will be centered at user coordinate ‘position’;
otherwise it will be aligned in the direction of ‘align’ and displaced
from ‘position’ by the ‘PostScript’ offset ‘align*labelmargin(p)’.

   Here, ‘real labelmargin(pen p=currentpen)’ is a quantity used to
align labels.  In the code below,
label("abcdefg",(0,0),align=up,basealign);
the baseline of the label will be exactly ‘labelmargin(currentpen)’
‘PostScript’ units above the center.

   The constant ‘Align’ can be used to align the bottom-left corner of
the label at ‘position’.  The Label ‘L’ can either be a string or the
structure obtained by calling one of the functions
Label Label(string s="", pair position, align align=NoAlign,
            pen p=nullpen, embed embed=Rotate, filltype filltype=NoFill);
Label Label(string s="", align align=NoAlign,
            pen p=nullpen, embed embed=Rotate, filltype filltype=NoFill);
Label Label(Label L, pair position, align align=NoAlign,
            pen p=nullpen, embed embed=L.embed, filltype filltype=NoFill);
Label Label(Label L, align align=NoAlign,
            pen p=nullpen, embed embed=L.embed, filltype filltype=NoFill);
   The text of a Label can be scaled, slanted, rotated, or shifted by
multiplying it on the left by an affine transform (*note Transforms::).
For example, ‘rotate(45)*xscale(2)*L’ first scales ‘L’ in the x
direction and then rotates it counterclockwise by 45 degrees.  The final
position of a Label can also be shifted by a ‘PostScript’ coordinate
translation: ‘shift(10,0)*L’.  An explicit pen specified within the
Label overrides other pen arguments.  The ‘embed’ argument determines
how the Label should transform with the embedding picture:
‘Shift’
     only shift with embedding picture;

‘Rotate’
     only shift and rotate with embedding picture (default);

‘Rotate(pair z)’
     rotate with (picture-transformed) vector ‘z’.

‘Slant’
     only shift, rotate, slant, and reflect with embedding picture;

‘Scale’
     shift, rotate, slant, reflect, and scale with embedding picture.

   To add a label to a path, use
void label(picture pic=currentpicture, Label L, path g, align align=NoAlign,
           pen p=currentpen, filltype filltype=NoFill);
   By default the label will be positioned at the midpoint of the path.
An alternative label position (in the sense of ‘point(path p, real t)’)
may be specified as a real value for ‘position’ in constructing the
Label.  The position ‘Relative(real)’ specifies a location relative to
the total arclength of the path.  These convenient abbreviations are
predefined:
position BeginPoint=Relative(0);
position MidPoint=Relative(0.5);
position EndPoint=Relative(1);

   Path labels are aligned in the direction ‘align’, which may be
specified as an absolute compass direction (pair) or a direction
‘Relative(pair)’ measured relative to a north axis in the local
direction of the path.  For convenience ‘LeftSide’, ‘Center’, and
‘RightSide’ are defined as ‘Relative(W)’, ‘Relative((0,0))’, and
‘Relative(E)’, respectively.  Multiplying ‘LeftSide’ and ‘RightSide’ on
the left by a real scaling factor will move the label further away from
or closer to the path.

   A label with a fixed-size arrow of length ‘arrowlength’ pointing to
‘b’ from direction ‘dir’ can be produced with the routine
void arrow(picture pic=currentpicture, Label L="", pair b, pair dir,
           real length=arrowlength, align align=NoAlign,
           pen p=currentpen, arrowbar arrow=Arrow, margin margin=EndMargin);
   If no alignment is specified (either in the Label or as an explicit
argument), the optional Label will be aligned in the direction ‘dir’,
using margin ‘margin’.

   The function ‘string graphic(string name, string options="")’ returns
a string that can be used to include an encapsulated ‘PostScript’ (EPS)
file.  Here, ‘name’ is the name of the file to include and ‘options’ is
a string containing a comma-separated list of optional bounding box
(‘bb=llx lly urx ury’), width (‘width=value’), height (‘height=value’),
rotation (‘angle=value’), scaling (‘scale=factor’), clipping
(‘clip=bool’), and draft mode (‘draft=bool’) parameters.  The ‘layer()’
function can be used to force future objects to be drawn on top of the
included image:
label(graphic("file.eps","width=1cm"),(0,0),NE);
layer();

   The ‘string baseline(string s, string template="\strut")’ function
can be used to enlarge the bounding box of a label to match a given
template, so that their baselines will be typeset on a horizontal line.
See ‘Pythagoras.asy’ for an example.  Alternatively, the pen ‘basealign’
may be used to force labels to respect the TeX baseline (*note
basealign::).

   One can prevent labels from overwriting one another with the
‘overwrite’ pen attribute (*note overwrite::).

   The structure ‘object’ defined in ‘plain_Label.asy’ allows Labels and
frames to be treated in a uniform manner.  A group of objects may be
packed together into single frame with the routine
frame pack(pair align=2S, ... object inset[]);
To draw or fill a box (or ellipse or other path) around a ‘Label’ and
return the bounding object, use one of the routines
object draw(picture pic=currentpicture, Label L, envelope e,
            real xmargin=0, real ymargin=xmargin, pen p=currentpen,
            filltype filltype=NoFill, bool above=true);
object draw(picture pic=currentpicture, Label L, envelope e, pair position,
            real xmargin=0, real ymargin=xmargin, pen p=currentpen,
            filltype filltype=NoFill, bool above=true);
Here ‘envelope’ is a boundary-drawing routine such as ‘box’, ‘roundbox’,
or ‘ellipse’ defined in ‘plain_boxes.asy’.

   The function ‘path[] texpath(Label L)’ returns the path array that
TeX would fill to draw the Label ‘L’.

   The ‘string minipage(string s, width=100pt)’ function can be used to
format string ‘s’ into a paragraph of width ‘width’.  This example uses
‘minipage’, ‘clip’, and ‘graphic’ to produce a CD label:

                              [./CDlabel]
size(11.7cm,11.7cm);

asy(nativeformat(),"logo");

fill(unitcircle^^(scale(2/11.7)*unitcircle),
     evenodd+rgb(124/255,205/255,124/255));
label(scale(1.1)*minipage(
                          "\centering\scriptsize \textbf{\LARGE {\tt Asymptote}\\
\smallskip
\small The Vector Graphics Language}\\
\smallskip
\textsc{Andy Hammerlindl, John Bowman, and Tom Prince}
https://asymptote.sourceforge.io\\
",8cm),(0,0.6));
label(graphic("logo","height=7cm"),(0,-0.22));
clip(unitcircle^^(scale(2/11.7)*unitcircle),evenodd);


File: asymptote.info,  Node: Bezier curves,  Next: Programming,  Prev: Drawing commands,  Up: Top

5 Bezier curves
***************

Each interior node of a cubic spline may be given a direction prefix or
suffix ‘{dir}’: the direction of the pair ‘dir’ specifies the direction
of the incoming or outgoing tangent, respectively, to the curve at that
node.  Exterior nodes may be given direction specifiers only on their
interior side.

   A cubic spline between the node z_0, with postcontrol point c_0, and
the node z_1, with precontrol point c_1, is computed as the Bezier curve

   [(1-t)^3*z_0+3t(1-t)^2*c_0+3t^2(1-t)*c_1+t^3*z_1 for 0 <=t <= 1.]

   As illustrated in the diagram below, the third-order midpoint (m_5)
constructed from two endpoints z_0 and z_1 and two control points c_0
and c_1, is the point corresponding to t=1/2 on the Bezier curve formed
by the quadruple (z_0, c_0, c_1, z_1).  This allows one to recursively
construct the desired curve, by using the newly extracted third-order
midpoint as an endpoint and the respective second- and first-order
midpoints as control points:

                              [./bezier2]

   Here m_0, m_1 and m_2 are the first-order midpoints, m_3 and m_4 are
the second-order midpoints, and m_5 is the third-order midpoint.  The
curve is then constructed by recursively applying the algorithm to (z_0,
m_0, m_3, m_5) and (m_5, m_4, m_2, z_1).

   In fact, an analogous property holds for points located at any
fraction t in [0,1] of each segment, not just for midpoints (t=1/2).

   The Bezier curve constructed in this manner has the following
properties:

   • It is entirely contained in the convex hull of the given four
     points.

   • It starts heading from the first endpoint to the first control
     point and finishes heading from the second control point to the
     second endpoint.

   The user can specify explicit control points between two nodes like
this:
draw((0,0)..controls (0,100) and (100,100)..(100,0));

   However, it is usually more convenient to just use the ‘..’ operator,
which tells ‘Asymptote’ to choose its own control points using the
algorithms described in Donald Knuth's monograph, The MetaFontbook,
Chapter 14.  The user can still customize the guide (or path) by
specifying direction, tension, and curl values.

   The higher the tension, the straighter the curve is, and the more it
approximates a straight line.  One can change the spline tension from
its default value of 1 to any real value greater than or equal to 0.75
[see John D. Hobby, Discrete and Computational Geometry 1, 123-140
(1986)]:
draw((100,0)..tension 2 ..(100,100)..(0,100));
draw((100,0)..tension 3 and 2 ..(100,100)..(0,100));
draw((100,0)..tension atleast 2 ..(100,100)..(0,100));

   In these examples there is a space between ‘2’ and ‘..’.  This is
needed as ‘2.’ is interpreted as a numerical constant.

   The curl parameter specifies the curvature at the endpoints of a path
(0 means straight; the default value of 1 means approximately circular):
draw((100,0){curl 0}..(100,100)..{curl 0}(0,100));

   The ‘MetaPost ...’ path connector, which requests, when possible, an
inflection-free curve confined to a triangle defined by the endpoints
and directions, is implemented in ‘Asymptote’ as the convenient
abbreviation ‘::’ for ‘..tension atleast 1 ..’ (the ellipsis ‘...’ is
used in ‘Asymptote’ to indicate a variable number of arguments; *note
Rest arguments::).  For example, compare
draw((0,0){up}..(100,25){right}..(200,0){down});

                               [./dots]
with
draw((0,0){up}::(100,25){right}::(200,0){down});

                              [./colons]

   The ‘---’ connector is an abbreviation for ‘..tension atleast
infinity..’ and the ‘&’ connector concatenates two paths, after first
stripping off the last node of the first path (which normally should
coincide with the first node of the second path).


File: asymptote.info,  Node: Programming,  Next: LaTeX usage,  Prev: Bezier curves,  Up: Top

6 Programming
*************

* Menu:

* Data types::                  void, bool, int, real, pair, triple, string
* Paths and guides::            Bezier curves
* Pens::                        Colors, line types, line widths, font sizes
* Transforms::                  Affine transforms
* Frames and pictures::         Canvases for immediate and deferred drawing
* Deferred drawing::            Withholding drawing until all data is available
* Files::                       Reading and writing your data
* Variable initializers::       Initialize your variables
* Structures::                  Organize your data
* Operators::                   Arithmetic and logical operators
* Implicit scaling::            Avoiding those ugly *s
* Functions::                   Traditional and higher-order functions
* Arrays::                      Dynamic vectors
* Casts::                       Implicit and explicit casts
* Import::                      Importing external ‘Asymptote’ modules
* Iterators::                   Iterating over structures
* Static::                      Where to allocate your variable?
* Autounravel::                 Adding associated libraries to structures
* Hash functions::              Hashing your data
* Collections (containers)::    Maps, sets, queues, and iterator tools

Here is a short introductory example to the ‘Asymptote’ programming
language that highlights the similarity of its control structures with
those of C, C++, and Java:
// This is a comment.

// Declaration: Declare x to be a real variable;
real x;

// Assignment: Assign the real variable x the value 1.
x=1.0;

// Conditional: Test if x equals 1 or not.
if(x == 1.0) {
  write("x equals 1.0");
} else {
  write("x is not equal to 1.0");
}

// Loop: iterate 10 times
for(int i=0; i < 10; ++i) {
  write(i);
}

   ‘Asymptote’ supports ‘while’, ‘do’, ‘break’, and ‘continue’
statements just as in C/C++.  It also supports range-based ‘for’ loops,
which are convenient for iterating over all elements of an array:

// Iterate over an array
int[] array={1,1,2,3,5};
for(int k : array) {
  write(k);
}
In addition, it supports many features beyond the ones found in those
languages.


File: asymptote.info,  Node: Data types,  Next: Paths and guides,  Up: Programming

6.1 Data types
==============

‘Asymptote’ supports the following data types (in addition to
user-defined types):

‘void’
     The void type is used only by functions that take or return no
     arguments.

‘bool’
     a boolean type that can only take on the values ‘true’ or ‘false’.
     For example:
     bool b=true;

     defines a boolean variable ‘b’ and initializes it to the value
     ‘true’.  If no initializer is given:
     bool b;

     the value ‘false’ is assumed.

‘bool3’
     an extended boolean type that can take on the values ‘true’,
     ‘default’, or ‘false’.  A bool3 type can be cast to or from a bool.
     The default initializer for bool3 is ‘default’.

‘int’
     an integer type; if no initializer is given, the implicit value ‘0’
     is assumed.  The minimum allowed value of an integer is ‘intMin’
     and the maximum value is ‘intMax’.

‘real’
     a real number; this should be set to the highest-precision native
     floating-point type on the architecture.  The implicit initializer
     for reals is ‘0.0’.  Real numbers have precision ‘realEpsilon’,
     with ‘realDigits’ significant digits.  The smallest positive real
     number is ‘realMin’ and the largest positive real number is
     ‘realMax’.  The variables ‘inf’ and ‘nan’, along with the function
     ‘bool isnan(real x)’ are useful when floating-point exceptions are
     masked with the ‘-mask’ command-line option (the default in
     interactive mode).

‘pair’
     complex number, that is, an ordered pair of real components
     ‘(x,y)’.  The real and imaginary parts of a pair ‘z’ can read as
     ‘z.x’ and ‘z.y’.  We say that ‘x’ and ‘y’ are virtual members of
     the data element pair; they cannot be directly modified, however.
     The implicit initializer for pairs is ‘(0.0,0.0)’.

     There are a number of ways to take the complex conjugate of a pair:
     pair z=(3,4);
     z=(z.x,-z.y);
     z=z.x-I*z.y;
     z=conj(z);

     Here ‘I’ is the pair ‘(0,1)’.  A number of built-in functions are
     defined for pairs:

     ‘pair conj(pair z)’
          returns the conjugate of ‘z’;

     ‘real length(pair z)’
          returns the complex modulus |‘z’| of its argument ‘z’.  For
          example,
          pair z=(3,4);
          length(z);
          returns the result 5.  A synonym for ‘length(pair)’ is
          ‘abs(pair)’.  The function ‘abs2(pair z)’ returns |‘z’|^2;

     ‘real angle(pair z, bool warn=true)’
          returns the angle of ‘z’ in radians in the interval
          [-‘pi’,‘pi’] or ‘0’ if ‘warn’ is ‘false’ and ‘z=(0,0)’ (rather
          than producing an error);

     ‘real degrees(pair z, bool warn=true)’
          returns the angle of ‘z’ in degrees in the interval [0,360) or
          ‘0’ if ‘warn’ is ‘false’ and ‘z=(0,0)’ (rather than producing
          an error);

     ‘pair unit(pair z)’
          returns a unit vector in the direction of the pair ‘z’;

     ‘pair expi(real angle)’
          returns a unit vector in the direction ‘angle’ measured in
          radians;

     ‘pair dir(real degrees)’
          returns a unit vector in the direction ‘degrees’ measured in
          degrees;

     ‘real xpart(pair z)’
          returns ‘z.x’;

     ‘real ypart(pair z)’
          returns ‘z.y’;

     ‘pair realmult(pair z, pair w)’
          returns the element-by-element product ‘(z.x*w.x,z.y*w.y)’;

     ‘real dot(explicit pair z, explicit pair w)’
          returns the dot product ‘z.x*w.x+z.y*w.y’;

     ‘real cross(explicit pair z, explicit pair w)’
          returns the 2D scalar product ‘z.x*w.y-z.y*w.x’;

     ‘real orient(pair a, pair b, pair c);’
          returns a positive (negative) value if ‘a--b--c--cycle’ is
          oriented counterclockwise (clockwise) or zero if all three
          points are colinear.  Equivalently, a positive (negative)
          value is returned if ‘c’ lies to the left (right) of the line
          through ‘a’ and ‘b’ or zero if ‘c’ lies on this line.  The
          value returned can be expressed in terms of the 2D scalar
          cross product as ‘cross(a-c,b-c)’, which is the determinant
          |a.x a.y 1|
          |b.x b.y 1|
          |c.x c.y 1|

     ‘real incircle(pair a, pair b, pair c, pair d);’
          returns a positive (negative) value if ‘d’ lies inside
          (outside) the circle passing through the
          counterclockwise-oriented points ‘a,b,c’ or zero if ‘d’ lies
          on the this circle.  The value returned is the determinant
          |a.x a.y a.x^2+a.y^2 1|
          |b.x b.y b.x^2+b.y^2 1|
          |c.x c.y c.x^2+c.y^2 1|
          |d.x d.y d.x^2+d.y^2 1|

     ‘pair minbound(pair z, pair w)’
          returns ‘(min(z.x,w.x),min(z.y,w.y))’;

     ‘pair maxbound(pair z, pair w)’
          returns ‘(max(z.x,w.x),max(z.y,w.y))’.

‘triple’
     an ordered triple of real components ‘(x,y,z)’ used for
     three-dimensional drawings.  The respective components of a triple
     ‘v’ can read as ‘v.x’, ‘v.y’, and ‘v.z’.  The implicit initializer
     for triples is ‘(0.0,0.0,0.0)’.

     Here are the built-in functions for triples:
     ‘real length(triple v)’
          returns the length |‘v’| of its argument ‘v’.  A synonym for
          ‘length(triple)’ is ‘abs(triple)’.  The function ‘abs2(triple
          v)’ returns |‘v’|^2;

     ‘real polar(triple v, bool warn=true)’
          returns the colatitude of ‘v’ measured from the z axis in
          radians or ‘0’ if ‘warn’ is ‘false’ and ‘v=O’ (rather than
          producing an error);

     ‘real azimuth(triple v, bool warn=true)’
          returns the longitude of ‘v’ measured from the x axis in
          radians or ‘0’ if ‘warn’ is ‘false’ and ‘v.x=v.y=0’ (rather
          than producing an error);

     ‘real colatitude(triple v, bool warn=true)’
          returns the colatitude of ‘v’ measured from the z axis in
          degrees or ‘0’ if ‘warn’ is ‘false’ and ‘v=O’ (rather than
          producing an error);

     ‘real latitude(triple v, bool warn=true)’
          returns the latitude of ‘v’ measured from the xy plane in
          degrees or ‘0’ if ‘warn’ is ‘false’ and ‘v=O’ (rather than
          producing an error);

     ‘real longitude(triple v, bool warn=true)’
          returns the longitude of ‘v’ measured from the x axis in
          degrees or ‘0’ if ‘warn’ is ‘false’ and ‘v.x=v.y=0’ (rather
          than producing an error);

     ‘triple unit(triple v)’
          returns a unit triple in the direction of the triple ‘v’;

     ‘triple expi(real polar, real azimuth)’
          returns a unit triple in the direction ‘(polar,azimuth)’
          measured in radians;

     ‘triple dir(real colatitude, real longitude)’
          returns a unit triple in the direction
          ‘(colatitude,longitude)’ measured in degrees;

     ‘real xpart(triple v)’
          returns ‘v.x’;

     ‘real ypart(triple v)’
          returns ‘v.y’;

     ‘real zpart(triple v)’
          returns ‘v.z’;

     ‘real dot(triple u, triple v)’
          returns the dot product ‘u.x*v.x+u.y*v.y+u.z*v.z’;

     ‘triple cross(triple u, triple v)’
          returns the cross product

          ‘(u.y*v.z-u.z*v.y,u.z*v.x-u.x*v.z,u.x*v.y-v.x*u.y)’;

     ‘triple minbound(triple u, triple v)’
          returns ‘(min(u.x,v.x),min(u.y,v.y),min(u.z,v.z))’;

     ‘triple maxbound(triple u, triple v)’
          returns ‘(max(u.x,v.x),max(u.y,v.y),max(u.z,v.z)’).

‘string’
     a character string, implemented using the STL ‘string’ class.

     Strings delimited by double quotes (‘"’) are subject to the
     following mappings to allow the use of double quotes in TeX (for
     using the ‘babel’ package, *note babel::):

        • \" maps to "
        • \\ maps to \\

     Strings delimited by single quotes (‘'’) have the same mappings as
     character strings in ANSI ‘C’:

        • \' maps to '
        • \" maps to "
        • \?  maps to ?
        • \\ maps to backslash
        • \a maps to alert
        • \b maps to backspace
        • \f maps to form feed
        • \n maps to newline
        • \r maps to carriage return
        • \t maps to tab
        • \v maps to vertical tab
        • \0-\377 map to corresponding octal byte
        • \x0-\xFF map to corresponding hexadecimal byte

     The implicit initializer for strings is the empty string ‘""’.
     Strings may be concatenated with the ‘+’ operator.  In the
     following string functions, position ‘0’ denotes the start of the
     string:

     ‘int length(string s)’
          returns the length of the string ‘s’;

     ‘int find(string s, string t, int pos=0)’
          returns the position of the first occurrence of string ‘t’ in
          string ‘s’ at or after position ‘pos’, or -1 if ‘t’ is not a
          substring of ‘s’;

     ‘int rfind(string s, string t, int pos=-1)’
          returns the position of the last occurrence of string ‘t’ in
          string ‘s’ at or before position ‘pos’ (if ‘pos’=-1, at the
          end of the string ‘s’), or -1 if ‘t’ is not a substring of
          ‘s’;

     ‘string insert(string s, int pos, string t)’
          returns the string formed by inserting string ‘t’ at position
          ‘pos’ in ‘s’;

     ‘string erase(string s, int pos, int n)’
          returns the string formed by erasing the string of length ‘n’
          (if ‘n’=-1, to the end of the string ‘s’) at position ‘pos’ in
          ‘s’;

     ‘string substr(string s, int pos, int n=-1)’
          returns the substring of ‘s’ starting at position ‘pos’ and of
          length ‘n’ (if ‘n’=-1, until the end of the string ‘s’);

     ‘string reverse(string s)’
          returns the string formed by reversing string ‘s’;

     ‘string replace(string s, string before, string after)’
          returns a string with all occurrences of the string ‘before’
          in the string ‘s’ changed to the string ‘after’;

     ‘string replace(string s, string[][] table)’
          returns a string constructed by translating in string ‘s’ all
          occurrences of the string ‘before’ in an array ‘table’ of
          string pairs {‘before’,‘after’} to the corresponding string
          ‘after’;

     ‘string[] split(string s, string delimiter="")’
          returns an array of strings obtained by splitting ‘s’ into
          substrings delimited by ‘delimiter’ (an empty delimiter
          signifies a space, but with duplicate delimiters discarded);

     ‘string[] array(string s)’
          returns an array of strings obtained by splitting ‘s’ into
          individual characters.  The inverse operation is provided by
          ‘operator +(...string[] a)’.

     ‘string format(string s, int n, string locale="")’
          returns a string containing ‘n’ formatted according to the
          C-style format string ‘s’ using locale ‘locale’ (or the
          current locale if an empty string is specified), following the
          behavior of the C function ‘fprintf’), except that only one
          data field is allowed.

     ‘string format(string s=defaultformat, bool forcemath=false, string s=defaultseparator, real x, string locale="")’
          returns a string containing ‘x’ formatted according to the
          C-style format string ‘s’ using locale ‘locale’ (or the
          current locale if an empty string is specified), following the
          behavior of the C function ‘fprintf’), except that only one
          data field is allowed, trailing zeros are removed by default
          (unless ‘#’ is specified), and if ‘s’ specifies math mode or
          ‘forcemath=true’, TeX is used to typeset scientific notation
          using the ‘defaultseparator="\!\times\!";’;

     ‘int hex(string s);’
          casts a hexadecimal string ‘s’ to an integer;

     ‘int ascii(string s);’
          returns the ASCII code for the first character of string ‘s’;

     ‘string string(real x, int digits=realDigits)’
          casts ‘x’ to a string using precision ‘digits’ and the C
          locale;

     ‘string locale(string s="")’
          sets the locale to the given string, if nonempty, and returns
          the current locale;

     ‘string time(string format="%a %b %d %T %Z %Y")’
          returns the current time formatted by the ANSI C routine
          ‘strftime’ according to the string ‘format’ using the current
          locale.  Thus
          time();
          time("%a %b %d %H:%M:%S %Z %Y");

          are equivalent ways of returning the current time in the
          default format used by the ‘UNIX’ ‘date’ command;

     ‘int seconds(string t="", string format="")’
          returns the time measured in seconds after the Epoch (Thu Jan
          01 00:00:00 UTC 1970) as determined by the ANSI C routine
          ‘strptime’ according to the string ‘format’ using the current
          locale, or the current time if ‘t’ is the empty string.  Note
          that the ‘"%Z"’ extension to the POSIX ‘strptime’
          specification is ignored by the current GNU C Library.  If an
          error occurs, the value -1 is returned.  Here are some
          examples:
          seconds("Mar 02 11:12:36 AM PST 2007","%b %d %r PST %Y");
          seconds(time("%b %d %r %z %Y"),"%b %d %r %z %Y");
          seconds(time("%b %d %r %Z %Y"),"%b %d %r "+time("%Z")+" %Y");
          1+(seconds()-seconds("Jan 1","%b %d"))/(24*60*60);
          The last example returns today's ordinal date, measured from
          the beginning of the year.

     ‘string time(int seconds, string format="%a %b %d %T %Z %Y")’
          returns the time corresponding to ‘seconds’ seconds after the
          Epoch (Thu Jan 01 00:00:00 UTC 1970) formatted by the ANSI C
          routine ‘strftime’ according to the string ‘format’ using the
          current locale.  For example, to return the date corresponding
          to 24 hours ago:
          time(seconds()-24*60*60);

     ‘int system(string s)’
     ‘int system(string[] s)’
          if the setting ‘safe’ is false, call the arbitrary system
          command ‘s’;

     ‘void asy(string format, bool overwrite=false, ... string[] s)’
          conditionally process each file name in array ‘s’ in a new
          environment, using format ‘format’, overwriting the output
          file only if ‘overwrite’ is true;

     ‘void abort(string s="")’
          aborts execution (with a non-zero return code in batch mode);
          if string ‘s’ is nonempty, a diagnostic message constructed
          from the source file, line number, and ‘s’ is printed;

     ‘void assert(bool b, string s="")’
          aborts execution with an error message constructed from ‘s’ if
          ‘b=false’;

     ‘void exit()’
          exits (with a zero error return code in batch mode);

     ‘void sleep(int seconds)’
          pauses for the given number of seconds;

     ‘void usleep(int microseconds)’
          pauses for the given number of microseconds;

     ‘void beep()’
          produces a beep on the console;

   As in C/C++, complicated types may be abbreviated with ‘typedef’ or
‘using’.  For instance, the line
using multipath = path[];
   will make ‘multipath’ a type equivalent ‘path[]’, as will the
equivalent line
typedef path[] multipath;
   For the most part such type aliasing is a convenience.  However, it
is required when declaring a function whose return type is a function
(see the example in *note Functions::).


File: asymptote.info,  Node: Paths and guides,  Next: Pens,  Prev: Data types,  Up: Programming

6.2 Paths and guides
====================

‘path’
     a cubic spline resolved into a fixed path.  The implicit
     initializer for paths is ‘nullpath’.

     For example, the routine ‘circle(pair c, real r)’, which returns a
     Bezier curve approximating a circle of radius ‘r’ centered on ‘c’,
     is based on ‘unitcircle’ (*note unitcircle::):
     path circle(pair c, real r)
     {
       return shift(c)*scale(r)*unitcircle;
     }
     If high accuracy is needed, a true circle may be produced with the
     routine ‘Circle’ defined in the module ‘graph’:
     import graph;
     path Circle(pair c, real r, int n=nCircle);

     A circular arc consistent with ‘circle’ centered on ‘c’ with radius
     ‘r’ from ‘angle1’ to ‘angle2’ degrees, drawing counterclockwise if
     ‘angle2 >= angle1’, can be constructed with
     path arc(pair c, real r, real angle1, real angle2);
     One may also specify the direction explicitly:
     path arc(pair c, real r, real angle1, real angle2, bool direction);
     Here the direction can be specified as CCW (counter-clockwise) or
     CW (clockwise).  For convenience, an arc centered at ‘c’ from pair
     ‘z1’ to ‘z2’ (assuming ‘|z2-c|=|z1-c|’) in the may also be
     constructed with
     path arc(pair c, explicit pair z1, explicit pair z2,
              bool direction=CCW)

     If high accuracy is needed, true arcs may be produced with routines
     in the module ‘graph’ that produce Bezier curves with ‘n’ control
     points:
     import graph;
     path Arc(pair c, real r, real angle1, real angle2, bool direction,
              int n=nCircle);
     path Arc(pair c, real r, real angle1, real angle2, int n=nCircle);
     path Arc(pair c, explicit pair z1, explicit pair z2,
              bool direction=CCW, int n=nCircle);

     An ellipse can be drawn with the routine
     path ellipse(pair c, real a, real b)
     {
       return shift(c)*scale(a,b)*unitcircle;
     }

     A brace can be constructed between pairs ‘a’ and ‘b’ with
     path brace(pair a, pair b, real amplitude=bracedefaultratio*length(b-a));

     This example illustrates the use of all five guide connectors
     discussed in *note Tutorial:: and *note Bezier curves:::
     size(300,0);
     pair[] z=new pair[10];
     
     z[0]=(0,100); z[1]=(50,0); z[2]=(180,0);
     
     for(int n=3; n <= 9; ++n)
       z[n]=z[n-3]+(200,0);
     
     path p=z[0]..z[1]---z[2]::{up}z[3]
     &z[3]..z[4]--z[5]::{up}z[6]
     &z[6]::z[7]---z[8]..{up}z[9];
     
     draw(p,grey+linewidth(4mm));
     
     dot(z);

                               [./join]

     Here are some useful functions for paths:

     ‘int length(path p);’
          This is the number of (linear or cubic) segments in path ‘p’.
          If ‘p’ is cyclic, this is the same as the number of nodes in
          ‘p’.

     ‘int size(path p);’
          This is the number of nodes in the path ‘p’.  If ‘p’ is
          cyclic, this is the same as ‘length(p)’.

     ‘bool cyclic(path p);’
          returns ‘true’ iff path ‘p’ is cyclic.

     ‘bool straight(path p, int i);’
          returns ‘true’ iff the segment of path ‘p’ between node ‘i’
          and node ‘i+1’ is straight.

     ‘bool piecewisestraight(path p)’
          returns ‘true’ iff the path ‘p’ is piecewise straight.

     ‘pair point(path p, int t);’
          If ‘p’ is cyclic, return the coordinates of node ‘t’ mod
          ‘length(p)’.  Otherwise, return the coordinates of node ‘t’,
          unless ‘t’ < 0 (in which case ‘point(0)’ is returned) or ‘t’ >
          ‘length(p)’ (in which case ‘point(length(p))’ is returned).

     ‘pair point(path p, real t);’
          This returns the coordinates of the point between node
          ‘floor(t)’ and ‘floor(t)+1’ corresponding to the cubic spline
          parameter ‘t-floor(t)’ (*note Bezier curves::).  If ‘t’ lies
          outside the range [0,‘length(p)’], it is first reduced modulo
          ‘length(p)’ in the case where ‘p’ is cyclic or else converted
          to the corresponding endpoint of ‘p’.

     ‘pair dir(path p, int t, int sign=0, bool normalize=true);’
          If ‘sign < 0’, return the direction (as a pair) of the
          incoming tangent to path ‘p’ at node ‘t’; if ‘sign > 0’,
          return the direction of the outgoing tangent.  If ‘sign=0’,
          the mean of these two directions is returned.

     ‘pair dir(path p, real t, bool normalize=true);’
          returns the direction of the tangent to path ‘p’ at the point
          between node ‘floor(t)’ and ‘floor(t)+1’ corresponding to the
          cubic spline parameter ‘t-floor(t)’ (*note Bezier curves::).

     ‘pair dir(path p)’
          returns dir(p,length(p)).

     ‘pair dir(path p, path q)’
          returns unit(dir(p)+dir(q)).

     ‘pair accel(path p, int t, int sign=0);’
          If ‘sign < 0’, return the acceleration of the incoming path
          ‘p’ at node ‘t’; if ‘sign > 0’, return the acceleration of the
          outgoing path.  If ‘sign=0’, the mean of these two
          accelerations is returned.

     ‘pair accel(path p, real t);’
          returns the acceleration of the path ‘p’ at the point ‘t’.

     ‘real radius(path p, real t);’
          returns the radius of curvature of the path ‘p’ at the point
          ‘t’.

     ‘pair precontrol(path p, int t);’
          returns the precontrol point of ‘p’ at node ‘t’.

     ‘pair precontrol(path p, real t);’
          returns the effective precontrol point of ‘p’ at parameter
          ‘t’.

     ‘pair postcontrol(path p, int t);’
          returns the postcontrol point of ‘p’ at node ‘t’.

     ‘pair postcontrol(path p, real t);’
          returns the effective postcontrol point of ‘p’ at parameter
          ‘t’.

     ‘real arclength(path p);’
          returns the length (in user coordinates) of the piecewise
          linear or cubic curve that path ‘p’ represents.

     ‘real arctime(path p, real L);’
          returns the path "time", a real number between 0 and the
          length of the path in the sense of ‘point(path p, real t)’, at
          which the cumulative arclength (measured from the beginning of
          the path) equals ‘L’.

     ‘pair arcpoint(path p, real L);’
          returns ‘point(p,arctime(p,L))’.

     ‘real dirtime(path p, pair z);’
          returns the first "time", a real number between 0 and the
          length of the path in the sense of ‘point(path, real)’, at
          which the tangent to the path has the direction of pair ‘z’,
          or -1 if this never happens.

     ‘real reltime(path p, real l);’
          returns the time on path ‘p’ at the relative fraction ‘l’ of
          its arclength.

     ‘pair relpoint(path p, real l);’
          returns the point on path ‘p’ at the relative fraction ‘l’ of
          its arclength.

     ‘pair midpoint(path p);’
          returns the point on path ‘p’ at half of its arclength.

     ‘path reverse(path p);’
          returns a path running backwards along ‘p’.

     ‘path subpath(path p, int a, int b);’
          returns the subpath of ‘p’ running from node ‘a’ to node ‘b’.
          If ‘a’ > ‘b’, the direction of the subpath is reversed.

     ‘path subpath(path p, real a, real b);’
          returns the subpath of ‘p’ running from path time ‘a’ to path
          time ‘b’, in the sense of ‘point(path, real)’.  If ‘a’ > ‘b’,
          the direction of the subpath is reversed.

     ‘real[] intersect(path p, path q, real fuzz=-1);’
          if ‘p’ and ‘q’ have at least one intersection point, return a
          real array of length 2 containing the times representing the
          respective path times along ‘p’ and ‘q’, in the sense of
          ‘point(path, real)’, for one such intersection point (as
          chosen by the algorithm described on page 137 of ‘The
          MetaFontbook’).  The computations are performed to the
          absolute error specified by ‘fuzz’, or if ‘fuzz < 0’, to
          machine precision.  If the paths do not intersect, return a
          real array of length 0.

     ‘real[][] intersections(path p, path q, real fuzz=-1);’
          returns all (unless there are infinitely many) intersection
          times of paths ‘p’ and ‘q’ as a sorted array of real arrays of
          length 2 (*note sort::).  The computations are performed to
          the absolute error specified by ‘fuzz’, or if ‘fuzz < 0’, to
          machine precision.

     ‘real[] intersections(path p, explicit pair a, explicit pair b, real fuzz=-1);’
          returns all (unless there are infinitely many) intersection
          times of path ‘p’ with the (infinite) line through points ‘a’
          and ‘b’ as a sorted array.  The intersections returned are
          guaranteed to be correct to within the absolute error
          specified by ‘fuzz’, or if ‘fuzz < 0’, to machine precision.

     ‘real[] times(path p, real x)’
          returns all intersection times of path ‘p’ with the vertical
          line through ‘(x,0)’.

     ‘real[] times(path p, explicit pair z)’
          returns all intersection times of path ‘p’ with the horizontal
          line through ‘(0,z.y)’.

     ‘real[] mintimes(path p)’
          returns an array of length 2 containing times at which path
          ‘p’ reaches its minimal horizontal and vertical extents,
          respectively.

     ‘real[] maxtimes(path p)’
          returns an array of length 2 containing times at which path
          ‘p’ reaches its maximal horizontal and vertical extents,
          respectively.

     ‘pair intersectionpoint(path p, path q, real fuzz=-1);’
          returns the intersection point
          ‘point(p,intersect(p,q,fuzz)[0])’.

     ‘pair[] intersectionpoints(path p, path q, real fuzz=-1);’
          returns an array containing all intersection points of the
          paths ‘p’ and ‘q’.

     ‘pair extension(pair P, pair Q, pair p, pair q);’
          returns the intersection point of the extensions of the line
          segments ‘P--Q’ and ‘p--q’, or if the lines are parallel,
          ‘(infinity,infinity)’.

     ‘slice cut(path p, path knife, int n);’
          returns the portions of path ‘p’ before and after the ‘n’th
          intersection of ‘p’ with path ‘knife’ as a structure ‘slice’
          (if no intersection exist is found, the entire path is
          considered to be 'before' the intersection):
          struct slice {
            path before,after;
          }
          The argument ‘n’ is treated as modulo the number of
          intersections.

     ‘slice firstcut(path p, path knife);’
          equivalent to ‘cut(p,knife,0);’ Note that ‘firstcut.after’
          plays the role of the ‘MetaPost cutbefore’ command.

     ‘slice lastcut(path p, path knife);’
          equivalent to ‘cut(p,knife,-1);’ Note that ‘lastcut.before’
          plays the role of the ‘MetaPost cutafter’ command.

     ‘path buildcycle(...path[] p);’
          This returns the path surrounding a region bounded by a list
          of two or more consecutively intersecting paths, following the
          behavior of the ‘MetaPost buildcycle’ command.

     ‘pair min(path p);’
          returns the pair (left,bottom) for the path bounding box of
          path ‘p’.

     ‘pair max(path p);’
          returns the pair (right,top) for the path bounding box of path
          ‘p’.

     ‘int windingnumber(path p, pair z);’
          returns the winding number of the cyclic path ‘p’ relative to
          the point ‘z’.  The winding number is positive if the path
          encircles ‘z’ in the counterclockwise direction.  If ‘z’ lies
          on ‘p’ the constant ‘undefined’ (defined to be the largest odd
          integer) is returned.

     ‘bool interior(int windingnumber, pen fillrule)’
          returns true if ‘windingnumber’ corresponds to an interior
          point according to ‘fillrule’.

     ‘bool inside(path p, pair z, pen fillrule=currentpen);’
          returns ‘true’ iff the point ‘z’ lies inside or on the edge of
          the region bounded by the cyclic path ‘p’ according to the
          fill rule ‘fillrule’ (*note fillrule::).

     ‘int inside(path p, path q, pen fillrule=currentpen);’
          returns ‘1’ if the cyclic path ‘p’ strictly contains ‘q’
          according to the fill rule ‘fillrule’ (*note fillrule::), ‘-1’
          if the cyclic path ‘q’ strictly contains ‘p’, and ‘0’
          otherwise.

     ‘pair inside(path p, pen fillrule=currentpen);’
          returns an arbitrary point strictly inside a nondegenerate
          cyclic path ‘p’ according to the fill rule ‘fillrule’ (*note
          fillrule::).

     ‘path[] strokepath(path g, pen p=currentpen);’
          returns the path array that ‘PostScript’ would fill in drawing
          path ‘g’ with pen ‘p’.

‘guide’
     an unresolved cubic spline (list of cubic-spline nodes and control
     points).  The implicit initializer for a guide is ‘nullpath’; this
     is useful for building up a guide within a loop.

     A guide is similar to a path except that the computation of the
     cubic spline is deferred until drawing time (when it is resolved
     into a path); this allows two guides with free endpoint conditions
     to be joined together smoothly.  The solid curve in the following
     example is built up incrementally as a guide, but only resolved at
     drawing time; the dashed curve is incrementally resolved at each
     iteration, before the entire set of nodes (shown in red) is known:

     size(200);
     
     real mexican(real x) {return (1-8x^2)*exp(-(4x^2));}
     
     int n=30;
     real a=1.5;
     real width=2a/n;
     
     guide hat;
     path solved;
     
     for(int i=0; i < n; ++i) {
       real t=-a+i*width;
       pair z=(t,mexican(t));
       hat=hat..z;
       solved=solved..z;
     }
     
     draw(hat);
     dot(hat,red);
     draw(solved,dashed);
     

                            [./mexicanhat]

     We point out an efficiency distinction in the use of guides and
     paths:
     guide g;
     for(int i=0; i < 10; ++i)
       g=g--(i,i);
     path p=g;

     runs in linear time, whereas
     path p;
     for(int i=0; i < 10; ++i)
       p=p--(i,i);

     runs in quadratic time, as the entire path up to that point is
     copied at each step of the iteration.

     The following routines can be used to examine the individual
     elements of a guide without actually resolving the guide to a fixed
     path (except for internal cycles, which are resolved):

     ‘int size(guide g);’
          Analogous to ‘size(path p)’.

     ‘int length(guide g);’
          Analogous to ‘length(path p)’.

     ‘bool cyclic(guide g);’
          Analogous to ‘cyclic(path p)’.

     ‘pair point(guide g, int t);’
          Analogous to ‘point(path p, int t)’.

     ‘guide reverse(guide g);’
          Analogous to ‘reverse(path p)’.  If ‘g’ is cyclic and also
          contains a secondary cycle, it is first solved to a path, then
          reversed.  If ‘g’ is not cyclic but contains an internal
          cycle, only the internal cycle is solved before reversal.  If
          there are no internal cycles, the guide is reversed but not
          solved to a path.

     ‘pair[] dirSpecifier(guide g, int i);’
          This returns a pair array of length 2 containing the outgoing
          (in element 0) and incoming (in element 1) direction
          specifiers (or ‘(0,0)’ if none specified) for the segment of
          guide ‘g’ between nodes ‘i’ and ‘i+1’.

     ‘pair[] controlSpecifier(guide g, int i);’
          If the segment of guide ‘g’ between nodes ‘i’ and ‘i+1’ has
          explicit outgoing and incoming control points, they are
          returned as elements 0 and 1, respectively, of a two-element
          array.  Otherwise, an empty array is returned.

     ‘tensionSpecifier tensionSpecifier(guide g, int i);’
          This returns the tension specifier for the segment of guide
          ‘g’ between nodes ‘i’ and ‘i+1’.  The individual components of
          the ‘tensionSpecifier’ type can be accessed as the virtual
          members ‘in’, ‘out’, and ‘atLeast’.

     ‘real[] curlSpecifier(guide g);’
          This returns an array containing the initial curl specifier
          (in element 0) and final curl specifier (in element 1) for
          guide ‘g’.

     As a technical detail we note that a direction specifier given to
     ‘nullpath’ modifies the node on the other side: the guides
     a..{up}nullpath..b;
     c..nullpath{up}..d;
     e..{up}nullpath{down}..f;
     are respectively equivalent to
     a..nullpath..{up}b;
     c{up}..nullpath..d;
     e{down}..nullpath..{up}f;


File: asymptote.info,  Node: Pens,  Next: Transforms,  Prev: Paths and guides,  Up: Programming

6.3 Pens
========

In ‘Asymptote’, pens provide a context for the four basic drawing
commands (*note Drawing commands::).  They are used to specify the
following drawing attributes: color, line type, line width, line cap,
line join, fill rule, text alignment, font, font size, pattern,
overwrite mode, and calligraphic transforms on the pen nib.  The default
pen used by the drawing routines is called ‘currentpen’.  This provides
the same functionality as the ‘MetaPost’ command ‘pickup’.  The implicit
initializer for pens is ‘defaultpen’.

   Pens may be added together with the nonassociative binary operator
‘+’.  This will add the colors of the two pens.  All other non-default
attributes of the rightmost pen will override those of the leftmost pen.
Thus, one can obtain a yellow dashed pen by saying ‘dashed+red+green’ or
‘red+green+dashed’ or ‘red+dashed+green’.  The binary operator ‘*’ can
be used to scale the color of a pen by a real number, until it saturates
with one or more color components equal to 1.

   • Colors are specified using one of the following colorspaces:
     ‘pen gray(real g);’
          This produces a grayscale color, where the intensity ‘g’ lies
          in the interval [0,1], with 0.0 denoting black and 1.0
          denoting white.

     ‘pen rgb(real r, real g, real b);’
          This produces an RGB color, where each of the red, green, and
          blue intensities ‘r’, ‘g’, ‘b’, lies in the interval [0,1].

     ‘pen RGB(int r, int g, int b);’
          This produces an RGB color, where each of the red, green, and
          blue intensities ‘r’, ‘g’, ‘b’, lies in the interval [0,255].

     ‘pen cmyk(real c, real m, real y, real k);’
          This produces a CMYK color, where each of the cyan, magenta,
          yellow, and black intensities ‘c’, ‘m’, ‘y’, ‘k’, lies in the
          interval [0,1].

     ‘pen invisible;’
          This special pen writes in invisible ink, but adjusts the
          bounding box as if something had been drawn (like the
          ‘\phantom’ command in TeX).  The function ‘bool
          invisible(pen)’ can be used to test whether a pen is
          invisible.

     The default color is ‘black’; this may be changed with the routine
     ‘defaultpen(pen)’.  The function ‘colorspace(pen p)’ returns the
     colorspace of pen ‘p’ as a string (‘"gray"’, ‘"rgb"’, ‘"cmyk"’, or
     ‘""’).

     The function ‘real[] colors(pen)’ returns the color components of a
     pen.  The functions ‘pen gray(pen)’, ‘pen rgb(pen)’, and ‘pen
     cmyk(pen)’ return new pens obtained by converting their arguments
     to the respective color spaces.  The function
     ‘colorless(pen=currentpen)’ returns a copy of its argument with the
     color attributes stripped (to avoid color mixing).

     A 6-character RGB hexadecimal string can be converted to a pen with
     the routine
     pen rgb(string s);
   • A pen can be converted to a hexadecimal string with ‘string hex(pen
     p);’

     Various shades and mixtures of the grayscale primary colors ‘black’
     and ‘white’, RGB primary colors ‘red’, ‘green’, and ‘blue’, and RGB
     secondary colors ‘cyan’, ‘magenta’, and ‘yellow’ are defined as
     named colors, along with the CMYK primary colors ‘Cyan’, ‘Magenta’,
     ‘Yellow’, and ‘Black’, in the module ‘plain’:

                              [./colors]

     The standard 140 RGB ‘X11’ colors can be imported with the command
     import x11colors;
     and the standard 68 CMYK TeX colors can be imported with the
     command
     import texcolors;
     Note that there is some overlap between these two standards and the
     definitions of some colors (such as ‘Green’) actually disagree.

     ‘Asymptote’ also comes with a ‘asycolors.sty’ LaTeX package that
     defines to LaTeX CMYK versions of ‘Asymptote’'s predefined colors,
     so that they can be used directly within LaTeX strings.  Normally,
     such colors are passed to LaTeX via a pen argument; however, to
     change the color of only a portion of a string, say for a slide
     presentation, (*note slide::) it may be desirable to specify the
     color directly to LaTeX.  This file can be passed to LaTeX with the
     ‘Asymptote’ command
     usepackage("asycolors");

     The structure ‘hsv’ defined in ‘plain_pens.asy’ may be used to
     convert between HSV and RGB spaces, where the hue ‘h’ is an angle
     in [0,360) and the saturation ‘s’ and value ‘v’ lie in ‘[0,1]’:
     pen p=hsv(180,0.5,0.75);
     write(p);           // ([default], red=0.375, green=0.75, blue=0.75)
     hsv q=p;
     write(q.h,q.s,q.v); // 180     0.5     0.75

   • Line types are specified with the function ‘pen linetype(real[] a,
     real offset=0, bool scale=true, bool adjust=true)’, where ‘a’ is an
     array of real array numbers.  The optional parameter ‘offset’
     specifies where in the pattern to begin.  The first number
     specifies how far (if ‘scale’ is ‘true’, in units of the pen line
     width; otherwise in ‘PostScript’ units) to draw with the pen on,
     the second number specifies how far to draw with the pen off, and
     so on.  If ‘adjust’ is ‘true’, these spacings are automatically
     adjusted by ‘Asymptote’ to fit the arclength of the path.  Here are
     the predefined line types:
     pen solid=linetype(new real[]);
     pen dotted=linetype(new real[] {0,4});
     pen dashed=linetype(new real[] {8,8});
     pen longdashed=linetype(new real[] {24,8});
     pen dashdotted=linetype(new real[] {8,8,0,8});
     pen longdashdotted=linetype(new real[] {24,8,0,8});
     pen Dotted(pen p=currentpen) {return linetype(new real[] {0,3})+2*linewidth(p);}
     pen Dotted=Dotted();

                             [./linetype]

     The default line type is ‘solid’; this may be changed with
     ‘defaultpen(pen)’.  The line type of a pen can be determined with
     the functions ‘real[] linetype(pen p=currentpen)’, ‘real offset(pen
     p)’, ‘bool scale(pen p)’, and ‘bool adjust(pen p)’.

   • The pen line width is specified in ‘PostScript’ units with ‘pen
     linewidth(real)’.  The default line width is 0.5 bp; this value may
     be changed with ‘defaultpen(pen)’.  The line width of a pen is
     returned by ‘real linewidth(pen p=currentpen)’.  For convenience,
     in the module ‘plain_pens’ we define
     void defaultpen(real w) {defaultpen(linewidth(w));}
     pen operator +(pen p, real w) {return p+linewidth(w);}
     pen operator +(real w, pen p) {return linewidth(w)+p;}
     so that one may set the line width like this:
     defaultpen(2);
     pen p=red+0.5;

   • A pen with a specific ‘PostScript’ line cap is returned on calling
     ‘linecap’ with an integer argument:
     pen squarecap=linecap(0);
     pen roundcap=linecap(1);
     pen extendcap=linecap(2);

     The default line cap, ‘roundcap’, may be changed with
     ‘defaultpen(pen)’.  The line cap of a pen is returned by ‘int
     linecap(pen p=currentpen)’.

   • A pen with a specific ‘PostScript’ join style is returned on
     calling ‘linejoin’ with an integer argument:
     pen miterjoin=linejoin(0);
     pen roundjoin=linejoin(1);
     pen beveljoin=linejoin(2);

     The default join style, ‘roundjoin’, may be changed with
     ‘defaultpen(pen)’.The join style of a pen is returned by ‘int
     linejoin(pen p=currentpen)’.

   • A pen with a specific ‘PostScript’ miter limit is returned by
     calling ‘miterlimit(real)’.  The default miterlimit, ‘10.0’, may be
     changed with ‘defaultpen(pen)’.  The miter limit of a pen is
     returned by ‘real miterlimit(pen p=currentpen)’.

   • A pen with a specific ‘PostScript’ fill rule is returned on calling
     ‘fillrule’ with an integer argument:
     pen zerowinding=fillrule(0);
     pen evenodd=fillrule(1);

     The fill rule, which identifies the algorithm used to determine the
     insideness of a path or array of paths, only affects the ‘clip’,
     ‘fill’, and ‘inside’ functions.  For the ‘zerowinding’ fill rule, a
     point ‘z’ is outside the region bounded by a path if the number of
     upward intersections of the path with the horizontal line
     ‘z--z+infinity’ minus the number of downward intersections is zero.
     For the ‘evenodd’ fill rule, ‘z’ is considered to be outside the
     region if the total number of such intersections is even.  The
     default fill rule, ‘zerowinding’, may be changed with
     ‘defaultpen(pen)’.  The fill rule of a pen is returned by ‘int
     fillrule(pen p=currentpen)’.

   • A pen with a specific text alignment setting is returned on calling
     ‘basealign’ with an integer argument:
     pen nobasealign=basealign(0);
     pen basealign=basealign(1);

     The default setting, ‘nobasealign’, which may be changed with
     ‘defaultpen(pen)’, causes the label alignment routines to use the
     full label bounding box for alignment.  In contrast, ‘basealign’
     requests that the TeX baseline be respected.  The base align
     setting of a pen is returned by ‘int basealign(pen p=currentpen)’.

     For example, in the following image, the baselines of green \pi and
     \gamma are aligned, while the bottom border of red -\pi and -\gamma
     are aligned.
     import fontsize;
     import three;
     
     settings.autobillboard=false;
     settings.embed=false;
     currentprojection=orthographic(Z);
     
     defaultpen(fontsize(100pt));
     
     dot(O);
     
     label("acg",O,align=N,basealign);
     label("ace",O,align=N,red);
     label("acg",O,align=S,basealign);
     label("ace",O,align=S,red);
     label("acg",O,align=E,basealign);
     label("ace",O,align=E,red);
     label("acg",O,align=W,basealign);
     label("ace",O,align=W,red);
     
     picture pic;
     dot(pic,(labelmargin(),0,0),blue);
     dot(pic,(-labelmargin(),0,0),blue);
     dot(pic,(0,labelmargin(),0),blue);
     dot(pic,(0,-labelmargin(),0),blue);
     add(pic,O);
     
     dot((0,0));
     
     label("acg",(0,0),align=N,basealign);
     label("ace",(0,0),align=N,red);
     label("acg",(0,0),align=S,basealign);
     label("ace",(0,0),align=S,red);
     label("acg",(0,0),align=E,basealign);
     label("ace",(0,0),align=E,red);
     label("acg",(0,0),align=W,basealign);
     label("ace",(0,0),align=W,red);
     
     picture pic;
     dot(pic,(labelmargin(),0),blue);
     dot(pic,(-labelmargin(),0),blue);
     dot(pic,(0,labelmargin()),blue);
     dot(pic,(0,-labelmargin()),blue);
     add(pic,(0,0));

                             [./basealign]

     Another method for aligning baselines is provided by the ‘baseline’
     function (*note baseline::).

   • The font size is specified in TeX points (1 pt = 1/72.27 inches)
     with the function ‘pen fontsize(real size, real
     lineskip=1.2*size)’.  The default font size, 12pt, may be changed
     with ‘defaultpen(pen)’.  Nonstandard font sizes may require
     inserting
     import fontsize;
     at the beginning of the file (this requires the ‘type1cm’ package
     available from
          <http://mirror.ctan.org/macros/latex/contrib/type1cm/>
     and included in recent LaTeX distributions).  The font size and
     line skip of a pen can be examined with the routines ‘real
     fontsize(pen p=currentpen)’ and ‘real lineskip(pen p=currentpen)’,
     respectively.

   • A pen using a specific LaTeX NFSS font is returned by calling the
     function ‘pen font(string encoding, string family, string series,
     string shape)’.  The default setting, ‘font("OT1","cmr","m","n")’,
     corresponds to 12pt Computer Modern Roman; this may be changed with
     ‘defaultpen(pen)’.  The font setting of a pen is returned by
     ‘string font(pen p=currentpen)’.

     Alternatively, one may select a fixed-size TeX font (on which
     ‘fontsize’ has no effect) like ‘"cmr12"’ (12pt Computer Modern
     Roman) or ‘"pcrr"’ (Courier) using the function ‘pen font(string
     name)’.  An optional size argument can also be given to scale the
     font to the requested size: ‘pen font(string name, real size)’.

     A nonstandard font command can be generated with ‘pen
     fontcommand(string)’.

     A convenient interface to the following standard ‘PostScript’ fonts
     is also provided:
     pen AvantGarde(string series="m", string shape="n");
     pen Bookman(string series="m", string shape="n");
     pen Courier(string series="m", string shape="n");
     pen Helvetica(string series="m", string shape="n");
     pen NewCenturySchoolBook(string series="m", string shape="n");
     pen Palatino(string series="m", string shape="n");
     pen TimesRoman(string series="m", string shape="n");
     pen ZapfChancery(string series="m", string shape="n");
     pen Symbol(string series="m", string shape="n");
     pen ZapfDingbats(string series="m", string shape="n");

   • Starting with the 2018/04/01 release, LaTeX takes UTF-8 as the new
     default input encoding.  However, you can still set different input
     encoding (so as the font, font encoding or even language context).
     Here is an example for ‘cp1251’ and Russian language in Cyrillic
     script (font encoding ‘T2A’):
     texpreamble("\usepackage[math]{anttor}");
     texpreamble("\usepackage[T2A]{fontenc}");
     texpreamble("\usepackage[cp1251]{inputenc}");
     texpreamble("\usepackage[russian]{babel}");
     Support for Chinese, Japanese, and Korean fonts is provided by the
     CJK package:
          <https://ctan.org/pkg/cjk>
     The following commands enable the CJK song family (within a label,
     you can also temporarily switch to another family, say kai, by
     prepending ‘"\CJKfamily{kai}"’ to the label string):
     texpreamble("\usepackage{CJK}
     \AtBeginDocument{\begin{CJK*}{GBK}{song}}
     \AtEndDocument{\clearpage\end{CJK*}}");

   • The transparency of a pen can be changed with the command:
     pen opacity(real opacity=1, string blend="Compatible");
     The opacity can be varied from ‘0’ (fully transparent) to the
     default value of ‘1’ (opaque), and ‘blend’ specifies one of the
     following foreground-background blending operations:
     "Compatible","Normal","Multiply","Screen","Overlay","SoftLight",
     "HardLight","ColorDodge","ColorBurn","Darken","Lighten","Difference",
     "Exclusion","Hue","Saturation","Color","Luminosity",
     as described in Tables 136 and 137 of
     <https://opensource.adobe.com/dc-acrobat-sdk-docs/standards/pdfstandards/pdf/PDF32000_2008.pdf>.
     Since ‘PostScript’ does not support transparency, this feature is
     only effective with the ‘-f pdf’ output format option; other
     formats can be produced from the resulting PDF file with the
     ‘ImageMagick’ ‘magick’ program.  Labels are always drawn with an
     ‘opacity’ of 1.  A simple example of transparent filling is
     provided in the example file ‘transparency.asy’.

   • ‘PostScript’ commands within a ‘picture’ may be used to create a
     tiling pattern, identified by the string ‘name’, for ‘fill’ and
     ‘draw’ operations by adding it to the global ‘PostScript’ frame
     ‘currentpatterns’, with optional left-bottom margin ‘lb’ and
     right-top margin ‘rt’.
     import patterns;
     void add(string name, picture pic, pair lb=0, pair rt=0);

     To ‘fill’ or ‘draw’ using pattern ‘name’, use the pen
     ‘pattern("name")’.  For example, rectangular tilings can be
     constructed using the routines ‘picture tile(real Hx=5mm, real
     Hy=0, pen p=currentpen, filltype filltype=NoFill)’, ‘picture
     checker(real Hx=5mm, real Hy=0, pen p=currentpen)’, and ‘picture
     brick(real Hx=5mm, real Hy=0, pen p=currentpen)’ defined in module
     ‘patterns’:
     size(0,90);
     import patterns;
     
     add("tile",tile());
     add("filledtilewithmargin",tile(6mm,4mm,red,Fill),(1mm,1mm),(1mm,1mm));
     add("checker",checker());
     add("brick",brick());
     
     real s=2.5;
     filldraw(unitcircle,pattern("tile"));
     filldraw(shift(s,0)*unitcircle,pattern("filledtilewithmargin"));
     filldraw(shift(2s,0)*unitcircle,pattern("checker"));
     filldraw(shift(3s,0)*unitcircle,pattern("brick"));

                               [./tile]

     Hatch patterns can be generated with the routines ‘picture
     hatch(real H=5mm, pair dir=NE, pen p=currentpen)’, ‘picture
     crosshatch(real H=5mm, pen p=currentpen)’:
     size(0,100);
     import patterns;
     
     add("hatch",hatch());
     add("hatchback",hatch(NW));
     add("crosshatch",crosshatch(3mm));
     
     real s=1.25;
     filldraw(unitsquare,pattern("hatch"));
     filldraw(shift(s,0)*unitsquare,pattern("hatchback"));
     filldraw(shift(2s,0)*unitsquare,pattern("crosshatch"));

                               [./hatch]

     You may need to turn off aliasing in your ‘PostScript’ viewer for
     patterns to appear correctly.  Custom patterns can easily be
     constructed, following the examples in module ‘patterns’.  The
     tiled pattern can even incorporate shading (*note gradient
     shading::), as illustrated in this example (not included in the
     manual because not all printers support ‘PostScript’ 3):
     size(0,100);
     import patterns;
     
     real d=4mm;
     picture tiling;
     path square=scale(d)*unitsquare;
     axialshade(tiling,square,white,(0,0),black,(d,d));
     fill(tiling,shift(d,d)*square,blue);
     add("shadedtiling",tiling);
     
     filldraw(unitcircle,pattern("shadedtiling"));
     

   • One can specify a custom pen nib as an arbitrary polygonal path
     with ‘pen makepen(path)’; this path represents the mark to be drawn
     for paths containing a single point.  This pen nib path can be
     recovered from a pen with ‘path nib(pen)’.  Unlike in ‘MetaPost’,
     the path need not be convex:

     size(200);
     pen convex=makepen(scale(10)*polygon(8))+grey;
     draw((1,0.4),convex);
     draw((0,0)---(1,1)..(2,0)--cycle,convex);
     
     pen nonconvex=scale(10)*
       makepen((0,0)--(0.25,-1)--(0.5,0.25)--(1,0)--(0.5,1.25)--cycle)+red;
     draw((0.5,-1.5),nonconvex);
     draw((0,-1.5)..(1,-0.5)..(2,-1.5),nonconvex);

                              [./makepen]

     The value ‘nullpath’ represents a circular pen nib (the default);
     an elliptical pen can be achieved simply by multiplying the pen by
     a transform: ‘yscale(2)*currentpen’.

   • One can prevent labels from overwriting one another by using the
     pen attribute ‘overwrite’, which takes a single argument:

     ‘Allow’
          Allow labels to overwrite one another.  This is the default
          behavior (unless overridden with ‘defaultpen(pen)’.

     ‘Suppress’
          Suppress, with a warning, each label that would overwrite
          another label.

     ‘SuppressQuiet’
          Suppress, without warning, each label that would overwrite
          another label.

     ‘Move’
          Move a label that would overwrite another out of the way and
          issue a warning.  As this adjustment is during the final
          output phase (in ‘PostScript’ coordinates) it could result in
          a larger figure than requested.

     ‘MoveQuiet’
          Move a label that would overwrite another out of the way,
          without warning.  As this adjustment is during the final
          output phase (in ‘PostScript’ coordinates) it could result in
          a larger figure than requested.

   The routine ‘defaultpen()’ returns the current default pen
attributes.  Calling the routine ‘resetdefaultpen()’ resets all pen
default attributes to their initial values.


File: asymptote.info,  Node: Transforms,  Next: Frames and pictures,  Prev: Pens,  Up: Programming

6.4 Transforms
==============

‘Asymptote’ makes extensive use of affine transforms.  A pair ‘(x,y)’ is
transformed by the transform ‘t=(t.x,t.y,t.xx,t.xy,t.yx,t.yy)’ to
‘(x',y')’, where
x' = t.x + t.xx * x + t.xy * y
y' = t.y + t.yx * x + t.yy * y
This is equivalent to the ‘PostScript’ transformation ‘[t.xx t.yx t.xy
t.yy t.x t.y]’.

   Transforms can be applied to pairs, guides, paths, pens, strings,
transforms, frames, and pictures by multiplication (via the binary
operator ‘*’) on the left (*note circle:: for an example).  Transforms
can be composed with one another and inverted with the function
‘transform inverse(transform t)’; they can also be raised to any integer
power with the ‘^’ operator.

   The built-in transforms are:

‘transform identity;’
     the identity transform;
‘transform shift(pair z);’
     translates by the pair ‘z’;
‘transform shift(real x, real y);’
     translates by the pair ‘(x,y)’;
‘transform xscale(real x);’
     scales by ‘x’ in the x direction;
‘transform yscale(real y);’
     scales by ‘y’ in the y direction;
‘transform scale(real s);’
     scale by ‘s’ in both x and y directions;
‘transform scale(real x, real y);’
     scale by ‘x’ in the x direction and by ‘y’ in the y direction;
‘transform slant(real s);’
     maps ‘(x,y)’ -> ‘(x+s*y,y)’;
‘transform rotate(real angle, pair z=(0,0));’
     rotates by ‘angle’ in degrees about ‘z’;
‘transform reflect(pair a, pair b);’
     reflects about the line ‘a--b’.
‘transform zeroTransform;’
     the zero transform;

   The implicit initializer for transforms is ‘identity()’.  The
routines ‘shift(transform t)’ and ‘shiftless(transform t)’ return the
transforms ‘(t.x,t.y,0,0,0,0)’ and ‘(0,0,t.xx,t.xy,t.yx,t.yy)’
respectively.  The function ‘bool isometry(transform t)’ can be used to
test if ‘t’ is an isometry (preserves distance).


File: asymptote.info,  Node: Frames and pictures,  Next: Deferred drawing,  Prev: Transforms,  Up: Programming

6.5 Frames and pictures
=======================

‘frame’
     Frames are canvases for drawing in ‘PostScript’ coordinates.  While
     working with frames directly is occasionally necessary for
     constructing deferred drawing routines The implicit initializer for
     frames is ‘newframe’.  The function ‘bool empty(frame f)’ returns
     ‘true’ only if the frame ‘f’ is empty.  A frame may be erased with
     the ‘erase(frame)’ routine.  The functions ‘pair min(frame)’ and
     ‘pair max(frame)’ return the (left,bottom) and (right,top)
     coordinates of the frame bounding box, respectively.  The contents
     of frame ‘src’ may be appended to frame ‘dest’ with the command
     void add(frame dest, frame src);
     or prepended with
     void prepend(frame dest, frame src);
     A frame obtained by aligning frame ‘f’ in the direction ‘align’, in
     a manner analogous to the ‘align’ argument of ‘label’ (*note
     label::), is returned by
     frame align(frame f, pair align);

‘picture’
     Pictures are high-level structures (*note Structures::) defined in
     the module ‘plain’ that provide canvases for drawing in user
     coordinates.  The default picture is called ‘currentpicture’.  A
     new picture can be created like this:
     picture pic;
     Anonymous pictures can be made by the expression ‘new picture’.

     The ‘size’ routine specifies the dimensions of the desired picture:

     void size(picture pic=currentpicture, real x, real y=x,
               bool keepAspect=pic.keepAspect);

     If the ‘x’ and ‘y’ sizes are both 0, user coordinates will be
     interpreted as ‘PostScript’ coordinates.  In this case, the
     transform mapping ‘pic’ to the final output frame is ‘identity()’.

     If exactly one of ‘x’ or ‘y’ is 0, no size restriction is imposed
     in that direction; it will be scaled the same as the other
     direction.

     If ‘keepAspect’ is set to ‘Aspect’ or ‘true’ (the default), the
     picture will be scaled with its aspect ratio preserved such that
     the final width is no more than ‘x’ and the final height is no more
     than ‘y’.

     If ‘keepAspect’ is set to ‘IgnoreAspect’ or ‘false’, the picture
     will be scaled in both directions so that the final width is ‘x’
     and the height is ‘y’.

     To make the user coordinates of picture ‘pic’ represent multiples
     of ‘x’ units in the x direction and ‘y’ units in the y direction,
     use
     void unitsize(picture pic=currentpicture, real x, real y=x);
     When nonzero, these ‘x’ and ‘y’ values override the corresponding
     size parameters of picture ‘pic’.

     The routine
     void size(picture pic=currentpicture, real xsize, real ysize,
               pair min, pair max);
     forces the final picture scaling to map the user coordinates
     ‘box(min,max)’ to a region of width ‘xsize’ and height ‘ysize’
     (when these parameters are nonzero).

     Alternatively, calling the routine
     transform fixedscaling(picture pic=currentpicture, pair min,
                            pair max, pen p=nullpen, bool warn=false);
     will cause picture ‘pic’ to use a fixed scaling to map user
     coordinates in ‘box(min,max)’ to the (already specified) picture
     size, taking account of the width of pen ‘p’.  A warning will be
     issued if the final picture exceeds the specified size.

     A picture ‘pic’ can be fit to a frame and output to a file
     ‘prefix’.‘format’ using image format ‘format’ by calling the
     ‘shipout’ function:
     void shipout(string prefix=defaultfilename, picture pic=currentpicture,
                  orientation orientation=orientation,
                  string format="", bool wait=false, bool view=true,
                  string options="", string script="",
                  light light=currentlight, projection P=currentprojection)
     The default output format, ‘PostScript’, may be changed with the
     ‘-f’ or ‘-tex’ command-line options.  The ‘options’, ‘script’, and
     ‘projection’ parameters are only relevant for 3D pictures.  If
     ‘defaultfilename’ is an empty string, the prefix ‘outprefix()’ will
     be used.

     A ‘shipout()’ command is added implicitly at file exit.  Explicit
     ‘shipout()’ commands to the same file as the final implicit shipout
     are ignored.  The default page orientation is ‘Portrait’; this may
     be modified by changing the variable ‘orientation’.  To output in
     landscape mode, simply set the variable ‘orientation=Landscape’ or
     issue the command
     shipout(Landscape);

     To rotate the page by -90 degrees, use the orientation ‘Seascape’.
     The orientation ‘UpsideDown’ rotates the page by 180 degrees.

     A picture ‘pic’ can be explicitly fit to a frame by calling
     frame pic.fit(real xsize=pic.xsize, real ysize=pic.ysize,
                   bool keepAspect=pic.keepAspect);
     The default size and aspect ratio settings are those given to the
     ‘size’ command (which default to ‘0’, ‘0’, and ‘true’,
     respectively).  The transformation that would currently be used to
     fit a picture ‘pic’ to a frame is returned by the member function
     ‘pic.calculateTransform()’.

     In certain cases (such as 2D graphs) where only an approximate size
     estimate for ‘pic’ is available, the picture fitting routine
     frame pic.scale(real xsize=this.xsize, real ysize=this.ysize,
                     bool keepAspect=this.keepAspect);
     (which scales the resulting frame, including labels and fixed-size
     objects) will enforce perfect compliance with the requested size
     specification, but should not normally be required.

     To draw a bounding box with margins around a picture, fit the
     picture to a frame using the function
     frame bbox(picture pic=currentpicture, real xmargin=0,
                real ymargin=xmargin, pen p=currentpen,
                filltype filltype=NoFill);
     Here ‘filltype’ specifies one of the following fill types:
     ‘FillDraw’
          Fill the interior and draw the boundary.

     ‘FillDraw(real xmargin=0, real ymargin=xmargin, pen fillpen=nullpen,’
          ‘pen drawpen=nullpen)’ If ‘fillpen’ is ‘nullpen’, fill with
          the drawing pen; otherwise fill with pen ‘fillpen’.  If
          ‘drawpen’ is ‘nullpen’, draw the boundary with ‘fillpen’;
          otherwise with ‘drawpen’.  An optional margin of ‘xmargin’ and
          ‘ymargin’ can be specified.

     ‘Fill’
          Fill the interior.

     ‘Fill(real xmargin=0, real ymargin=xmargin, pen p=nullpen)’
          If ‘p’ is ‘nullpen’, fill with the drawing pen; otherwise fill
          with pen ‘p’.  An optional margin of ‘xmargin’ and ‘ymargin’
          can be specified.

     ‘NoFill’
          Do not fill.

     ‘Draw’
          Draw only the boundary.

     ‘Draw(real xmargin=0, real ymargin=xmargin, pen p=nullpen)’
          If ‘p’ is ‘nullpen’, draw the boundary with the drawing pen;
          otherwise draw with pen ‘p’.  An optional margin of ‘xmargin’
          and ‘ymargin’ can be specified.

     ‘UnFill’
          Clip the region.

     ‘UnFill(real xmargin=0, real ymargin=xmargin)’
          Clip the region and surrounding margins ‘xmargin’ and
          ‘ymargin’.

     ‘RadialShade(pen penc, pen penr)’
          Fill varying radially from ‘penc’ at the center of the
          bounding box to ‘penr’ at the edge.

     ‘RadialShadeDraw(real xmargin=0, real ymargin=xmargin, pen penc,’
          ‘pen penr, pen drawpen=nullpen)’ Fill with RadialShade and
          draw the boundary.

     For example, to draw a bounding box around a picture with a 0.25 cm
     margin and output the resulting frame, use the command:
     shipout(bbox(0.25cm));
     A ‘picture’ may be fit to a frame with the background color pen
     ‘p’, using the function ‘bbox(p,Fill)’.

     To pad a picture to a precise size in both directions, fit the
     picture to a frame using the function
     frame pad(picture pic=currentpicture, real xsize=pic.xsize,
               real ysize=pic.ysize, filltype filltype=NoFill);

     The functions
     pair min(picture pic, user=false);
     pair max(picture pic, user=false);
     pair size(picture pic, user=false);
     calculate the bounds that picture ‘pic’ would have if it were
     currently fit to a frame using its default size specification.  If
     ‘user’ is ‘false’ the returned value is in ‘PostScript’
     coordinates, otherwise it is in user coordinates.

     The function
     pair point(picture pic=currentpicture, pair dir, bool user=true);
     is a convenient way of determining the point on the bounding box of
     ‘pic’ in the direction ‘dir’ relative to its center, ignoring the
     contributions from fixed-size objects (such as labels and
     arrowheads).  If ‘user’ is ‘true’ the returned value is in user
     coordinates, otherwise it is in ‘PostScript’ coordinates.

     The function
     pair truepoint(picture pic=currentpicture, pair dir, bool user=true);
     is identical to ‘point’, except that it also accounts for
     fixed-size objects, using the scaling transform that picture ‘pic’
     would have if currently fit to a frame using its default size
     specification.  If ‘user’ is ‘true’ the returned value is in user
     coordinates, otherwise it is in ‘PostScript’ coordinates.

     Sometimes it is useful to draw objects on separate pictures and add
     one picture to another using the ‘add’ function:
     void add(picture src, bool group=true,
              filltype filltype=NoFill, bool above=true);
     void add(picture dest, picture src, bool group=true,
              filltype filltype=NoFill, bool above=true);
     The first example adds ‘src’ to ‘currentpicture’; the second one
     adds ‘src’ to ‘dest’.  The ‘group’ option specifies whether or not
     the graphical user interface should treat all of the elements of
     ‘src’ as a single entity (*note GUI::), ‘filltype’ requests
     optional background filling or clipping, and ‘above’ specifies
     whether to add ‘src’ above or below existing objects.

     There are also routines to add a fixed-size picture or frame ‘src’
     to another picture ‘dest’ (or ‘currentpicture’) about the user
     coordinate ‘position’:
     void add(picture src, pair position, bool group=true,
              filltype filltype=NoFill, bool above=true);
     void add(picture dest, picture src, pair position,
              bool group=true, filltype filltype=NoFill, bool above=true);
     void add(picture dest=currentpicture, frame src, pair position=0,
              bool group=true, filltype filltype=NoFill, bool above=true);
     void add(picture dest=currentpicture, frame src, pair position,
              pair align, bool group=true, filltype filltype=NoFill,
              bool above=true);

     The optional ‘align’ argument in the last form specifies a
     direction to use for aligning the frame, in a manner analogous to
     the ‘align’ argument of ‘label’ (*note label::).  However, one key
     difference is that when ‘align’ is not specified, labels are
     centered, whereas frames and pictures are aligned so that their
     origin is at ‘position’.  Illustrations of frame alignment can be
     found in the examples *note errorbars:: and *note image::.  If you
     want to align three or more subpictures, group them two at a time:

     picture pic1;
     real size=50;
     size(pic1,size);
     fill(pic1,(0,0)--(50,100)--(100,0)--cycle,red);
     
     picture pic2;
     size(pic2,size);
     fill(pic2,unitcircle,green);
     
     picture pic3;
     size(pic3,size);
     fill(pic3,unitsquare,blue);
     
     picture pic;
     add(pic,pic1.fit(),(0,0),N);
     add(pic,pic2.fit(),(0,0),10S);
     
     add(pic.fit(),(0,0),N);
     add(pic3.fit(),(0,0),10S);
     

                            [./subpictures]

     Alternatively, one can use ‘attach’ to automatically increase the
     size of picture ‘dest’ to accommodate adding a frame ‘src’ about
     the user coordinate ‘position’:
     void attach(picture dest=currentpicture, frame src,
                      pair position=0, bool group=true,
                      filltype filltype=NoFill, bool above=true);
     void attach(picture dest=currentpicture, frame src,
                      pair position, pair align, bool group=true,
                      filltype filltype=NoFill, bool above=true);

     To erase the contents of a picture (but not the size
     specification), use the function
     void erase(picture pic=currentpicture);

     To save a snapshot of ‘currentpicture’, ‘currentpen’, and
     ‘currentprojection’, use the function ‘save()’.

     To restore a snapshot of ‘currentpicture’, ‘currentpen’, and
     ‘currentprojection’, use the function ‘restore()’.

     Many further examples of picture and frame operations are provided
     in the base module ‘plain’.

     It is possible to insert verbatim ‘PostScript’ commands in a
     picture with one of the routines
     void postscript(picture pic=currentpicture, string s);
     void postscript(picture pic=currentpicture, string s, pair min,
                     pair max)
     Here ‘min’ and ‘max’ can be used to specify explicit bounds
     associated with the resulting ‘PostScript’ code.

     Verbatim TeX commands can be inserted in the intermediate LaTeX
     output file with one of the functions
     void tex(picture pic=currentpicture, string s);
     void tex(picture pic=currentpicture, string s, pair min, pair max)
     Here ‘min’ and ‘max’ can be used to specify explicit bounds
     associated with the resulting TeX code.

     To issue a global TeX command (such as a TeX macro definition) in
     the TeX preamble (valid for the remainder of the top-level module)
     use:
     void texpreamble(string s);

     The TeX environment can be reset to its initial state, clearing all
     macro definitions, with the function
     void texreset();

     The routine
     void usepackage(string s, string options="");
     provides a convenient abbreviation for
     texpreamble("\usepackage["+options+"]{"+s+"}");
     that can be used for importing LaTeX packages.


File: asymptote.info,  Node: Deferred drawing,  Next: Files,  Prev: Frames and pictures,  Up: Programming

6.6 Deferred drawing
====================

It is sometimes desirable to have elements of a fixed absolute size,
independent of the picture scaling from user to ‘PostScript’
coordinates.  For example, normally one would want the size of a dot
created with ‘dot’, the size of the arrowheads created with ‘arrow’
(*note arrows::), and labels to be drawn independent of the scaling.

   However, because of ‘Asymptote’'s automatic scaling feature (*note
Figure size::), the translation between user coordinate and ‘PostScript’
coordinate is not determined until shipout time:
size(1cm);
dot((0,0));
dot((1,1));
shipout("x"); // at this point, 1 unit coordinate = 1cm
dot((2,2));
shipout("y"); // at this point, 1 unit coordinate = 0.5cm
It is therefore necessary to defer the drawing of these elements until
shipout time.

   For example, a frame can be added at a specified user coordinate of a
picture with the deferred drawing routine ‘add(picture
dest=currentpicture, frame src, pair position)’:
frame f;
fill(f,circle((0,0),1.5pt));
add(f,position=(1,1),align=(0,0));

   A deferred drawing routine is an object of type ‘drawer’, which is a
function with signature ‘void(frame f, transform t)’.  For example, if
the drawing routine
void d(frame f, transform t){
  fill(f,shift(3cm,0)*circle(t*(1,1),2pt));
}
is added to ‘currentpicture’ with
currentpicture.add(d);
a filled circle of radius 2pt will be drawn on currentpicture centered
3cm to the right of user coordinate ‘(1,1)’.  The parameter ‘t’ is the
affine transformation from user coordinates to ‘PostScript’ coordinates.

   Even though the actual drawing is deferred, you still need to specify
to the ‘Asymptote’ scaling routines that ultimately a fixed-size circle
of radius 2pt will be drawn 3cm to the right of user-coordinate (1,1),
by adding a frame about (1,1) that extends beyond (1,1) from
‘(3cm-2pt,-2pt)+min(currentpen)’ to ‘(3cm+2pt,2pt)+max(currentpen)’,
where we have even accounted for the pen linewidth.  The following
example will then produce a PDF file 10 cm wide:

settings.outformat="pdf";
size(10cm);
dot((0,0));
dot((1,1),red);
add(new void(frame f, transform t) {
  fill(f,shift(3cm,0)*circle(t*(1,1),2pt));
});
pair trueMin=(3cm-2pt,-2pt)+min(currentpen);
pair trueMax=(3cm+2pt,2pt)+max(currentpen);
currentpicture.addPoint((1,1),trueMin);
currentpicture.addPoint((1,1),trueMax);
Here we specified the minimum and maximum user and truesize (fixed)
coordinates of the drawer with the ‘picture’ routine
void addPoint(pair user, pair truesize);
Alternatively, one can use
void addBox(pair userMin, pair userMax, pair trueMin=0, pair trueMax=0) {
to specify a bounding box with bottom-left corner ‘t*(1,1)+trueMin’ and
top-right corner ‘t*(1,1)+trueMax’, where ‘t’ is the transformation that
transforms from user coordinates to ‘PostScript’ coordinates.

   For more details about deferred drawing, see "Asymptote: A vector
graphics language," John C. Bowman and Andy Hammerlindl, TUGBOAT: The
Communications of the TeX Users Group, 29:2, 288-294 (2008),

<https://www.math.ualberta.ca/~bowman/publications/asyTUG.pdf>.


File: asymptote.info,  Node: Files,  Next: Variable initializers,  Prev: Deferred drawing,  Up: Programming

6.7 Files
=========

‘Asymptote’ can read and write text files (including comma-separated
value) files and portable XDR (External Data Representation) binary
files.

   An input file can be opened with
input(string name=""
   reading is then done by assignment:
file fin=input("test.txt");
real a=fin;

   If the optional boolean argument ‘check’ is ‘false’, no check will be
made that the file exists.  If the file does not exist or is not
readable, the function ‘bool error(file)’ will return ‘true’.  The first
character of the string ‘comment’ specifies a comment character.  If
this character is encountered in a data file, the remainder of the line
is ignored.  When reading strings, a comment character followed
immediately by another comment character is treated as a single literal
comment character.  If ‘Asymptote’ is compiled with support for
‘libcurl’, ‘name’ can be a URL.

   Unless the ‘-noglobalread’ command-line option is specified, one can
change the current working directory for read operations to the contents
of the string ‘s’ with the function ‘string cd(string s)’, which returns
the new working directory.  If ‘string s’ is empty, the path is reset to
the value it had at program startup.

   When reading pairs, the enclosing parenthesis are optional.  Strings
are also read by assignment, by reading characters up to but not
including a newline.  In addition, ‘Asymptote’ provides the function
‘string getc(file)’ to read the next character (treating the comment
character as an ordinary character) and return it as a string.

   A file named ‘name’ can be open for output with
file output(string name="", bool update=false, string comment="#", string mode="");
If ‘update=false’, any existing data in the file will be erased and only
write operations can be used on the file.  If ‘update=true’, any
existing data will be preserved, the position will be set to the
end-of-file, and both reading and writing operations will be enabled.
For security reasons, writing to files in directories other than the
current directory is allowed only if the ‘-globalwrite’ (or ‘-nosafe’)
command-line option is specified.  Reading from files in other
directories is allowed unless the ‘-noglobalread’ command-line option is
specified.  The function ‘string mktemp(string s)’ may be used to create
and return the name of a unique temporary file in the current directory
based on the string ‘s’.

   There are two special files: ‘stdin’, which reads from the keyboard,
and ‘stdout’, which writes to the terminal.  The implicit initializer
for files is ‘null’.

   Data of a built-in type ‘T’ can be written to an output file by
calling one of the functions
write(string s="", T x, suffix suffix=endl, ... T[]);
write(file file, string s="", T x, suffix suffix=none, ... T[]);
write(file file=stdout, string s="", explicit T[] x, ... T[][]);
write(file file=stdout, T[][]);
write(file file=stdout, T[][][]);
write(suffix suffix=endl);
write(file file, suffix suffix=none);
   If ‘file’ is not specified, ‘stdout’ is used and terminated by
default with a newline.  If specified, the optional identifying string
‘s’ is written before the data ‘x’.  An arbitrary number of data values
may be listed when writing scalars or one-dimensional arrays.  The
‘suffix’ may be one of the following: ‘none’ (do nothing), ‘flush’
(output buffered data), ‘endl’ (terminate with a newline and flush),
‘newl’ (terminate with a newline), ‘DOSendl’ (terminate with a DOS
newline and flush), ‘DOSnewl’ (terminate with a DOS newline), ‘tab’
(terminate with a tab), or ‘comma’ (terminate with a comma).  Here are
some simple examples of data output:
file fout=output("test.txt");
write(fout,1);                  // Writes "1"
write(fout, endl);              // Writes a new line
write(fout,"List: ",1,2,3);     // Writes "List: 1     2     3"

   A file may be opened with ‘mode="xdr"’ to read or write double
precision (64-bit) reals and single precision (32-bit) integers in Sun
Microsystem's XDR (External Data Representation) portable binary format.
A file may instead be opened with ‘mode="binary"’ to read or write
double precision reals and single precision integers in the native
(nonportable) machine binary format.  The virtual member functions ‘file
singlereal(bool b=true)’ and ‘file singleint(bool b=true)’ may be used
to change the precision of real and integer I/O operations,
respectively, for an XDR or binary file.  Similarly, the function ‘file
signedint(bool b=true)’ can be used to modify the signedness of integer
reads and writes for an XDR or binary file.  Since there are no
end-of-line terminators in these formats, reading a string from an XDR
or binary file will by default read in the remainder of the file as a
string.  The virtual member function ‘file word(bool b=true)’ may be
used to change this behaviour: a double precision integer specifying the
length of the string is first read or written, followed by that many
characters.

   The virtual fields ‘name’, ‘mode’, ‘singlereal’, ‘singleint’, and
‘signedint’ may be used to query these respective states of a file.

   One can test a file for end-of-file with the boolean function
‘eof(file)’, end-of-line with ‘eol(file)’, and for I/O errors with
‘error(file)’.  One can flush the output buffers with ‘flush(file)’,
clear a previous I/O error with ‘clear(file)’, and close the file with
‘close(file)’.  The function ‘int precision(file file=stdout, int
digits=0)’ sets the number of digits of output precision for ‘file’ to
‘digits’, provided ‘digits’ is nonzero, and returns the previous
precision setting.  The function ‘int tell(file)’ returns the current
position in a file relative to the beginning.  The routine ‘seek(file
file, int pos)’ can be used to change this position, where a negative
value for the position ‘pos’ is interpreted as relative to the
end-of-file.  For example, one can rewind a file ‘file’ with the command
‘seek(file,0)’ and position to the final character in the file with
‘seek(file,-1)’.  The command ‘seekeof(file)’ sets the position to the
end of the file.

   Assigning ‘settings.scroll=n’ for a positive integer ‘n’ requests a
pause after every ‘n’ output lines to ‘stdout’.  One may then press
‘Enter’ to continue to the next ‘n’ output lines, ‘s’ followed by
‘Enter’ to scroll without further interruption, or ‘q’ followed by
‘Enter’ to quit the current output operation.  If ‘n’ is negative, the
output scrolls a page at a time (by one less than the current number of
display lines).  The default value, ‘settings.scroll=0’, specifies
continuous scrolling.

   The routines
string getstring(string name="", string default="", string prompt="",
                 bool store=true);
int getint(string name="", int default=0, string prompt="",
           bool store=true);
real getreal(string name="", real default=0, string prompt="",
             bool store=true);
pair getpair(string name="", pair default=0, string prompt="",
             bool store=true);
triple gettriple(string name="", triple default=(0,0,0), string prompt="",
                 bool store=true);
defined in the module ‘plain’ may be used to prompt for a value from
‘stdin’ using the GNU ‘readline’ library.  If ‘store=true’, the history
of values for ‘name’ is stored in the file ‘".asy_history_"+name’ (*note
history::).  The most recent value in the history will be used to
provide a default value for subsequent runs.  The default value
(initially ‘default’) is displayed after ‘prompt’.  These functions are
based on the internal routines
string readline(string prompt="", string name="", bool tabcompletion=false);
void saveline(string name, string value, bool store=true);
   Here, ‘readline’ prompts the user with the default value formatted
according to ‘prompt’, while ‘saveline’ is used to save the string
‘value’ in a local history named ‘name’, optionally storing the local
history in a file ‘".asy_history_"+name’.

   The routine ‘history(string name, int n=1)’ can be used to look up
the ‘n’ most recent values (or all values up to ‘historylines’ if ‘n=0’)
entered for string ‘name’.  The routine ‘history(int n=0)’ returns the
interactive history.  For example,
write(output("transcript.asy"),history());
outputs the interactive history to the file ‘transcript.asy’.

   The function ‘int delete(string s)’ deletes the file named by the
string ‘s’.  Unless the ‘-globalwrite’ (or ‘-nosafe’) option is enabled,
the file must reside in the current directory.  The function ‘int
rename(string from, string to)’ may be used to rename file ‘from’ to
file ‘to’.  Unless the ‘-globalwrite’ (or ‘-nosafe’) option is enabled,
this operation is restricted to the current directory.  The functions
int convert(string args="", string file="", string format="");
int animate(string args="", string file="", string format="");
call the ‘ImageMagick’ commands ‘magick’ and ‘animate’, respectively,
with the arguments ‘args’ and the file name constructed from the strings
‘file’ and ‘format’.


File: asymptote.info,  Node: Variable initializers,  Next: Structures,  Prev: Files,  Up: Programming

6.8 Variable initializers
=========================

A variable can be assigned a value when it is declared ‘int x=3;’ where
the variable ‘x’ is assigned the value ‘3’.  As well as literal
constants such as ‘3’, arbitary expressions can be used as initializers,
as in ‘real x=2*sin(pi/2);’.

   A variable is not added to the namespace until after the initializer
is evaluated, so for example, in
int x=2;
int x=5*x;
the ‘x’ in the initializer on the second line refers to the variable ‘x’
declared on the first line.  The second line, then, declares a variable
‘x’ shadowing the original ‘x’ and initializes it to the value ‘10’.

   Variables of most types can be declared without an explicit
initializer and they will be initialized by the default initializer of
that type:

   • Variables of the numeric types ‘int’, ‘real’, and ‘pair’ are all
     initialized to zero; variables of type ‘triple’ are initialized to
     ‘O=(0,0,0)’.
   • ‘boolean’ variables are initialized to ‘false’.
   • ‘string’ variables are initialized to the empty string.
   • ‘transform’ variables are initialized to the identity
     transformation.
   • ‘path’ and ‘guide’ variables are initialized to ‘nullpath’.
   • ‘pen’ variables are initialized to the default pen.
   • ‘frame’ and ‘picture’ variables are initialized to empty frames and
     pictures, respectively.
   • ‘file’ variables are initialized to ‘null’.

   The default initializers for user-defined array, structure, and
function types are explained in their respective sections.  Some types,
such as ‘code’, do not have default initializers.  When a variable of
such a type is introduced, the user must initialize it by explicitly
giving it a value.

   The default initializer for any type ‘T’ can be redeclared by
defining the function ‘T operator init()’.  For instance, ‘int’
variables are usually initialized to zero, but in
int operator init() {
  return 3;
}
int y;

the variable ‘y’ is initialized to ‘3’.  This example was given for
illustrative purposes; redeclaring the initializers of built-in types is
not recommended.  Typically, ‘operator init’ is used to define sensible
defaults for user-defined types.

   The special type ‘var’ may be used to infer the type of a variable
from its initializer.  If the initializer is an expression of a unique
type, then the variable will be defined with that type.  For instance,
var x=5;
var y=4.3;
var reddash=red+dashed;
is equivalent to
int x=5;
real y=4.3;
pen reddash=red+dashed;

   ‘var’ may also be used with the extended ‘for’ loop syntax.

int[] a = {1,2,3};
for (var x : a)
  write(x);


File: asymptote.info,  Node: Structures,  Next: Operators,  Prev: Variable initializers,  Up: Programming

6.9 Structures
==============

Users may also define their own data types as structures, along with
user-defined operators, much as in C++.  By default, structure members
are ‘public’ (may be read and modified anywhere in the code), but may be
optionally declared ‘restricted’ (readable anywhere but writeable only
inside the structure where they are defined) or ‘private’ (readable and
writable only inside the structure).  In a structure definition, the
keyword ‘this’ can be used as an expression to refer to the enclosing
structure.  Any code at the top-level scope within the structure is
executed on initialization.

   Variables hold references to structures.  That is, in the example:
struct T {
  int x;
}

T foo;
T bar=foo;
bar.x=5;

   The variable ‘foo’ holds a reference to an instance of the structure
‘T’.  When ‘bar’ is assigned the value of ‘foo’, it too now holds a
reference to the same instance as ‘foo’ does.  The assignment ‘bar.x=5’
changes the value of the field ‘x’ in that instance, so that ‘foo.x’
will also be equal to ‘5’.

   The expression ‘new T’ creates a new instance of the structure ‘T’
and returns a reference to that instance.  In creating the new instance,
any code in the body of the record definition is executed.  For example:
int Tcount=0;
struct T {
  int x;
  ++Tcount;
}

T foo=new T;
T foo;
Here, ‘new T’ produces a new instance of the struct, which causes
‘Tcount’ to be incremented, tracking the number of instances produced.
The declarations ‘T foo=new T’ and ‘T foo’ are equivalent: the second
form implicitly creates a new instance of ‘T’.  That is, after the
definition of a structure ‘T’, a variable of type ‘T’ is initialized to
a new instance (‘new T’) by default.  During the definition of the
structure, however, variables of type ‘T’ are initialized to ‘null’ by
default.  This special behavior is to avoid infinite recursion of
creating new instances in code such as
struct tree {
  int value;
  tree left;
  tree right;
}

   The expression ‘null’ can be cast to any structure type to yield a
null reference, a reference that does not actually refer to any instance
of the structure.  Trying to use a field of a null reference will cause
an error.

   The function ‘bool alias(T,T)’ checks to see if two structure
references refer to the same instance of the structure (or both to
‘null’).  In the example at the beginning of this section,
‘alias(foo,bar)’ would return true, but ‘alias(foo,new T)’ would return
false, as ‘new T’ creates a new instance of the structure ‘T’.  The
boolean operators ‘==’ and ‘!=’ are by default equivalent to ‘alias’ and
‘!alias’ respectively, but may be overwritten for a particular type (for
example, to do a deep comparison).

   Here is a simple example that illustrates the use of structures:
struct S {
  real a=1;
  real f(real a) {return a+this.a;}
}

S s;                            // Initializes s with new S;

write(s.f(2));                  // Outputs 3

S operator + (S s1, S s2)
{
  S result;
  result.a=s1.a+s2.a;
  return result;
}

write((s+s).f(0));              // Outputs 2

   It is often convenient to have functions that construct new instances
of a structure.  Say we have a ‘Person’ structure:
struct Person {
  string firstname;
  string lastname;
}

Person joe;
joe.firstname="Joe";
joe.lastname="Jones";
Creating a new Person is a chore; it takes three lines to create a new
instance and to initialize its fields (that's still considerably less
effort than creating a new person in real life, though).

   We can reduce the work by defining ‘operator init’:
struct Person {
  string firstname;
  string lastname;

  void operator init(string firstname, string lastname) {
    this.firstname=firstname;
    this.lastname=lastname;
  }
}

Person joe=Person("Joe", "Jones");

   The use of ‘operator init’ to implicitly define constructors should
not be confused with its use to define default values for variables
(*note Variable initializers::).  Indeed, in the first case, the return
type of the ‘operator init’ must be ‘void’ while in the second, it must
be the (non-‘void’) type of the variable.

   Internally, ‘operator init’ implicitly defines a constructor function
‘Person(string,string)’ as follows, where ARGS is ‘string firstname,
string lastname’ in this case:
     struct Person {
       string firstname;
       string lastname;

       static Person Person(ARGS) {
         Person p=new Person;
         p.operator init(ARGS);
         return p;
       }
     }
which then can be used as:
Person joe=Person.Person("Joe", "Jones");

   The following is also implicitly generated in the enclosing scope,
after the end of the structure definition.
from Person unravel Person;
It allows us to use the constructor without qualification, otherwise we
would have to refer to the constructor by the qualified name
‘Person.Person’.

   Much like in C++, casting (*note Casts::) provides for an elegant
implementation of structure inheritance, including a virtual function
‘v’:
struct parent {
  real x;
  void operator init(int x) {this.x=x;}
  void v(int) {write(0);}
  void f() {v(1);}
}

void write(parent p) {write(p.x);}

struct child {
  parent parent;
  real y=3;
  void operator init(int x) {parent.operator init(x);}
  void v(int x) {write(x);}
  parent.v=v;
  void f()=parent.f;
}

parent operator cast(child child) {return child.parent;}

parent p=parent(1);
child c=child(2);

write(c);                       // Outputs 2;

p.f();                          // Outputs 0;
c.f();                          // Outputs 1;

write(c.parent.x);              // Outputs 2;
write(c.y);                     // Outputs 3;

   For further examples of structures, see ‘Legend’ and ‘picture’ in the
‘Asymptote’ base module ‘plain’.


File: asymptote.info,  Node: Operators,  Next: Implicit scaling,  Prev: Structures,  Up: Programming

6.10 Operators
==============

* Menu:

* Arithmetic & logical::        Basic mathematical operators
* Self & prefix operators::     Increment and decrement
* User-defined operators::      Overloading operators
* Bracket operators::           Bracket operators
* Cast operators::              Type conversion


File: asymptote.info,  Node: Arithmetic & logical,  Next: Self & prefix operators,  Up: Operators

6.10.1 Arithmetic & logical operators
-------------------------------------

‘Asymptote’ uses the standard binary arithmetic operators.  However,
when one integer is divided by another, both arguments are converted to
real values before dividing and a real quotient is returned (since this
is typically what is intended; otherwise one can use the function ‘int
quotient(int x, int y)’, which returns greatest integer less than or
equal to ‘x/y’).  In all other cases both operands are promoted to the
same type, which will also be the type of the result:
‘+’
     addition
‘-’
     subtraction
‘*’
     multiplication
‘/’
     division
‘#’
     integer division; equivalent to ‘quotient(x,y)’.  Noting that the
     ‘Python3’ community adopted our comment symbol (‘//’) for integer
     division, we decided to reciprocate and use their comment symbol
     for integer division in ‘Asymptote’!
‘%’
     modulo; the result always has the same sign as the divisor.  In
     particular, this makes ‘q*(p # q)+p % q == p’ for all integers ‘p’
     and nonzero integers ‘q’.
‘^’
     power; if the exponent (second argument) is an int, recursive
     multiplication is used; otherwise, logarithms and exponentials are
     used (‘**’ is a synonym for ‘^’).

   The usual boolean operators are also defined:
‘==’
     equals
‘!=’
     not equals
‘<’
     less than
‘<=’
     less than or equals
‘>=’
     greater than or equals
‘>’
     greater than
‘&&’
     and (with conditional evaluation of right-hand argument)
‘&’
     and
‘||’
     or (with conditional evaluation of right-hand argument)
‘|’
     or
‘^’
     xor
‘!’
     not

   ‘Asymptote’ also supports the C-like conditional syntax:
bool positive=(pi > 0) ? true : false;

   The function ‘T interp(T a, T b, real t)’ returns ‘(1-t)*a+t*b’ for
nonintegral built-in arithmetic types ‘T’.  If ‘a’ and ‘b’ are pens,
they are first promoted to the same color space.

   ‘Asymptote’ also defines bitwise functions ‘int AND(int,int)’, ‘int
OR(int,int)’, ‘int XOR(int,int)’, ‘int NOT(int)’, ‘int CLZ(int)’ (count
leading zeros), ‘int CTZ(int)’ (count trailing zeros), ‘int
popcount(int)’ (count bits populated by ones), and ‘int bitreverse(int
a, int bits)’ (reverse bits within a word of length bits).


File: asymptote.info,  Node: Self & prefix operators,  Next: User-defined operators,  Prev: Arithmetic & logical,  Up: Operators

6.10.2 Self & prefix operators
------------------------------

As in C, each of the arithmetic operators ‘+’, ‘-’, ‘*’, ‘/’, ‘#’, ‘%’,
and ‘^’ can be used as a self operator.  The prefix operators ‘++’
(increment by one) and ‘--’ (decrement by one) are also defined.  For
example,
int i=1;
i += 2;
int j=++i;

is equivalent to the code
int i=1;
i=i+2;
int j=i=i+1;

   However, postfix operators like ‘i++’ and ‘i--’ are not defined
(because of the inherent ambiguities that would arise with the ‘--’
path-joining operator).  In the rare instances where ‘i++’ and ‘i--’ are
really needed, one can substitute the expressions ‘(++i-1)’ and
‘(--i+1)’, respectively.


File: asymptote.info,  Node: User-defined operators,  Next: Bracket operators,  Prev: Self & prefix operators,  Up: Operators

6.10.3 User-defined operators
-----------------------------

The following symbols may be used with ‘operator’ to define or redefine
operators on structures and built-in types:
- + * / % ^ ! < > == != <= >= & | ^^ .. :: -- --- ++
<< >> $ $$ @ @@ <>
The operators on the second line have precedence one higher than the
boolean operators ‘<’, ‘>’, ‘<=’, and ‘>=’.

   Guide operators like ‘..’ may be overloaded, say, to write a user
function that produces a new guide from a given guide:
guide dots(...guide[] g)=operator ..;

guide operator ..(...guide[] g) {
  guide G;
  if(g.length > 0) {
    write(g[0]);
    G=g[0];
  }
  for(int i=1; i < g.length; ++i) {
    write(g[i]);
    write();
    G=dots(G,g[i]);
  }
  return G;
}

guide g=(0,0){up}..{SW}(100,100){NE}..{curl 3}(50,50)..(10,10);
write("g=",g);

   *Note Autounravel:: for an example of overloading the ‘+’ operator.


File: asymptote.info,  Node: Bracket operators,  Next: Cast operators,  Prev: User-defined operators,  Up: Operators

6.10.4 Bracket operators
------------------------

If ‘a’ is not an array, the notation ‘a[i]’ is equivalent to
‘a.operator[](i)’.  Thus, the ‘[]’ operator can be defined to provide
this notation for a user-defined structure.  Similarly, the notation
‘a[i]=b’ is equivalent to ‘a.operator[=](i,b)’ (except that the former
always returns ‘b’).  Note that there are some restrictions on how these
operators can be defined:

   • Neither operator can be overloaded within a single structure
     (although different structures can have different definitions for
     these operators).
   • ‘operator []’ must take a single argument and have a non-void
     return type.
   • ‘operator [=]’ can only be defined if ‘operator[]’ is also defined.
   • ‘operator[=]’ must take two arguments and have a void return type.
     Furthermore, if ‘K’ and ‘V’ are the types of the first and second
     arguments to ‘operator[=]’, respectively, then ‘K’ must be the type
     of the argument to ‘operator[]’ and ‘V’ must be the return type of
     ‘operator[]’.

   Here is a simple example.  (*Note Maps:: for data structures similar
to ‘slowMap’ but more flexible, more powerful, and faster.)

struct slowMap {
  string[] keys;
  real[] values;
  real operator [](string key) {
    for(int i=0; i < keys.length; ++i) {
      if(keys[i] == key) return values[i];
    }
    return -inf;
  }
  void operator [=](string key, real value) {
    assert(all(keys != key), 'duplicate key');
    keys.push(key);
    values.push(value);
  }
}

slowMap m;
m['pi']=3.14159;
m['e']=2.71828;
assert(m['pi']==3.14159);
assert(m['e']==2.71828);
assert(m['sqrt(2)'] == -inf);
write("m['pi']=", m['pi']);


File: asymptote.info,  Node: Cast operators,  Prev: Bracket operators,  Up: Operators

6.10.5 Cast operators
---------------------

*Note Casts:: for information on ‘operator cast’ and ‘operator ecast’.


File: asymptote.info,  Node: Implicit scaling,  Next: Functions,  Prev: Operators,  Up: Programming

6.11 Implicit scaling
=====================

If a numeric literal is in front of certain types of expressions, then
the two are multiplied:
int x=2;
real y=2.0;
real cm=72/2.540005;

write(3x);
write(2.5x);
write(3y);
write(-1.602e-19 y);
write(0.5(x,y));
write(2x^2);
write(3x+2y);
write(3(x+2y));
write(3sin(x));
write(3(sin(x))^2);
write(10cm);

   This produces the output
6
5
6
-3.204e-19
(1,1)
8
10
18
2.72789228047704
2.48046543129542
283.464008929116


File: asymptote.info,  Node: Functions,  Next: Arrays,  Prev: Implicit scaling,  Up: Programming

6.12 Functions
==============

* Menu:

* Default arguments::           Default values can appear anywhere
* Named arguments::             Assigning function arguments by keyword
* Rest arguments::              Functions with a variable number of arguments
* Mathematical functions::      Standard libm functions

‘Asymptote’ functions are treated as variables with a signature
(non-function variables have null signatures).  Variables with the same
name are allowed, so long as they have distinct signatures.

   Function arguments are passed by value.  To pass an argument by
reference, simply enclose it in a structure (*note Structures::).

   Here are some significant features of ‘Asymptote’ functions:

  1. Variables with signatures (functions) and without signatures
     (nonfunction variables) are distinct:
     int x, x();
     x=5;
     x=new int() {return 17;};
     x=x();              // calls x() and puts the result, 17, in the scalar x

  2. Traditional function definitions are allowed:
     int sqr(int x)
     {
       return x*x;
     }
     sqr=null;           // but the function is still just a variable.

  3. Casting can be used to resolve ambiguities:
     int a, a(), b, b(); // Valid: creates four variables.
     a=b;                // Invalid: assignment is ambiguous.
     a=(int) b;          // Valid: resolves ambiguity.
     (int) (a=b);        // Valid: resolves ambiguity.
     (int) a=b;          // Invalid: cast expressions cannot be L-values.
     
     int c();
     c=a;                // Valid: only one possible assignment.

  4. "Higher-order" functions, in other words functions that return
     functions, are allowed.  The return type must be given a
     signature-free alias with ‘using’ or ‘typedef’:
     using intop = int(int);
     // typedef int intop(int); // Equivalent to previous line
     intop adder(int m)
     {
       return new int(int n) {return m+n;};
     }
     intop addby7=adder(7);
     write(addby7(1));       // Writes 8.

  5. One may redefine a function ‘f’, even for calls to ‘f’ in
     previously declared functions, by assigning another (anonymous or
     named) function to it.  However, if ‘f’ is overloaded by a new
     function definition, previous calls will still access the original
     version of ‘f’, as illustrated in this example:
     void f() {
       write("hi");
     }
     
     void g() {
       f();
     }
     
     g(); // writes "hi"
     
     f=new void() {write("bye");};
     
     g(); // writes "bye"
     
     void f() {write("overloaded");};
     
     f(); // writes "overloaded"
     g(); // writes "bye"

  6. Anonymous functions can be used to redefine a function variable
     that has been declared (and implicitly initialized to the null
     function) but not yet explicitly defined:
     void f(bool b);
     
     void g(bool b) {
       if(b) f(b);
       else write(b);
     }
     
     f=new void(bool b) {
       write(b);
       g(false);
     };
     
     g(true); // Writes true, then writes false.

   ‘Asymptote’ is the only language we know of that treats functions as
variables, but allows overloading by distinguishing variables based on
their signatures.

   Functions are allowed to call themselves recursively.  As in C++,
infinite nested recursion will generate a stack overflow (reported as a
segmentation fault or illegal instruction, unless a fully working
version of the GNU library ‘libsigsegv’ is installed at configuration
time).


File: asymptote.info,  Node: Default arguments,  Next: Named arguments,  Up: Functions

6.12.1 Default arguments
------------------------

‘Asymptote’ supports a more flexible mechanism for default function
arguments than C++: they may appear anywhere in the function prototype.
Because certain data types are implicitly cast to more sophisticated
types (*note Casts::) one can often avoid ambiguities by ordering
function arguments from the simplest to the most complicated.  For
example, given
real f(int a=1, real b=0) {return a+b;}
then ‘f(1)’ returns 1.0, but ‘f(1.0)’ returns 2.0.

   The value of a default argument is determined by evaluating the given
‘Asymptote’ expression in the scope where the called function is
defined.


File: asymptote.info,  Node: Named arguments,  Next: Rest arguments,  Prev: Default arguments,  Up: Functions

6.12.2 Named arguments
----------------------

It is sometimes difficult to remember the order in which arguments
appear in a function declaration.  Named (keyword) arguments make
calling functions with multiple arguments easier.  Unlike in the C and
C++ languages, an assignment in a function argument is interpreted as an
assignment to a parameter of the same name in the function signature,
_not within the local scope_.  The command-line option ‘-d’ may be used
to check ‘Asymptote’ code for cases where a named argument may be
mistaken for a local assignment.

   When matching arguments to signatures, first all of the keywords are
matched, then the arguments without names are matched against the
unmatched formals as usual.  For example,
int f(int x, int y) {
  return 10x+y;
}
write(f(4,x=3));
outputs 34, as ‘x’ is already matched when we try to match the unnamed
argument ‘4’, so it gets matched to the next item, ‘y’.

   For the rare occasions where it is desirable to assign a value to
local variable within a function argument (generally _not_ a good
programming practice), simply enclose the assignment in parentheses.
For example, given the definition of ‘f’ in the previous example,
int x;
write(f(4,(x=3)));
is equivalent to the statements
int x;
x=3;
write(f(4,3));
and outputs 43.

   Parameters can be specified as "keyword-only" by putting ‘keyword’
immediately before the parameter name, as in ‘int f(int keyword x)’ or
‘int f(int keyword x=77)’.  This forces the caller of the function to
use a named argument to give a value for this parameter.  That is,
‘f(x=42)’ is legal, but ‘f(25)’ is not.  Keyword-only parameters must be
listed after normal parameters in a function definition.

   As a technical detail, we point out that, since variables of the same
name but different signatures are allowed in the same scope, the code
int f(int x, int x()) {
  return x+x();
}
int seven() {return 7;}
is legal in ‘Asymptote’, with ‘f(2,seven)’ returning 9.  A named
argument matches the first unmatched formal of the same name, so
‘f(x=2,x=seven)’ is an equivalent call, but ‘f(x=seven,2)’ is not, as
the first argument is matched to the first formal, and ‘int ()’ cannot
be implicitly cast to ‘int’.  Default arguments do not affect which
formal a named argument is matched to, so if ‘f’ were defined as
int f(int x=3, int x()) {
  return x+x();
}
then ‘f(x=seven)’ would be illegal, even though ‘f(seven)’ obviously
would be allowed.


File: asymptote.info,  Node: Rest arguments,  Next: Mathematical functions,  Prev: Named arguments,  Up: Functions

6.12.3 Rest arguments
---------------------

Rest arguments allow one to write functions that take a variable number
of arguments:
// This function sums its arguments.
int sum(...int[] nums) {
  int total=0;
  for(int i=0; i < nums.length; ++i)
    total += nums[i];
  return total;
}

sum(1,2,3,4);                       // returns 10
sum();                              // returns 0

// This function subtracts subsequent arguments from the first.
int subtract(int start, ... int[] subs) {
  for(int i=0; i < subs.length; ++i)
    start -= subs[i];
  return start;
}

subtract(10,1,2);                   // returns 7
subtract(10);                       // returns 10
subtract();                         // illegal

   The comma before an ellipsis is optional, so that the following two
declarations are equivalent:
int subtract(int start, ... int[] subs);
int subtract(int start ... int[] subs);
   Note, however, that a comma cannot directly follow an open
parenthesis: ‘int sum(, ... int[] nums)’ is a syntax error.

   Putting an argument into a rest array is called _packing_.  One can
give an explicit list of arguments for the rest argument, so ‘subtract’
could alternatively be implemented as
int subtract(int start, ... int[] subs) {
  return start - sum(... subs);
}

   One can even combine normal arguments with rest arguments:
sum(1,2,3, ... new int[] {4,5,6});   // returns 21
This builds a new six-element array that is passed to ‘sum’ as ‘nums’.
The opposite operation, _unpacking_, is not allowed:
subtract(...new int[] {10, 1, 2});
is illegal, as the start formal is not matched.

   If no arguments are packed, then a zero-length array (as opposed to
‘null’) is bound to the rest parameter.  Note that default arguments are
ignored for rest formals and the rest argument is not bound to a
keyword.

   In some cases, keyword-only parameters are helpful to avoid arguments
intended for the rest parameter to be assigned to other parameters.  For
example, here the use of ‘keyword’ is to avoid ‘pnorm(1.0,2.0,0.3)’
matching ‘1.0’ to ‘p’.
real pnorm(real keyword p=2.0, ... real[] v)
{
  return sum(v^p)^(1/p);
}

   The overloading resolution in ‘Asymptote’ is similar to the function
matching rules used in C++.  Every argument match is given a score.
Exact matches score better than matches with casting, and matches with
formals (regardless of casting) score better than packing an argument
into the rest array.  A candidate is maximal if all of the arguments
score as well in it as with any other candidate.  If there is one unique
maximal candidate, it is chosen; otherwise, there is an ambiguity error.

int f(path g);
int f(guide g);
f((0,0)--(100,100)); // matches the second; the argument is a guide

int g(int x, real y);
int g(real x, int x);

g(3,4); // ambiguous; the first candidate is better for the first argument,
        // but the second candidate is better for the second argument

int h(... int[] rest);
int h(real x, ... int[] rest);

h(1,2); // the second definition matches, even though there is a cast,
        // because casting is preferred over packing

int i(int x, ... int[] rest);
int i(real x, real y, ... int[] rest);

i(3,4); // ambiguous; the first candidate is better for the first argument,
        // but the second candidate is better for the second one


File: asymptote.info,  Node: Mathematical functions,  Prev: Rest arguments,  Up: Functions

6.12.4 Mathematical functions
-----------------------------

‘Asymptote’ has built-in versions of the standard ‘libm’ mathematical
real(real) functions ‘sin’, ‘cos’, ‘tan’, ‘asin’, ‘acos’, ‘atan’, ‘exp’,
‘log’, ‘pow10’, ‘log10’, ‘sinh’, ‘cosh’, ‘tanh’, ‘asinh’, ‘acosh’,
‘atanh’, ‘sqrt’, ‘cbrt’, ‘fabs’, ‘expm1’, ‘log1p’, as well as the
identity function ‘identity’.  ‘Asymptote’ also defines the order ‘n’
Bessel functions of the first kind ‘Jn(int n, real)’ and second kind
‘Yn(int n, real)’, as well as the gamma function ‘gamma’, the error
function ‘erf’, and the complementary error function ‘erfc’.  The
standard real(real, real) functions ‘atan2’, ‘hypot’, ‘fmod’,
‘remainder’ are also included.

   The functions ‘degrees(real radians)’ and ‘radians(real degrees)’ can
be used to convert between radians and degrees.  The function
‘Degrees(real radians)’ returns the angle in degrees in the interval
[0,360).  For convenience, ‘Asymptote’ defines variants ‘Sin’, ‘Cos’,
‘Tan’, ‘aSin’, ‘aCos’, and ‘aTan’ of the standard trigonometric
functions that use degrees rather than radians.  We also define complex
versions of the ‘sqrt’, ‘sin’, ‘cos’, ‘exp’, ‘log’, and ‘gamma’
functions.

   The functions ‘floor’, ‘ceil’, and ‘round’ differ from their usual
definitions in that they all return an int value rather than a real
(since that is normally what one wants).  The functions ‘Floor’, ‘Ceil’,
and ‘Round’ are respectively similar, except that if the result cannot
be converted to a valid int, they return ‘intMax’ for positive arguments
and ‘intMin’ for negative arguments, rather than generating an integer
overflow.  We also define a function ‘sgn’, which returns the sign of
its real argument as an integer (-1, 0, or 1).

   There is an ‘abs(int)’ function, as well as an ‘abs(real)’ function
(equivalent to ‘fabs(real)’), an ‘abs(pair)’ function (equivalent to
‘length(pair)’).

   Asymptote's random number generators can be seeded with ‘srand(int)’.
If a negative argument is given to ‘srand’, an internal entropy source
is used as a seed.  Random numbers are initially seeded with
‘srand(-1)’.  The function ‘int rand(int a=0, int b=randMax)’ returns a
random integer in [a,b].  The ‘unitrand()’ function returns a random
number uniformly distributed in the interval [0,1).  The ‘randString(int
len, string chars)’ function returns a random string of length ‘len’
composed of characters from the string ‘chars’.  If ‘chars’ is the empty
string, the string
‘'abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789'’ is
used.  In each position of the output, the probability of a character
appearing is proportional to the number of times it appears in ‘chars’.

   A Gaussian random number generator ‘Gaussrand’ and a collection of
statistics routines, including ‘histogram’, are provided in the module
‘stats’.

   The functions ‘factorial(int n)’, which returns n!, and ‘choose(int
n, int k)’, which returns n!/(k!(n-k)!), are also defined.

   When configured with the GNU Scientific Library (GSL), available from
<https://www.gnu.org/software/gsl/>, ‘Asymptote’ contains an internal
module ‘gsl’ that defines the airy functions ‘Ai(real)’, ‘Bi(real)’,
‘Ai_deriv(real)’, ‘Bi_deriv(real)’, ‘zero_Ai(int)’, ‘zero_Bi(int)’,
‘zero_Ai_deriv(int)’, ‘zero_Bi_deriv(int)’, the Bessel functions ‘I(int,
real)’, ‘K(int, real)’, ‘j(int, real)’, ‘y(int, real)’, ‘i_scaled(int,
real)’, ‘k_scaled(int, real)’, ‘J(real, real)’, ‘Y(real, real)’,
‘I(real, real)’, ‘K(real, real)’, ‘zero_J(real, int)’, the elliptic
functions ‘F(real, real)’, ‘E(real, real)’, and ‘P(real, real)’, the
Jacobi elliptic functions ‘real[] sncndn(real,real)’, the
exponential/trigonometric integrals ‘Ei’, ‘Si’, and ‘Ci’, the Legendre
polynomials ‘Pl(int, real)’, and the Riemann zeta function ‘zeta(real)’.
For example, to compute the sine integral ‘Si’ of 1.0:
import gsl;
write(Si(1.0));

   ‘Asymptote’ also provides a few general purpose numerical routines:

‘real newton(int iterations=100, real f(real), real fprime(real), real x, bool verbose=false);’
     Use Newton-Raphson iteration to solve for a root of a real-valued
     differentiable function ‘f’, given its derivative ‘fprime’ and an
     initial guess ‘x’.  Diagnostics for each iteration are printed if
     ‘verbose=true’.  If the iteration fails after the maximum allowed
     number of loops (‘iterations’), ‘realMax’ is returned.

‘real newton(int iterations=100, real f(real), real fprime(real), real x1, real x2, bool verbose=false);’
     Use bracketed Newton-Raphson bisection to solve for a root of a
     real-valued differentiable function ‘f’ within an interval
     [‘x1’,‘x2’] (on which the endpoint values of ‘f’ have opposite
     signs), given its derivative ‘fprime’.  Diagnostics for each
     iteration are printed if ‘verbose=true’.  If the iteration fails
     after the maximum allowed number of loops (‘iterations’), ‘realMax’
     is returned.

‘real simpson(real f(real), real a, real b, real acc=realEpsilon, real dxmax=b-a)’
     returns the integral of ‘f’ from ‘a’ to ‘b’ using adaptive Simpson
     integration.

‘cputime cputime()’
     returns a structure ‘cputime’ with cumulative CPU times broken down
     into the fields ‘parent.user’, ‘parent.system’, ‘child.user’, and
     ‘child.system’, along with the cumulative wall clock time in
     ‘parent.clock’, all measured in seconds.  For convenience, the
     incremental fields ‘change.user’, ‘change.system’, and
     ‘change.clock’ indicate the change in the corresponding fields
     since the last call to ‘cputime()’.  The function
     void write(file file=stdout, string s="", cputime c,
                string format=cputimeformat, suffix suffix=none);
     displays the incremental user cputime followed by "u", the
     incremental system cputime followed by "s", the total user cputime
     followed by "U", and the total system cputime followed by "S".


File: asymptote.info,  Node: Arrays,  Next: Casts,  Prev: Functions,  Up: Programming

6.13 Arrays
===========

* Menu:

* Slices::                      Python-style array slices

Appending ‘[]’ to a built-in or user-defined type yields an array.  The
array element ‘i’ of an array ‘A’ can be accessed as ‘A[i]’.  By
default, attempts to access or assign to an array element using a
negative index generates an error.  Reading an array element with an
index beyond the length of the array also generates an error; however,
assignment to an element beyond the length of the array causes the array
to be resized to accommodate the new element.  One can also index an
array ‘A’ with an integer array ‘B’: the array ‘A[B]’ is formed by
indexing array ‘A’ with successive elements of array ‘B’.  A convenient
shorthand exists for iterating over all elements of an array; see *note
array iteration::.

   The declaration
real[] A;

initializes ‘A’ to be an empty (zero-length) array.  Empty arrays should
be distinguished from null arrays.  If we say
real[] A=null;

then ‘A’ cannot be dereferenced at all (null arrays have no length and
cannot be read from or assigned to).

   Arrays can be explicitly initialized like this:
real[] A={0,1,2};
An explicit array initialization can also end with a copy of another
array:
real[] A={0,1};
real[] B={...A};  // Now B contains {0,1}.
real[] C={4 ...A};  // Now C contains {4,0,1}.
(Side note: without the space, ‘4...A’ would be parsed as ‘4.0 .. A’ and
thus give a syntax error.)

   Array assignment in ‘Asymptote’ does a shallow copy: only the pointer
is copied (if one copy if modified, the other will be too).
real[] A = {0,1,2};
real[] B = A;
B[0] = 3;
write(A[0]); // Outputs 3.
The ‘copy’ function listed below provides a deep copy of an array.

   Every array ‘A’ of type ‘T[]’ has the virtual members
   • ‘int length’,
   • ‘bool cyclic’,
   • ‘int[] keys’,
   • ‘T push(T x)’,
   • ‘void append(T[] a)’,
   • ‘T pop()’,
   • ‘void insert(int i, ... T[] x)’,
   • ‘void delete(int i, int j=i)’,
   • ‘void delete()’, and
   • ‘bool initialized(int n)’.

   The member ‘A.length’ evaluates to the length of the array.  Setting
‘A.cyclic=true’ signifies that array indices should be reduced modulo
the current array length.  Reading from or writing to a nonempty cyclic
array never leads to out-of-bounds errors or array resizing.

   The member ‘A.keys’ evaluates to an array of integers containing the
indices of initialized entries in the array in ascending order.  Hence,
for an array of length ‘n’ with all entries initialized, ‘A.keys’
evaluates to ‘{0,1,...,n-1}’.  A new keys array is produced each time
‘A.keys’ is evaluated.

   The functions ‘A.push’ and ‘A.append’ append their arguments onto the
end of the array, while ‘A.insert(int i, ... T[] x)’ inserts ‘x’ into
the array at index ‘i’.  For convenience ‘A.push’ returns the pushed
item.  The function ‘A.pop()’ pops and returns the last element, while
‘A.delete(int i, int j=i)’ deletes elements with indices in the range
[‘i’,‘j’], shifting the position of all higher-indexed elements down.
If no arguments are given, ‘A.delete()’ provides a convenient way of
deleting all elements of ‘A’.  The routine ‘A.initialized(int n)’ can be
used to examine whether the element at index ‘n’ is initialized.  Like
all ‘Asymptote’ functions, ‘push’, ‘append’, ‘pop’, ‘insert’, ‘delete’,
and ‘initialized’ can be "pulled off" of the array and used on their
own.  For example,
int[] A={1};
A.push(2);         // A now contains {1,2}.
A.append(A);       // A now contains {1,2,1,2}.
int f(int)=A.push;
f(3);              // A now contains {1,2,1,2,3}.
int g()=A.pop;
write(g());        // Outputs 3.
A.delete(0);       // A now contains {2,1,2}.
A.delete(0,1);       // A now contains {2}.
A.insert(1,3);     // A now contains {2,3}.
A.insert(1, ... A); // A now contains {2,2,3,3}
A.insert(2,4,5);   // A now contains {2,2,4,5,3,3}.

   The ‘[]’ suffix can also appear after the variable name; this is
sometimes convenient for declaring a list of variables and arrays of the
same type:
real a,A[];
This declares ‘a’ to be ‘real’ and implicitly declares ‘A’ to be of type
‘real[]’.

   In the following list of built-in array functions, ‘T’ represents a
generic type.  Note that the internal functions ‘alias’, ‘array’,
‘copy’, ‘concat’, ‘sequence’, ‘map’, and ‘transpose’, which depend on
type ‘T[]’, are defined only after the first declaration of a variable
of type ‘T[]’.

‘new T[]’
     returns a new empty array of type ‘T[]’;

‘new T[] {list}’
     returns a new array of type ‘T[]’ initialized with ‘list’ (a comma
     delimited list of elements);

‘new T[n]’
     returns a new array of ‘n’ elements of type ‘T[]’.  These ‘n’ array
     elements are not initialized unless they are arrays themselves (in
     which case they are each initialized to empty arrays);

‘T[] array(int n, T value, int depth=intMax)’
     returns an array consisting of ‘n’ copies of ‘value’.  If ‘value’
     is itself an array, a deep copy of ‘value’ is made for each entry.
     If ‘depth’ is specified, this deep copying only recurses to the
     specified number of levels;

‘int[] sequence(int n)’
     if ‘n >= 1’ returns the array ‘{0,1,...,n-1}’ (otherwise returns a
     null array);

‘int[] sequence(int n, int m)’
     if ‘m >= n’ returns an array ‘{n,n+1,...,m}’ (otherwise returns a
     null array); see also *note range::;

‘int[] sequence(int n, int m, int skip)’
     if ‘(m-n)/skip >= 0’ returns an array ‘{n,n+skip,...,m}’ skipping
     by ‘skip’ (otherwise returns a null array); see also *note range::;

‘T[] sequence(T f(int), int n)’
     if ‘n >= 1’ returns the sequence ‘{f_i :i=0,1,...n-1}’ given a
     function ‘T f(int)’ and integer ‘int n’ (otherwise returns a null
     array);

‘T[] map(T f(T), T[] a)’
     returns the array obtained by applying the function ‘f’ to each
     element of the array ‘a’.  This is equivalent to ‘sequence(new
     T(int i) {return f(a[i]);},a.length)’;

‘T2[] map(T2 f(T1), T1[] a)’
     constructed by running ‘from mapArray(Src=T1, Dst=T2) access map;’,
     returns the array obtained by applying the function ‘f’ to each
     element of the array ‘a’;

‘int[] reverse(int n)’
     if ‘n >= 1’ returns the array ‘{n-1,n-2,...,0}’ (otherwise returns
     a null array);

‘int[] complement(int[] a, int n)’
     returns the complement of the integer array ‘a’ in
     ‘{0,1,2,...,n-1}’, so that ‘b[complement(a,b.length)]’ yields the
     complement of ‘b[a]’;

‘real[] uniform(real a, real b, int n)’
     if ‘n >= 1’ returns a uniform partition of ‘[a,b]’ into ‘n’
     subintervals (otherwise returns a null array);

‘int find(bool[] a, int n=1)’
     returns the index of the ‘n’th ‘true’ value in the boolean array
     ‘a’ or -1 if not found.  If ‘n’ is negative, search backwards from
     the end of the array for the ‘-n’th value;

‘int[] findall(bool[] a)’
     returns the indices of all ‘true’ values in the boolean array ‘a’;

‘int search(T[] a, T key)’
     For built-in ordered types ‘T’, searches a sorted array ‘a’ of ‘n’
     elements for ‘key’, returning the index ‘i’ if ‘a[i] <= key <
     a[i+1]’, ‘-1’ if ‘key’ is less than all elements of ‘a’, or ‘n-1’
     if ‘key’ is greater than or equal to the last element of ‘a’;

‘int search(T[] a, T key, bool less(T i, T j))’
     searches an array ‘a’ for ‘key’ sorted in ascending order such that
     element ‘i’ precedes element ‘j’ if ‘less(i,j)’ is true;

‘T[] copy(T[] a)’
     returns a deep copy of the array ‘a’;

‘T[] concat(...T[][] a)’
     returns a new array formed by concatenating the given
     one-dimensional arrays given as arguments;

‘bool alias(T[] a, T[] b)’
     returns ‘true’ if the arrays ‘a’ and ‘b’ are identical;

‘T[] sort(T[] a)’
     For built-in ordered types ‘T’, returns a copy of ‘a’ sorted in
     ascending order;

‘T[][] sort(T[][] a)’
     For built-in ordered types ‘T’, returns a copy of ‘a’ with the rows
     sorted by the first column, breaking ties with successively higher
     columns.  For example:
     string[][] a={{"bob","9"},{"alice","5"},{"pete","7"},
                   {"alice","4"}};
     // Row sort (by column 0, using column 1 to break ties):
     write(sort(a));

     produces
     alice   4
     alice   5
     bob     9
     pete    7

‘T[] sort(T[] a, bool less(T i, T j), bool stable=true)’
     returns a copy of ‘a’ sorted in ascending order such that element
     ‘i’ precedes element ‘j’ if ‘less(i,j)’ is true, subject to (if
     ‘stable’ is ‘true’) the stability constraint that the original
     order of elements ‘i’ and ‘j’ is preserved if ‘less(i,j)’ and
     ‘less(j,i)’ are both ‘false’;

‘T[][] transpose(T[][] a)’
     returns the transpose of ‘a’;

‘T[][][] transpose(T[][][] a, int[] perm)’
     returns the 3D transpose of ‘a’ obtained by applying the
     permutation ‘perm’ of ‘new int[]{0,1,2}’ to the indices of each
     entry;

‘T sum(T[] a)’
     for arithmetic types ‘T’, returns the sum of ‘a’.  In the case
     where ‘T’ is ‘bool’, the number of true elements in ‘a’ is
     returned;

‘T min(T[] a)’
‘T min(T[][] a)’
‘T min(T[][][] a)’
     for built-in ordered types ‘T’, returns the minimum element of ‘a’;

‘T max(T[] a)’
‘T max(T[][] a)’
‘T max(T[][][] a)’
     for built-in ordered types ‘T’, returns the maximum element of ‘a’;

‘T[] min(T[] a, T[] b)’
     for built-in ordered types ‘T’, and arrays ‘a’ and ‘b’ of the same
     length, returns an array composed of the minimum of the
     corresponding elements of ‘a’ and ‘b’;

‘T[] max(T[] a, T[] b)’
     for built-in ordered types ‘T’, and arrays ‘a’ and ‘b’ of the same
     length, returns an array composed of the maximum of the
     corresponding elements of ‘a’ and ‘b’;

‘pair[] pairs(real[] x, real[] y);’
     for arrays ‘x’ and ‘y’ of the same length, returns the pair array
     ‘sequence(new pair(int i) {return (x[i],y[i]);},x.length)’;

‘pair[] fft(pair[] a, int sign=1)’
     returns the unnormalized Fast Fourier Transform of ‘a’ (if the
     optional ‘FFTW’ package is installed), using the given ‘sign’.
     Here is a simple example:
     int n=4;
     pair[] f=sequence(n);
     write(f);
     pair[] g=fft(f,-1);
     write();
     write(g);
     f=fft(g,1);
     write();
     write(f/n);

‘pair[][] fft(pair[][] a, int sign=1)’
     returns the unnormalized two-dimensional Fourier transform of ‘a’
     using the given ‘sign’;

‘pair[][][] fft(pair[][][] a, int sign=1)’
     returns the unnormalized three-dimensional Fourier transform of ‘a’
     using the given ‘sign’;

‘realschur schur(real[][] a)’
     returns a structure ‘realschur’ containing a unitary matrix ‘U’ and
     a quasitriangular matrix ‘T’ such that ‘a=U*T*transpose(U)’;

‘schur schur(pair[][] a)’
     returns a structure ‘schur’ containing a unitary matrix ‘U’ and a
     triangular matrix ‘T’ such that ‘a=U*T*conj(transpose(U))’;

‘real dot(real[] a, real[] b)’
     returns the dot product of the vectors ‘a’ and ‘b’;

‘pair dot(pair[] a, pair[] b)’
     returns the complex dot product ‘sum(a*conj(b))’ of the vectors ‘a’
     and ‘b’;

‘real[] tridiagonal(real[] a, real[] b, real[] c, real[] f);’
     Solve the periodic tridiagonal problem L‘x’=‘f’ and return the
     solution ‘x’, where ‘f’ is an n vector and L is the n \times n
     matrix
     [ b[0] c[0]           a[0]   ]
     [ a[1] b[1] c[1]             ]
     [      a[2] b[2] c[2]        ]
     [                ...         ]
     [ c[n-1]       a[n-1] b[n-1] ]
     For Dirichlet boundary conditions (denoted here by ‘u[-1]’ and
     ‘u[n]’), replace ‘f[0]’ by ‘f[0]-a[0]u[-1]’ and
     ‘f[n-1]-c[n-1]u[n]’; then set ‘a[0]=c[n-1]=0’;

‘real[] solve(real[][] a, real[] b, bool warn=true)’
     Solve the linear equation ‘a’x=‘b’ by LU decomposition and return
     the solution x, where ‘a’ is an n \times n matrix and ‘b’ is an
     array of length n.  For example:
     import math;
     real[][] a={{1,-2,3,0},{4,-5,6,2},{-7,-8,10,5},{1,50,1,-2}};
     real[] b={7,19,33,3};
     real[] x=solve(a,b);
     write(a); write();
     write(b); write();
     write(x); write();
     write(a*x);
     If ‘a’ is a singular matrix and ‘warn’ is ‘false’, return an empty
     array.  If the matrix ‘a’ is tridiagonal, the routine ‘tridiagonal’
     provides a more efficient algorithm (*note tridiagonal::);

‘real[][] solve(real[][] a, real[][] b, bool warn=true)’
     Solve the linear equation ‘a’x=‘b’ and return the solution x, where
     ‘a’ is an n \times n matrix and ‘b’ is an n \times m matrix.  If
     ‘a’ is a singular matrix and ‘warn’ is ‘false’, return an empty
     matrix;

‘real[][] identity(int n);’
     returns the n \times n identity matrix;

‘real[][] diagonal(...real[] a)’
     returns the diagonal matrix with diagonal entries given by a;

‘real[][] inverse(real[][] a)’
     returns the inverse of a square matrix ‘a’;

‘real[] quadraticroots(real a, real b, real c);’
     This numerically robust solver returns the real roots of the
     quadratic equation ax^2+bx+c=0, in ascending order.  Multiple roots
     are listed separately;

‘pair[] quadraticroots(explicit pair a, explicit pair b, explicit pair c);’
     This numerically robust solver returns the complex roots of the
     quadratic equation ax^2+bx+c=0;

‘real[] cubicroots(real a, real b, real c, real d);’
     This numerically robust solver returns the real roots of the cubic
     equation ax^3+bx^2+cx+d=0.  Multiple roots are listed separately.

   ‘Asymptote’ includes a full set of vectorized array instructions for
arithmetic (including self) and logical operations.  These
element-by-element instructions are implemented in C++ code for speed.
Given
real[] a={1,2};
real[] b={3,2};
then ‘a == b’ and ‘a >= 2’ both evaluate to the vector ‘{false, true}’.
To test whether all components of ‘a’ and ‘b’ agree, use the boolean
function ‘all(a == b)’.  One can also use conditionals like ‘(a >= 2) ?
a : b’, which returns the array ‘{3,2}’, or ‘write((a >= 2) ? a : null’,
which returns the array ‘{2}’.

   All of the standard built-in ‘libm’ functions of signature
‘real(real)’ also take a real array as an argument, effectively like an
implicit call to ‘map’.

   As with other built-in types, arrays of the basic data types can be
read in by assignment.  In this example, the code
file fin=input("test.txt");
real[] A=fin;

reads real values into ‘A’ until the end-of-file is reached (or an I/O
error occurs).

   The virtual member functions ‘line()’, ‘word()’, ‘csv()’,
‘dimension()’, and ‘read()’ of a file are useful for qualifying array
reads, while the virtual fields ‘line’, ‘word’, ‘csv’, and ‘dimension’
may be used to query these respective states.

   If line mode is set with ‘file line(bool b=true)’, then reading will
stop once the end of the line is reached:
file fin=input("test.txt");
real[] A=fin.line();

   Since string reads by default read up to the end of line anyway, line
mode normally has no effect on string array reads.  However, there is a
white-space delimiter mode for reading strings, ‘file word(bool
b=true)’, which causes string reads to respect white-space delimiters,
instead of the default end-of-line delimiter:
file fin=input("test.txt").line().word();
real[] A=fin;

   Another useful mode is comma-separated-value mode, ‘file csv(bool
b=true)’, which causes reads to respect comma delimiters:
file fin=input("test.txt").csv();
real[] A=fin;

   To restrict the number of values read, use the ‘file dimension(int)’
virtual member function:
file fin=input("test.txt");
real[] A=fin.dimension(10);

   This reads 10 values into A, unless end-of-file (or end-of-line in
line mode) occurs first.  Attempting to read beyond the end of the file
will produce a runtime error message.  Specifying a value of 0 for the
integer limit is equivalent to the previous example of reading until
end-of-file (or end-of-line in line mode) is encountered.

   Two- and three-dimensional arrays of the basic data types can be read
in like this:
file fin=input("test.txt");
real[][] A=fin.dimension(2,3);
real[][][] B=fin.dimension(2,3,4);

   Sometimes the array dimensions are stored with the data as integer
fields at the beginning of an array.  Such 1, 2, or 3 dimensional arrays
can be read in with the virtual member functions ‘read(1)’, ‘read(2)’,
or ‘read(3)’, respectively:
file fin=input("test.txt");
real[] A=fin.read(1);
real[][] B=fin.read(2);
real[][][] C=fin.read(3);

   One, two, and three-dimensional arrays of the basic data types can be
output with the functions ‘write(file,T[])’, ‘write(file,T[][])’,
‘write(file,T[][][])’, respectively.


File: asymptote.info,  Node: Slices,  Up: Arrays

6.13.1 Slices
-------------

Asymptote allows a section of an array to be addressed as a slice using
a Python-like syntax.  If ‘A’ is an array, the expression ‘A[m:n]’
returns a new array consisting of the elements of ‘A’ with indices from
‘m’ up to but not including ‘n’.  For example,
int[] x={0,1,2,3,4,5,6,7,8,9};
int[] y=x[2:6];  // y={2,3,4,5};
int[] z=x[5:10]; // z={5,6,7,8,9};

   If the left index is omitted, it is taken be ‘0’.  If the right index
is omitted it is taken to be the length of the array.  If both are
omitted, the slice then goes from the start of the array to the end,
producing a non-cyclic deep copy of the array.  For example:
int[] x={0,1,2,3,4,5,6,7,8,9};
int[] y=x[:4];  // y={0,1,2,3}
int[] z=x[5:];  // z={5,6,7,8,9}
int[] w=x[:];   // w={0,1,2,3,4,5,6,7,8,9}, distinct from array x.

   If A is a non-cyclic array, it is illegal to use negative values for
either of the indices.  If the indices exceed the length of the array,
however, they are politely truncated to that length.

   For cyclic arrays, the slice ‘A[m:n]’ still consists of the cells
with indices in the set [‘m’,‘n’), but now negative values and values
beyond the length of the array are allowed.  The indices simply wrap
around.  For example:

int[] x={0,1,2,3,4,5,6,7,8,9};
x.cyclic=true;
int[] y=x[8:15];  // y={8,9,0,1,2,3,4}.
int[] z=x[-5:5];  // z={5,6,7,8,9,0,1,2,3,4}
int[] w=x[-3:17]; // w={7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6}

   Notice that with cyclic arrays, it is possible to include the same
element of the original array multiple times within a slice.  Regardless
of the original array, arrays produced by slices are always non-cyclic.

   If the left and right indices of a slice are the same, the result is
an empty array.  If the array being sliced is empty, the result is an
empty array.  Any slice with a left index greater than its right index
will yield an error.

   Slices can also be assigned to, changing the value of the original
array.  If the array being assigned to the slice has a different length
than the slice itself, elements will be inserted or removed from the
array to accommodate it.  For instance:
string[] toppings={"mayo", "salt", "ham", "lettuce"};
toppings[0:2]=new string[] {"mustard", "pepper"};
    // Now toppings={"mustard", "pepper", "ham", "lettuce"}
toppings[2:3]=new string[] {"turkey", "bacon" };
    // Now toppings={"mustard", "pepper", "turkey", "bacon", "lettuce"}
toppings[0:3]=new string[] {"tomato"};
    // Now toppings={"tomato", "bacon", "lettuce"}

   If an array is assigned to a slice of itself, a copy of the original
array is assigned to the slice.  That is, code such as ‘x[m:n]=x’ is
equivalent to ‘x[m:n]=copy(x)’.  One can use the shorthand ‘x[m:m]=y’ to
insert the contents of the array ‘y’ into the array ‘x’ starting at the
location just before ‘x[m]’.

   For a cyclic array, a slice is bridging if it addresses cells up to
the end of the array and then continues on to address cells at the start
of the array.  For instance, if ‘A’ is a cyclic array of length 10,
‘A[8:12]’, ‘A[-3:1]’, and ‘A[5:25]’ are bridging slices whereas
‘A[3:7]’, ‘A[7:10]’, ‘A[-3:0]’ and ‘A[103:107]’ are not.  Bridging
slices can only be assigned to if the number of elements in the slice is
exactly equal to the number of elements we are assigning to it.
Otherwise, there is no clear way to decide which of the new entries
should be ‘A[0]’ and an error is reported.  Non-bridging slices may be
assigned an array of any length.

   For a cyclic array ‘A’ an expression of the form
‘A[A.length:A.length]’ is equivalent to the expression ‘A[0:0]’ and so
assigning to this slice will insert values at the start of the array.
‘A.append()’ can be used to insert values at the end of the array.

   It is illegal to assign to a slice of a cyclic array that repeats any
of the cells.


File: asymptote.info,  Node: Casts,  Next: Import,  Prev: Arrays,  Up: Programming

6.14 Casts
==========

‘Asymptote’ implicitly casts ‘int’ to ‘real’, ‘int’ to ‘pair’, ‘real’ to
‘pair’, ‘pair’ to ‘path’, ‘pair’ to ‘guide’, ‘path’ to ‘guide’, ‘guide’
to ‘path’, ‘real’ to ‘pen’, ‘pair[]’ to ‘guide[]’, ‘pair[]’ to ‘path[]’,
‘path’ to ‘path[]’, and ‘guide’ to ‘path[]’, along with various
three-dimensional casts defined in module ‘three’.  Implicit casts are
automatically attempted on assignment and when trying to match function
calls with possible function signatures.  Implicit casting can be
inhibited by declaring individual arguments ‘explicit’ in the function
signature, say to avoid an ambiguous function call in the following
example, which outputs 0:
int f(pair a) {return 0;}
int f(explicit real x) {return 1;}

write(f(0));

   Other conversions, say ‘real’ to ‘int’ or ‘real’ to ‘string’, require
an explicit cast:
int i=(int) 2.5;
string s=(string) 2.5;

real[] a={2.5,-3.5};
int[] b=(int []) a;
write(stdout,b);     // Outputs 2,-3
   In situations where casting from a string to a type ‘T’ fails, an
uninitialized variable is returned; this condition can be detected with
the function ‘bool initialized(T);’
int i=(int) "2.5";
assert(initialized(i),"Invalid cast.");

real x=(real) "2.5a";
assert(initialized(x),"Invalid cast.");

   Casting to user-defined types is also possible using ‘operator cast’:
struct rpair {
  real radius;
  real angle;
}

pair operator cast(rpair x) {
  return (x.radius*cos(x.angle),x.radius*sin(x.angle));
}

rpair x;
x.radius=1;
x.angle=pi/6;

write(x);            // Outputs (0.866025403784439,0.5)

   One must use care when defining new cast operators.  Suppose that in
some code one wants all integers to represent multiples of 100.  To
convert them to reals, one would first want to multiply them by 100.
However, the straightforward implementation
real operator cast(int x) {return x*100;}
is equivalent to an infinite recursion, since the result ‘x*100’ needs
itself to be cast from an integer to a real.  Instead, we want to use
the standard conversion of int to real:
real convert(int x) {return x*100;}
real operator cast(int x)=convert;

   Explicit casts are implemented similarly, with ‘operator ecast’.


File: asymptote.info,  Node: Import,  Next: Iterators,  Prev: Casts,  Up: Programming

6.15 Import
===========

While ‘Asymptote’ provides many features by default, some applications
require specialized features contained in external ‘Asymptote’ modules.
For instance, the lines
access graph;
graph.axes();
draw x and y axes on a two-dimensional graph.  Here, the command looks
up the module under the name ‘graph’ in a global dictionary of modules
and puts it in a new variable named ‘graph’.  The module is a structure,
and we can refer to its fields as we usually would with a structure.

   Often, one wants to use module functions without having to specify
the module name.  The code
from graph access axes;
adds the ‘axes’ field of ‘graph’ into the local name space, so that
subsequently, one can just write ‘axes()’.  If the given name is
overloaded, all types and variables of that name are added.  To add more
than one name, just use a comma-separated list:
from graph access axes, xaxis, yaxis;
Wild card notation can be used to add all non-private fields and types
of a module to the local name space:

from graph access *;

   Similarly, one can add the non-private fields and types of a
structure to the local environment with the ‘unravel’ keyword:
struct matrix {
  real a,b,c,d;
}

real det(matrix m) {
  unravel m;
  return a*d-b*c;
}
   Alternatively, one can unravel selective fields:
real det(matrix m) {
  from m unravel a,b,c as C,d;
  return a*d-b*C;
}

   The command
import graph;
   is a convenient abbreviation for the commands
access graph;
unravel graph;
   That is, ‘import graph’ first loads a module into a structure called
‘graph’ and then adds its non-private fields and types to the local
environment.  This way, if a member variable (or function) is
overwritten with a local variable (or function of the same signature),
the original one can still be accessed by qualifying it with the module
name.

   Wild card importing will work fine in most cases, but one does not
usually know all of the internal types and variables of a module, which
can also change as the module writer adds or changes features of the
module.  As such, it is prudent to add ‘import’ commands at the start of
an ‘Asymptote’ file, so that imported names won't shadow locally defined
functions.  Still, imported names may shadow other imported names,
depending on the order in which they were imported, and imported
functions may cause overloading resolution problems if they have the
same name as local functions defined later.

   To rename modules or fields when adding them to the local
environment, use ‘as’:
access graph as graph2d;
from graph access xaxis as xline, yaxis as yline;

   The command
import graph as graph2d;
   is a convenient abbreviation for the commands
access graph as graph2d;
unravel graph2d;

   Except for a few built-in modules, such as ‘settings’, all modules
are implemented as ‘Asymptote’ files.  When looking up a module that has
not yet been loaded, ‘Asymptote’ searches the standard search paths
(*note Search paths::) for the matching file.  The file corresponding to
that name is read and the code within it is interpreted as the body of a
structure defining the module.

   If the file is contained in a subdirectory, the directory structure
can be specified with a dot-separated name.  For instance, the command
import directoryName.moduleName;
access directoryName.moduleName;
will load the file ‘directoryName/moduleName.asy’ as module
‘moduleName’.  Likewise,
from directoryName.moduleName access functionName;
   will access ‘functionName’ from ‘directoryName/moduleName.asy’.  The
directory ‘directoryName’ is relative to the current directory, the
‘base’ directory, or any directory in the search path (*note Search
paths::).  Nested directory specifications are allowed, as long as the
directory and module names are also valid as variable names.  Otherwise
(for example if the directory is an absolute path), the filename must be
enclosed with quotation marks:

‘import '/usr/local/share/asymptote/graph.asy' as graph;’

   If ‘Asymptote’ is compiled with support for ‘libcurl’, the file name
can even be a URL: ‘import
'https://raw.githubusercontent.com/vectorgraphics/asymptote/HEAD/doc/axis3.asy'
as axis3;’

   It is an error if modules import themselves (or each other in a
cycle).  The module name to be imported must be known at compile time.

   However, you can import an ‘Asymptote’ module determined by the
string ‘s’ at runtime like this:
eval('import '+s,true);

   To conditionally execute an array of asy files, use
void asy(string format, bool overwrite, ... string[] s);
   The file will only be processed, using output format ‘format’, if
overwrite is ‘true’ or the output file is missing.

   One can evaluate an ‘Asymptote’ expression (without any return value,
however) contained in the string ‘s’ with:
void eval(string s, bool embedded=false);
   It is not necessary to terminate the string ‘s’ with a semicolon.  If
‘embedded’ is ‘true’, the string will be evaluated at the top level of
the current environment.  If ‘embedded’ is ‘false’ (the default), the
string will be evaluated in an independent environment, sharing the same
‘settings’ module (*note settings::).

   One can evaluate arbitrary ‘Asymptote’ code (which may contain
unescaped quotation marks) with the command
void eval(code s, bool embedded=false);
   Here ‘code’ is a special type used with ‘quote {}’ to enclose
‘Asymptote code’ like this:
real a=1;
code s=quote {
  write(a);
};
eval(s,true);        // Outputs 1

   To include the contents of an existing file ‘graph’ verbatim (as if
the contents of the file were inserted at that point), use one of the
forms:
include graph;

‘include '/usr/local/share/asymptote/graph.asy';’

   To list all global functions and variables defined in a module named
by the contents of the string ‘s’, use the function
void list(string s, bool imports=false);
Imported global functions and variables are also listed if ‘imports’ is
‘true’.

* Menu:

* Templated imports::           Imports parametrized by generic types


File: asymptote.info,  Node: Templated imports,  Up: Import

6.15.1 Templated imports
------------------------

In Asymptote types are specified when they are imported.  The first
executable line of any such module must be of the form ‘typedef
import(<types>)’, where ‘<types>’ is a list of required type parameters.
For instance,
typedef import(T, S, Number);
could be the first line of a module that requires three type parameters.
The remaining code in the module can then use ‘T’, ‘S’, and ‘Number’ as
types.

   To import such a module, one must specify the types to be used.  For
instance, if the module above were named ‘templatedModule’, it could be
accessed for types ‘string’, ‘int[]’, and ‘real’ with the import command
access templatedModule(T=string, S=int[], Number=real)
    as templatedModule_string_int_real;
Note that this is actually an _access_ command rather than an _import_
command, so a type, function, or variable ‘A’ defined in
‘templatedModule.asy’ would need to be accessed qualified as
‘templatedModule_string_int_real.A’.

   Alternatively, the module could be imported via a command like
from templatedModule(T=string, S=int[], Number=real) access
    Wrapper_Number as Wrapper_real,
    operator ==;
This command would automatically rename ‘Wrapper_Number’ to
‘Wrapper_real’ and would also allow the use of any ‘operator ==’
overloads defined in the module.

   Further examples can be found in the ‘tests/template’ subdirectory of
the ‘Asymptote’ source directory.

   Issues: Certain expected operators (such as ‘operator ==’) may only
be available for type arguments that are builtin or defined in module
‘plain’.


File: asymptote.info,  Node: Iterators,  Next: Static,  Prev: Import,  Up: Programming

6.16 Iterators
==============

If a structure ‘a’ has a member function ‘operator iter()’, the code
for(var i : a) {
  <statements>
}
is syntactic sugar for
for(var it=a.operator iter(); it.valid(); it.advance()) {
  var i=it.get();
  <statements>
}
   Thus, we can make a structure iterable by defining an ‘operator iter’
method that returns an object with the following methods:
‘T get()’
     returns the value at the iterator's current position (without
     changing the position).  Note that ‘T’ can be any type, builtin or
     user-defined;

‘void advance()’
     advances the iterator to the next position;

‘bool valid()’
     returns ‘true’ if the iterator is at a valid position or ‘false’ if
     the iterator has advanced past the last item.

   Although the return type of ‘operator iter’ can theoretically be any
type that has these three methods, it is strongly recommended to use the
‘Iter_T’ structure defined in the templated module ‘collections.iter(T)’
so that other utilities (*note Iterators and utilities::) can be used
with the iterator.  The three methods of this structure can be set as
fields.

   As an example, here is an iterator for the even-numbered elements of
an array of strings:
from collections.iter(T=string) access
    Iter_T as Iter_string,
    Iterable_T as Iterable_string;
struct EvenStrings {
  string[] a;
  void operator init(string[] a) {
    this.a=a;
  }
  Iter_string operator iter() {
    int i = 0;
    Iter_string result;
    result.get = new string() { return a[i]; };
    result.advance = new void() { i += 2; };
    result.valid = new bool() { return i < a.length; };
    return result;
  }
  autounravel Iterable_string operator cast(EvenStrings es) {
    Iterable_string result;
    result.operator iter = es.operator iter;
    return result;
  }
}
string[] a = {'a', 'b', 'c', 'd', 'e', 'f'};
for (string s : EvenStrings(a)) {
  write(s);
}
   Running this code prints out the following:
a
c
e

The autounraveled (*note Autounravel::) implicit cast to
‘Iterable_string’ is not strictly necessary, but it allows the use of
iterable utilities; *note Iterators and utilities::.  This enables code
like the following, which uses the ‘enumerate’ utility to attach
counters to the iterated strings:
from collections.enumerate(T=string) access enumerate;
for (var kv : enumerate(EvenStrings(a))) {
  write(string(kv.k) + ": " + kv.v);
}
   Running this code prints out the following:
0: a
1: c
2: e

* Menu:

* range::                       Iterating over integers


File: asymptote.info,  Node: range,  Up: Iterators

6.16.1 range
------------

The following functions produce iterators over integers:
‘Iterable_int range(int n)’
     returns an iterable that produces the integers ‘0,...,n-1’;
‘Iterable_int range(int n, int m, int skip=1)’
     returns an iterable that produces the integers ‘n, n+skip, ..., m’
     if ‘(m-n)/skip >= 0’, and otherwise produces no integers.

   Generally speaking, ‘for (int i : range(n))’ is equivalent to ‘for
(int i : sequence(n))’.  The latter uses O(‘n’) memory but is usually
faster.  The former uses O(1) memory but is slower.  If an iterator is
specifically required, however, ‘range(n)’ is faster than
‘range(sequence(n))’ (as well as being more memory-efficient).


File: asymptote.info,  Node: Static,  Next: Autounravel,  Prev: Iterators,  Up: Programming

6.17 Static
===========

Static qualifiers allocate the memory address of a variable in a higher
enclosing level.

   For a function body, the variable is allocated in the block where the
function is defined; so in the code
struct s {
  int count() {
    static int c=0;
    ++c;
    return c;
  }
}

there is one instance of the variable ‘c’ for each object ‘s’ (as
opposed to each call of ‘count’).

   Similarly, in
int factorial(int n) {
  int helper(int k) {
    static int x=1;
    x *= k;
    return k == 1 ? x : helper(k-1);
  }
  return helper(n);
}

there is one instance of ‘x’ for every call to ‘factorial’ (and not for
every call to ‘helper’), so this is a correct, but ugly, implementation
of factorial.

   Similarly, a static variable declared within a structure is allocated
in the block where the structure is defined.  Thus,
struct A {
  struct B {
    static pair z;
  }
}

creates one object ‘z’ for each object of type ‘A’ created.

   In this example,
int pow(int n, int k) {
  struct A {
    static int x=1;
    void helper() {
      x *= n;
    }
  }
  for(int i=0; i < k; ++i) {
    A a;
    a.helper();
  }
  return A.x;
}

there is one instance of ‘x’ for each call to ‘pow’, so this is an ugly
implementation of exponentiation.

   Loop constructs allocate a new frame in every iteration.  This is so
that higher-order functions can refer to variables of a specific
iteration of a loop:
void f();
for(int i=0; i < 10; ++i) {
  int x=i;
  if(x==5) {
    f=new void() {write(x);};
  }
}
f();

   Here, every iteration of the loop has its own variable ‘x’, so ‘f()’
will write ‘5’.  If a variable in a loop is declared static, it will be
allocated where the enclosing function or structure was defined (just as
if it were declared static outside of the loop).  For instance, in:
void f() {
  static int x;
  for(int i=0; i < 10; ++i) {
    static int y;
  }
}
both ‘x’ and ‘y’ will be allocated in the same place, which is also
where ‘f’ is allocated.

   Statements may also be declared static, in which case they are run at
the place where the enclosing function or structure is defined.
Declarations or statements not enclosed in a function or structure
definition are already at the top level, so static modifiers are
meaningless.  A warning is given in such a case.

   Since structures can have static fields, it is not always clear for a
qualified name whether the qualifier is a variable or a type.  For
instance, in:

struct A {
  static int x;
}
pair A;

int y=A.x;
does the ‘A’ in ‘A.x’ refer to the structure or to the pair variable.
It is the convention in Asymptote that, if there is a non-function
variable with the same name as the qualifier, the qualifier refers to
that variable, and not to the type.  This is regardless of what fields
the variable actually possesses.


File: asymptote.info,  Node: Autounravel,  Next: Hash functions,  Prev: Static,  Up: Programming

6.18 Autounravel
================

The ‘autounravel’ modifier can be used to automatically unravel a field.
This is useful when building an associated library of functions that
operate on a structure.  For instance, consider a simple implementation
of the ‘rational’ structure defined in ‘rational.asy’:

struct rational {
  int p=0, q=1;
  void operator init(int p, int q) {
    this.p=p;
    this.q=q;
  }
}
rational operator +(rational a, rational b) {
  return rational(a.p*b.q+b.p*a.q, a.q*b.q);
}

   To allow ‘rational’ to be used as a type parameter for a templated
import that adds instances of ‘rational’, we should move ‘operator +’
into the body of the ‘rational’ structure and add the ‘autounravel’
modifier:

struct rational {
  int p=0, q=1;
  void operator init(int p, int q) {
    this.p=p;
    this.q=q;
  }
  autounravel rational operator +(rational a, rational b) {
    return rational(a.p*b.q+b.p*a.q, a.q*b.q);
  }
}

   This is almost equivalent to the previous code, but now the ‘+’
operator will be accessible wherever the ‘rational’ structure is.

   *Currently, types cannot be autounraveled.*  Users who encounter a
case where this might be useful can submit a feature request at
<https://github.com/vectorgraphics/asymptote/issues>.

* Menu:

* Where fields are autounraveled::  Where autounraveling happens
* Where autounravel is legal::  Valid autounravel locations


File: asymptote.info,  Node: Where fields are autounraveled,  Next: Where autounravel is legal,  Up: Autounravel

6.18.1 Where fields are autounraveled
-------------------------------------

If a ‘struct’ contains fields (including functions) that are declared
with ‘autounravel’, these fields will be unraveled from the ‘struct’ at:

   • the end of the ‘struct’ definition;
   • a ‘typedef import’ statement, if the ‘struct’ is the argument for
     one of the type parameters;
   • an ‘unravel’ or ‘access’ statement that unravels the ‘struct’ from
     a module or outer ‘struct’ (for instance, ‘from rational access
     rational;’ would make the ‘+’ operator available from the ‘struct’
     ‘rational’ defined in ‘rational.asy’);

   • A ‘typedef’ statement like ‘typedef rational.rational rat;’ that
     renames a ‘struct’.


File: asymptote.info,  Node: Where autounravel is legal,  Prev: Where fields are autounraveled,  Up: Autounravel

6.18.2 Where ‘autounravel’ is legal
-----------------------------------

The ‘autounravel’ modifier implies ‘static’ and can be used in many of
the same places as ‘static’.  However, specifying ‘autounravel’ at the
top level of a module (outside of any structure or function) is an
error, whereas ‘static’ gives only a warning.  (1)  In front of a
‘struct’ definition or ‘typedef’ statement, ‘autounravel’ is forbidden
because types cannot be autounraveled.  While ‘static static’ results in
an error, ‘static autounravel’ and ‘autounravel static’ are both legal
and have exactly the same effect as ‘autounravel’ alone.

   ---------- Footnotes ----------

   (1) If top-level ‘autounravel’ were allowed, a user might incorrectly
assume that the field would be unraveled whenever the module is
‘access’ed.  The ‘static’ modifier is allowed at the top level because,
while it does nothing, it does not mislead the user.


File: asymptote.info,  Node: Hash functions,  Next: Collections (containers),  Prev: Autounravel,  Up: Programming

6.19 Hash functions
===================

For an object to be used as a key in a hashmap (*note Maps::) or a
member of a hashset (*note Sets::), it must have a method ‘int hash()’
that returns a nonnegative integer.  For correctness, this method must
return the same value for two objects of the same type that are equal
according to the ‘==’ operator.  For efficiency, it should behave like a
pseudorandom function, so that the hash values of distinct objects are
unrelated.  Also note that it is not necessary for hash functions to be
consistent across different runs of the program; ‘a.hash() == a.hash()’
should always be true, but ‘write(a.hash())’ may produce a different
output each time the program is run.

   In ‘Asymptote’, the ‘hash’ method is defined for three of the builtin
types: ‘int’, ‘real’, and ‘string’.  Note that things like ‘3.hash()’
and ‘'hello'.hash()’ are _not_ valid syntax; the correct version of
these would be ‘(3).hash()’ and ‘('hello').hash()’, with parentheses
around the literal.  If you want to hash a ‘real’, there are two
important things to note: first, even if two ‘real’s are extremely
close, their hash values will almost certainly not be; second, the same
number will almost certainly have different hash values when hashed as
an ‘int’ versus a ‘real’.

* Menu:

* Hashing user-defined structures::  Implementing hash methods for custom types


File: asymptote.info,  Node: Hashing user-defined structures,  Up: Hash functions

6.19.1 Hashing user-defined structures
--------------------------------------

Additionally, there is a function ‘hash(int[])’ that can be helpful when
constructing hash methods for user-defined structures: convert all the
fields to integers (possibly by hashing them), put these integers into
an array, and hash the array.

   Let's return to the example of the ‘Person’ structure from *note
Structures::, adding an ‘age’ field to make it more interesting.  For
this structure, there are two reasonable ways we might define a hash
function for it.  The first is to create a hash function based on all
the fields of the structure:
struct Person {
  string firstname;
  string lastname;
  int age;
  int hash() {
    return hash(new int[] {firstname.hash(), lastname.hash(), age});
  }
  autounravel bool operator ==(Person a, Person b) {
    if (alias(a, null)) return alias(b, null);
    if (alias(b, null)) return false;
    return a.firstname == b.firstname &&
           a.lastname == b.lastname &&
           a.age == b.age;
  }
  autounravel bool operator !=(Person a, Person b) {
    return !(a == b);
  }
}
   This approach is necessary if we are redefining the ‘==’ operator.
Two ‘Person’s are equal if they have the same first name, last name, and
age.

   The default ‘==’ operator for user-defined structs considers two
objects equal if they are the same object, meaning that changes to one
will be reflected in the other.  A hash method that plays well with the
default ‘==’ operator must remain the same even if the fields change.
In this case, we can add a never-changing ‘id’ field that uniquely
identifies a ‘Person’ object, and use that for hashing:

struct Person {
  string firstname;
  string lastname;
  int age;
  private static int lastID=-1;
  restricted int id = ++lastID;
  int hash() {
    return id.hash();
  }
}

   The builtin hash functions are designed to remove undesirable
regularity (with respect to modular arithmetic) when applied exactly
once.  This explains why the second example returns ‘id.hash()’ instead
of ‘id’, while ‘age’ is not independently hashed in the first example.
It may sometimes be most practical to feed the hash function's output
back into it, as in ‘firstname’ and ‘lastname’ above, but this does not
increase the hash quality; in theory, if we could serialize a ‘Person’
object to a string and then hash the string, the hash quality would be
even higher.


File: asymptote.info,  Node: Collections (containers),  Prev: Hash functions,  Up: Programming

6.20 Collections (containers)
=============================

The ‘collections’ subdirectory provides a number of useful containers
and utilities via templated modules.

   *Note Maps:: for information on hashmaps (sometimes called
dictionaries or associative arrays), which are arguably the single most
useful container in the ‘collections’ module.

* Menu:

* Pairs::                       Generic pair of two types with hash support
* Iterators and utilities::     Iterable structures and manipulation tools
* Maps::                        Hashmaps and b-tree maps (dictionaries/associative arrays)
* Using Maps as Sets::          ‘Map_K_bool’ for set operations
* Sets::                        Hash sets and sorted b-tree sets
* Using Sets as Maps::          ‘Set_Pair_K_V’ to map keys to values
* Queues::                      First-in-first-out (FIFO) queue
* Wrapping arrays::             Arrays that can be used as keys or values


File: asymptote.info,  Node: Pairs,  Next: Iterators and utilities,  Up: Collections (containers)

6.20.1 Pairs
------------

‘Asymptote’ has a builtin type for pairs of ‘real’s.  To obtain a pair
type for _any_ two types, one can use the ‘collections.genericpair(K,V)’
templated module.  The fields are called ‘k’ and ‘v’ rather than ‘x’ and
‘y’.  The pair type is called ‘Pair_K_V’, which users are encouraged to
rename based on the types being used.  There is also a function
‘makePair’ that creates a pair from two values; because of function
overloading, the name of the function does not need to specify the two
types.

   Here's an example:
from collections.genericpair(K=int, V=string) access
    Pair_K_V as Pair_int_string,
    makePair;
var p=makePair(3, 'hello');
write(p.k);  // Outputs 3
write(p.v);  // Outputs hello
var q = makePair(4, 'hello');
var s = makePair(3, 'world');
Pair_int_string t = makePair(3, 'hello');
assert(p != q);
assert(p != s);
assert(p == t);
   Note that the ‘==’ and ‘!=’ operators are overloaded for pairs, and
the overloads are based on the ‘==’ operators of the two types in the
pair.  An unfortunate consequence of this is that this module cannot be
used for pairs of arrays, because the ‘==’ operator for arrays returns a
‘bool[]’ rather than a ‘bool’.  *Note Wrapping arrays:: for a
workaround.

   The ‘Pair_K_V’ type also has an ‘int hash()’ method initialized to
‘null’.  If you want to use pairs as keys in a hashmap, you should
define this method, something like this:
from collections.genericpair(K=int, V=string) access
    Pair_K_V as Pair_int_string;
Pair_int_string p = makePair(3, 'hello');
p.hash = new int() {
  return hash(new int[] {p.k, p.v.hash()});
};
   *Note Hash functions:: for more information.


File: asymptote.info,  Node: Iterators and utilities,  Next: Maps,  Prev: Pairs,  Up: Collections (containers)

6.20.2 Iterators and utilities
------------------------------

* Menu:

* collections.iter(T)::  Iterable structures
* Combining (zipping) iterators::  Iterating over multiple containers simultaneously
* Attaching a counter to an iterator::  Enumerating elements with indices


File: asymptote.info,  Node: collections.iter(T),  Next: Combining (zipping) iterators,  Up: Iterators and utilities

6.20.2.1 ‘collections.iter(T)’
..............................

The ‘collections.iter(T)’ templated module provides structs ‘Iter_T’ and
‘Iterable_T’ that can be convenient for defining iterators.
Additionally, there are several utilities for manipulating iterators of
these types.  Technically, the syntax ‘for(T x : a)’ does not require
anything from this module, but several utilities only work for iterables
of type ‘Iterable_T’.

   The ‘Iter_T’ structure has the following fields, all of which are
null by default:

   • ‘T get()’
   • ‘void advance()’
   • ‘bool valid()’

   ‘Iter_T’ is intended as the return type of an ‘operator iter’ method;
*Note Iterators:: for more information on how to use these fields.

   ‘Iterable_T’ is a generic structure for something that has an
‘operator iter’ method returning an ‘Iter_T’.  This structure, not
‘Iter_T’, is the type that can appear inside a range-based ‘for’ loop.
Consequently, most of the utilities below manipulate instances of
‘Iterable_T’ rather than ‘Iter_T’.

   It is good practice for a structure that defines an ‘operator iter’
method to also define an implicit cast to ‘Iterable_T’ for better
compatibility with the utilities in this module.  Additionally, an
instance ‘a’ of ‘Iterable_T’ can be cast to a ‘T[]’ with the syntax
‘(T[]) a’.  (This is an explicit cast, so it requires parentheses.)
Going the other way, a ‘T[]’ can be implicitly cast to an ‘Iterable_T’
(no parentheses required) as long as the appropriate ‘Iterable_T’ type
has been ‘access’ed (*note Autounravel::).  This can be done explicitly
using the ‘range’ function listed below.

   The ‘collections.iter’ module provides the following functions:

‘Iterable_T Iterable(Iter_T iter())’
     creates an ‘Iterable_T’ with its ‘operator iter’ method set to
     ‘iter’ (autounraveled whenever ‘Iterable_T’ is accessed).  This
     function is an alias for the constructor ‘Iterable_T(Iter_T
     iter())’.
‘Iterable_T range(T[] items)’
     creates an ‘Iterable_T’ that iterates over the elements of ‘items’.
     Separate calls to ‘operator iter’ return independent iterators.
‘Iter_T Iter_T(T[] items)’
     creates an ‘Iter_T’ that iterates over the elements of ‘items’.


File: asymptote.info,  Node: Combining (zipping) iterators,  Next: Attaching a counter to an iterator,  Prev: collections.iter(T),  Up: Iterators and utilities

6.20.2.2 Combining (zipping) iterators
......................................

The ‘collections.zip2(K,V)’ templated module provides a way of iterating
over two containers of potentially different types in a single ‘for’
loop, for example,
from collections.zip2(K=int, V=string) access zip, operator cast;
int[] x = {0, 1, 2, 3};
string[] y = {'0', '1', '2', '3'};
for (var ab : zip(x, y)) {
  assert(string(ab.k) == ab.v);
}
where
Iterable_Pair_K_V zip(Iterable_K a, Iterable_V b,
                      Pair_K_V keyword default=null)
returns an iterable that produces pairs of elements from ‘a’ and ‘b’.
If ‘default’ is not provided, the iterable stops when either ‘a’ or ‘b’
runs out of elements.  If ‘default’ is provided, the returned iterable
will be the length of the longer of ‘a’ and ‘b’, and the default value
will be used for any missing elements.

   Additionally, the module exposes the following types and all their
autounravels:

type               from
--------------------------------------------------------------------------
‘Pair_K_V’         ‘collections.genericpair(K,V)’
‘Iterable_K’       ‘collections.iter(T=K)’
‘Iterable_V’       ‘collections.iter(T=V)’
‘Iterable_Pair_K_V’‘collections.iter(T=Pair_K_V)’
‘Iter_K’           ‘collections.iter(T=K)’
‘Iter_V’           ‘collections.iter(T=V)’
‘Iter_Pair_K_V’    ‘collections.iter(T=Pair_K_V)’

   The ‘collections.zip(T)’ templated module provides a way to iterate
over multiple containers in a single ‘for’ loop.  It is similar to
‘collections.zip2(K,V)’, but condenses the iterables into an iterable
over arrays rather than over pairs.  This requires that all the
iterables be over the same type (for example, all string iterators or
all integer iterators):
from collections.zip(T=int) access zip;
int[] x = {0, 1, 2, 3};
int[] y = {0, 1, 2};
int[] z = {0, 1, 2, 3, 4};
for (int[] abc : zip(default=-1, x, y, z)) {
  write(string(abc[0]) + '\t' + string(abc[1]) + '\t' + string(abc[2]));
}
which outputs
0       0       0
1       1       1
2       2       2
3       -1      3
-1      -1      4

   The ‘collections.zip(T)’ module exposes the types
type               from
--------------------------------------------------------------------------
‘Iterable_array_T’ ‘collections.iter(T=T[])’
‘Iter_array_T’     ‘collections.iter(T=T[])’
‘Iterable_T’       ‘collections.iter(T)’
‘Iter_T’           ‘collections.iter(T)’
and defines the following functions (all called ‘zip’):
‘Iterable_array_T zip(...Iterable_T[] iterables)’
     zips together one or more iterables of ‘T’s.  If ‘a’ is an item
     from the result, then ‘a[0]’ is from the first iterable, ‘a[1]’ is
     from the second iterable, and so on.  If the iterables are not all
     the same length, the result will be the length of the shortest
     iterable.
‘Iterable_array_T zip(T keyword default ...Iterable_T[] iterables)’
     zips together one or more iterables of ‘T’s.  The result will be
     the length of the longest iterable, with the default value filling
     in any missing elements.
‘T[][] zip(...T[][] arrays)’
     zips together one or more arrays.  If the arrays are not all the
     same length, the result will be the length of the shortest array.
‘T[][] zip(T keyword default ...T[][] arrays)’
     zips together one or more arrays.  The result will be the length of
     the longest array, with the default value filling in any missing
     elements.  This is basically equivalent to applying ‘transpose’ and
     then replacing any uninitialized elements with the default value.


File: asymptote.info,  Node: Attaching a counter to an iterator,  Prev: Combining (zipping) iterators,  Up: Iterators and utilities

6.20.2.3 Attaching a counter to an iterator
...........................................

The ‘collections.enumerate(T)’ module provides a way to attach a counter
to an iterator.  There are two functions, both called ‘enumerate’:
   • ‘Iterable_Pair_int_T enumerate(Iterable_T iterable)’
   • ‘Iterable_Pair_int_T enumerate(T[] array)’
   Here's a simple example:
from collections.enumerate(T=int) access enumerate;
int[] x = {1, 4, 9, 16};
for (var ab : enumerate(x))
  write('Element ' + string(ab.k) + ' is ' + string(ab.v));
which outputs
Element 0 is 1
Element 1 is 4
Element 2 is 9
Element 3 is 16
   Note that ‘enumerate’ is not necessarily worthwhile for arrays since
they are already indexed by integers:
int[] x = {1, 4, 9, 16};
for (int i=0; i < x.length; ++i)
  write('Element ' + string(i) + ' is ' + string(x[i]));
The ‘enumerate’ utility is more useful for structures that do not have a
builtin integer index (such as ‘Map’ and ‘Set’).


File: asymptote.info,  Node: Maps,  Next: Using Maps as Sets,  Prev: Iterators and utilities,  Up: Collections (containers)

6.20.3 Maps
-----------

Many modern programming languages have a builtin datastructure that is
variously called a map, dictionary, or associative array.  This
structure is based on the mathematical concept of a function, but
differs from the programming functions discussed previously (*note
Functions::):

   • A programming function computes an output value for each valid
     input.  Unlike a mathematical function, it can have side effects.
   • A ‘Map’ stores a set of input-output pairs.  It has a finite domain
     consisting of those inputs whose outputs were explicitly added.
     Typically, the inputs to a ‘Map’ are called keys, and the outputs
     are called values.

This example of a ‘Map’ assigns integer values to string keys:
from collections.hashmap(K=string, V=int) access
    HashMap_K_V as HashMap_string_int;

// Create a Map called h.
HashMap_string_int h;

// Add some key-value pairs to h.
h['hello'] = 5;
h['world'] = 3;

// Iterate over the keys (the domain) of h.
for (string key : h) {

  // Compute the output for the given input.
  int value = h[key];
  write(key + ' ' + string(value));
}
producing the output
hello 5
world 3

* Menu:

* Initializing a Map::   Creating hashmaps and b-tree maps
* The Map API::          Methods for map operations


File: asymptote.info,  Node: Initializing a Map,  Next: The Map API,  Up: Maps

6.20.3.1 Initializing a ‘Map’
.............................

The ‘Map_K_V’ structure itself is defined in the templated module
‘collections.map(K,V)’.  To initialize a ‘Map_K_V’, you need to use a
function from ‘collections.hashmap’ or ‘collections.btreemap’:

‘HashMap_K_V()’
‘HashMap_K_V(V keyword nullValue, bool keyword isNullValue(V v) =
                                new bool(V v) {return v == nullValue; })’
     These two functions, which create a hash map, must be accessed from
     the ‘collections.hashmap(K,V)’ templated module.  The key type must
     have a ‘hash’ method, *note Hash functions::.
‘BTreeMap_K_V()’
‘BTreeMap_K_V(V keyword nullValue, bool keyword isNullValue(V v) =
                                 new bool(V v) {return v == nullValue; })’
     These two functions must be accessed from the
     ‘collections.btreemap(K,V)’ templated module.  They create a b-tree
     map, which stores its keys in sorted order.  The key type must have
     a ‘<’ operator.

*Note:* ‘HashMap_K_V’ is an appropriate choice for most purposes.

   If ‘nullValue’ is not provided, looking up a missing key will throw
an error.  The ‘nullValue’ parameter gives the map something to return
when a nonexistent key is looked up.  This is useful in scenarios where
missing keys are expected and should be handled gracefully.  The
‘isNullValue’ parameter defaults to a function that checks whether its
argument is equal to ‘nullValue’ via ‘operator ==’.  This is usually the
desired behavior, but it is occasionally necessary to override it; for
instance, if you have overridden ‘operator ==’ assuming its parameters
are not null, then something like ‘isNullValue=new bool(V v){return
alias(v,null);}’ might be appropriate.

   Both ‘HashMap_K_V’ and ‘BTreeMap_K_V’ can be implicitly cast to
‘Map_K_V’.  Even without being cast, the two types have the same methods
as ‘Map_K_V’.  Similarly, the ‘nullValue’ and ‘isNullValue’ optional
parameters behave the same way for both types.  *Note The Map API::.

   In theory, adding, removing, or looking up an element should require,
on average, O(1) time for ‘HashMap_K_V’ and O(\log n) time for
‘BTreeMap_K_V’ (where n represents the number of elements in the
container).  In practice, ‘HashMap_K_V’ is faster than ‘BTreeMap_K_V’ by
about a factor of 2, assuming no more than a few million entries.  (This
is a very rough estimate that depends on the key type ‘K’ having cheap
hash, equality, and comparison functions.)  The relative performance of
these map implementations could change in the future as the code is
optimized.


File: asymptote.info,  Node: The Map API,  Prev: Initializing a Map,  Up: Maps

6.20.3.2 The ‘Map’ API
......................

The ‘Map_K_V’ structure has the following methods.  Note that its
"subclasses" ‘HashMap_K_V’ and ‘BTreeMap_K_V’ each have the same fields.

   Depending on which constructor is used, a ‘Map’ may or may not have
valid instances of ‘nullValue’ and ‘isNullValue(V)’.  If one of these is
valid, the other will be too.  In some methods below, an error will be
thrown unless ‘nullValue’ can be returned to indicate an element is not
present.  When ‘nullValue’ is defined for a map ‘m’, the expression
‘m.isNullValue(m[k])’ is equivalent to ‘m.contains(k)’.

‘int size()’
     returns the number of distinct keys in the map.

‘bool empty()’
     returns true iff ‘size() == 0’.

‘bool contains(K key)’
     returns true if ‘key’ is present as a key in the map.

‘V operator [] (K key)’
     returns the value associated with the element ‘key’.  If ‘key’ is
     not present, this method either returns ‘nullValue’ or, if
     ‘nullValue’ does not exist, throws an error.  *Note Bracket
     operators::.

‘void operator [=] (K key, V value)’
     adds the pair ‘(key, value)’ to the map.  If a key equivalent to
     ‘key’ is already present, its associated value is replaced (as is
     the key itself, but that should rarely matter since the old and new
     keys are equivalent).  However, if ‘isNullValue(value)’ is true,
     then the assignment instead removes ‘key’ from the map if it was
     already there.  *Note Bracket operators::.

‘void delete(K key)’
     removes ‘key’ from the map.  If ‘key’ was not already in the map,
     an error is thrown.  (An alternative that will not throw an error
     is to set the value of ‘key’ to ‘nullValue’.)

‘Iter_K operator iter()’
     returns an iterator over the keys of the map.  It allows the map to
     be used in range-based ‘for’ loops, as in
     for (K key : m) {<statements>}
     If a key is deleted or a new key is inserted, then any iterators
     created before that change are undermined and may behave
     unpredictably.  The current implementations attempt to throw errors
     if an iterator is used after being undermined, but this behavior is
     not guaranteed and might be removed in future versions for
     performance reasons.

‘Iterable_K_V pairs()’
     returns an iterable over the key-value pairs.  For example,
     for (Pair_K_V p : m.pairs()) {
       K key = p.k;
       V value = p.v;
       <statements>
     }

‘K[] keys()’
     returns an array of all the keys.  The array is a shallow copy;
     changes to it will not affect the map (and vice versa), but changes
     to the entries of the keys will undermine the integrity of the map.

‘void add(Iterable_K_V other)’
     adds the key-value pairs from ‘other’.  Running ‘m.add(o);’ should
     be equivalent to running
     for (var pr : o) { m[pr.k] = pr.v; }

   In addition to these methods, there is also an autounraveled implicit
cast from ‘Map_K_V’ to ‘Iterable_K’.  This cast allows a ‘Map_K_V’ to be
passed as a parameter to utilities like ‘zip’ or ‘enumerate’.


File: asymptote.info,  Node: Using Maps as Sets,  Next: Sets,  Prev: Maps,  Up: Collections (containers)

6.20.4 Using Maps as Sets
-------------------------

Sometimes, you want a collection of objects of type ‘K’ that lets you
quickly add items, delete items, and determine whether an item is
present.  In Asymptote, the most straightforward option for such a
collection is a ‘Map’ from keys of type ‘K’ to ‘bool’ values, with
‘nullValue’ set to ‘false’.

   One needs to access some version of ‘Map_K_bool’.  If ‘K’ is
hashable, one could write
from collections.hashmap(K=K,V=bool) access
    Map_K_V as Map_K_bool,
    HashMap_K_V as HashMap_K_bool;
Map_K_bool mset = HashMap_K_bool(nullValue=false);
If ‘K’ is not hashable but does have ‘operator <’, something like this
may be used instead:
from collections.btreemap(K=K,V=bool) access
    Map_K_V as Map_K_bool,
    BTreeMap_K_V as BTreeMap_K_bool;
Map_K_bool mset = BTreeMap_K_bool(nullValue=false);

   To add an item ‘k’ to ‘mset’, assign it the value ‘true’:
mset[k] = true;
If the item was already in the set, this statement will have no effect.

   To test whether an item is in the set, you can either use the
‘contains’ method or use brackets to access the value associated with
‘k’:
if (mset.contains(k))
  write('membership verified');
or, equivalently,
if (mset[k])
  write('membership verified');

   To remove an item from the set, you can either use the ‘delete’
method (which throws an error if the item was not already present) or
set its value to ‘false’ (which does nothing if the item was not already
present):
mset.delete(k);  // guaranteed to reduce mset.size() by 1 or throw an error
mset[k] = false; // would delete k if we had not already deleted it

   Once a key has its value set to ‘false’, it will not show up in
iterators over the set, nor will it contribute to the ‘size()’ of the
set:
mset[k] = true;
int size1 = mset.size();
mset[k] = false;
assert(mset.size() == size1 - 1);

   One can iterate over the set or check its size:
int count = 0;
for (K k : mset) { ++count; }
assert(mset.size() == count);


File: asymptote.info,  Node: Sets,  Next: Using Sets as Maps,  Prev: Using Maps as Sets,  Up: Collections (containers)

6.20.5 Sets
-----------

There is also a dedicated ‘Set_T’ datastructure which is a bit more
efficient (not storing an unneeded ‘bool’) and allows more precise
methods (especially when it comes to sorted sets).  The ‘Set_T’ API
attempts to respect the principle of reversibility, which states that a
method that mutates the data structure should ideally return precisely
the amount of information needed to reverse the modification.  For
instance, a method that removes an element should return that element --
unless the element is already known to the caller (e.g., passed as an
argument), in which case it suffices to return a boolean to indicate
whether the removal was successful.  (By contrast, the ‘Map_K_V’ API
follows the principle that a method should either return something or
mutate the datastructure, but not both.)

* Menu:

* Initializing a Set::   Creating hash sets and b-tree sets
* The Set API::          Methods for set operations
* The SortedSet API::    Additional methods for sorted sets


File: asymptote.info,  Node: Initializing a Set,  Next: The Set API,  Up: Sets

6.20.5.1 Initializing a ‘Set’
.............................

The ‘Set_T’ datastructure itself is defined in the templated module
‘collections.set(T)’.  However, the ‘Set_T’ structure is abstract, with
many of its methods defaulting to ‘null’.  To get a working
implementation, you need either a ‘HashSet_T’ or a ‘BTreeSet_T’:

‘HashSet_T()’
‘HashSet_T(T nullT,
          bool equiv(T a, T b) = operator ==,
          bool isNullT(T) = new bool(T t) { return equiv(t, nullT); }
         )’
     These two functions, which create a hash set, must be accessed from
     the ‘collections.hashset(T)’ templated module.  The type ‘T’ must
     have a ‘hash’ method, *note Hash functions::.  If ‘isNullT’ is not
     defined, or if ‘!isNullT(null)’, then passing ‘null’ to most
     methods will throw an error.

‘BTreeSet_T()’
‘BTreeSet_T(T nullT, bool isNullT(T) = new bool(T t) { return t == nullT; })’
     These two functions must be accessed from the
     ‘collections.btree(T)’ module.  They create a b-tree set, which
     stores its elements in order based on the ‘<’ operator.  The ‘<’
     operator must define a strict weak ordering on all items of type
     ‘T’, except those satisfying ‘isNullT’.  Two items are considered
     equivalent if neither is less than the other.

     Accessing the ‘collections.btree(T)’ module requires that ‘operator
     <’ is defined for the type ‘T’; if not one can use the constructors
     defined in ‘collections.btreegeneral(T)’ module, as discussed next.

‘BTreeSet_T(bool lessThan(T,T))’
‘BTreeSet_T(bool lessThan(T,T),
           T nullT,
           bool isNullT(T) = new bool(T t) { return t == nullT; })’
     These two functions must be accessed from the
     ‘collections.btreegeneral(T)’ module.  They create a BTreeSet that
     stores its elements in sorted order based on the ‘lessThan’
     function.  The ‘lessThan’ function must define a strict weak
     ordering on all items of type ‘T’, except those satisfying
     ‘isNullT’.  Two items are considered equivalent if neither is less
     than the other.

   Both ‘HashSet_T’ and ‘BTreeSet_T’ can be implicitly cast to ‘Set_T’
and have all of the methods of ‘Set_T’.  Similarly, the ‘nullT’ and
‘isNullT’ optional parameters behave the same way for both types.  *Note
The Set API::.

   Additionally, ‘BTreeSet_T’ can be implicitly cast to ‘SortedSet_T’
and has all its methods, which provide additional functionality specific
to sorted sets (such as retrieving the least element or the next element
greater than a given element).  *Note The SortedSet API::.


File: asymptote.info,  Node: The Set API,  Next: The SortedSet API,  Prev: Initializing a Set,  Up: Sets

6.20.5.2 The ‘Set’ API
......................

Depending on which constructor is used, ‘nullT’ and ‘isNullT’ may or may
not be defined for a ‘Set’.  If one of these is defined the other will
be too.  In some methods below, an error will be thrown unless ‘nullT’
can be returned to indicate an element is not present.

   The ‘Set_T’ structure has the following methods:

‘int size()’
     returns the number of distinct elements in the set.

‘bool empty()’
     returns true iff ‘size() == 0’.

‘bool contains(T item)’
     returns true if ‘item’ (or an equivalent item) is present in the
     set.  If ‘isNullT’ is defined and ‘isNullT(item)’ is true, this
     method always returns false.

‘bool add(T item)’
     adds ‘item’ to the set if it is not already present.  This method
     does nothing if ‘isNullT(item)’ is defined and true, or if ‘item’
     or an equivalent item is already present.  The return value is true
     iff ‘item’ was added.

‘void add(Iterable_T other)’
     adds the elements from ‘other’ to the set.  This method violates
     the principle of reversibility.

‘bool delete(T item)’
     removes ‘item’ from the set if it is present, otherwise does
     nothing.  This method does nothing if ‘isNullT(item)’ is defined
     and true.  The return value is true iff the set was modified.

‘void delete(Iterable_T other)’
     removes the elements from ‘other’ from the set.  Elements of
     ‘other’ that are not in the set are ignored.  This method violates
     the principle of reversibility.

‘Iter_T operator iter()’
     returns an iterator over the elements of the set.  It allows a set
     ‘s’ to be used in a range-based ‘for’ loop:
     for (T item : s) {<statements>}
     If an item is deleted or a new item is inserted, then any iterators
     created before that change are undermined and may behave
     unpredictably.  The current implementations attempt to throw errors
     if an iterator is used after being undermined, but this behavior is
     not guaranteed and might be removed in future versions for
     performance reasons.

‘Set_T newEmpty()’
     returns a new empty set with the same ‘nullT’, ‘isNullT’,
     equivalence relation, and (if applicable) ordering as the current
     set.  Implementors of new set types should implement this method,
     otherwise the binary operators that are supposed to return a set
     will instead throw errors.

   The preceeding methods are sufficient assuming that equivalent
elements are interchangeable.  However, on rare occasions, one may care
which of two equivalent elements is in the set.  For example, if the set
was configured such that two ordered pairs are considered equivalent if
they have the same first element, then the methods above would provide
no way to look up a pair based on its first element.  For this
situation, the ‘Set_T’ API provides the following methods:

‘T get(T item)’
     returns the item in the set equivalent to ‘item’.  If no such item
     exists, ‘nullT’ is returned; if ‘nullT’ is not defined, an error is
     thrown.

‘T push(T item)’
     inserts ‘item’ into the set even if it is equivalent to an existing
     item.  If ‘item’ is equivalent to an existing item, the replaced
     item is returned; otherwise ‘nullT’ is returned (unless ‘nullT’ is
     not defined, in which case an error is thrown).  If ‘isNullT(item)’
     is true, the set is not modified and ‘nullT’ is returned.

     The ‘push’ method can be used to add a new item to the set, but
     only if ‘nullT’ is defined.

‘T extract(T item)’
     removes and returns an item in the set equivalent to ‘item’.  If no
     such item exists, ‘nullT’ is returned; if ‘nullT’ is not defined,
     an error is thrown.  If ‘isNullT(item)’ is true, the set is not
     modified and ‘nullT’ is returned.

   In addition to the methods above, there are a number of autounraveled
functions that deal with sets.

‘Iterable_T operator cast(Set_T set)’
     implicitly casts a set to an iterable so that it can be passed as a
     parameter to utilities like ‘zip’ or ‘enumerate’.

‘bool operator <= (Set_T a, Set_T b)’
     returns true if ‘a’ is a subset of ‘b’.

‘bool operator >= (Set_T a, Set_T b)’
     returns true if ‘b’ is a subset of ‘a’.

‘bool operator == (Set_T a, Set_T b)’
     returns true if ‘a’ and ‘b’ are subsets of each other.

‘bool operator != (Set_T a, Set_T b)’
     returns true if either ‘a’ or ‘b’ contains an element not in the
     other.

‘bool sameElementsInOrder(Set_T a, Set_T b)’
     returns true if iterating over ‘a’ and ‘b’ produces the same
     elements in the same order.

   The following operators leave their two arguments unchanged; the
result is initialized by calling the ‘newEmpty()’ method of the first
argument to produce a new set with the same underlying implementation as
the first argument.

‘Set_T operator + (Set_T a, Iterable_T b)’
     returns the union of ‘a’ and ‘b’.

‘Set_T operator - (Set_T a, Set_T b)’
     returns a new set containing the elements of ‘a’ that are not in
     ‘b’.

‘Set_T operator & (Set_T a, Set_T b)’
     returns the intersection of ‘a’ and ‘b’.

‘Set_T operator ^ (Set_T a, Set_T b)’
     returns the symmetric difference of ‘a’ and ‘b’, in other words,
     the set of all items in exactly one of ‘a,b’.

*Note:* While it is possible to use the ‘+=’ and ‘-=’ operators, they
may not function as expected.  In particular, these two operators will
construct a new set, rather than modifying the set in place.  For most
purposes, the ‘add(Iterable_T)’ and ‘delete(Iterable_T)’ methods are
preferred.


File: asymptote.info,  Node: The SortedSet API,  Prev: The Set API,  Up: Sets

6.20.5.3 The ‘SortedSet’ API
............................

The ‘BTreeSet_T’ implementation of ‘Set_T’ (*note Initializing a Set::)
maintains its elements in sorted order.  For the ‘Set_T’ API, this has
no effect other than the iteration order.  However, the ‘SortedSet_T’
API provides additional methods to take advantage of this ordering.  The
‘SortedSet_T’ structure is defined in the ‘collections.sortedset(T)’
module.  Here is a list of the relations among the various APIs
implemented by ‘BTreeSet_T’:
   • ‘BTreeSet_T’ has all the methods of ‘Set_T’ and can be implicitly
     cast to ‘Set_T’.

   • ‘BTreeSet_T’ has all the methods of ‘SortedSet_T’ and can be
     implicitly cast to ‘SortedSet_T’.

   • ‘SortedSet_T’ has all the methods of ‘Set_T’ and can be implicitly
     cast to ‘Set_T’.

   • All of these types can be implicitly cast to ‘Iterable_T’.

   In addition to the methods of ‘Set_T’, ‘SortedSet_T’ has the
following methods:

‘T min()’
     returns the least element in the set, or ‘nullT’ if the set is
     empty.  If the set is empty and ‘nullT’ is not defined, an error is
     thrown.

‘T max()’
     returns the greatest element in the set, or ‘nullT’ if the set is
     empty.  If the set is empty and ‘nullT’ is not defined, an error is
     thrown.

‘T popMin()’
     removes and returns the least element in the set, or returns
     ‘nullT’ if the set is empty.  If the set is empty and ‘nullT’ is
     not defined, an error is thrown.

‘T popMax()’
     removes and returns the greatest element in the set, or returns
     ‘nullT’ if the set is empty.  If the set is empty and ‘nullT’ is
     not defined, an error is thrown.

‘T after(T item)’
     returns the least element in the set that is strictly greater than
     ‘item’.  If no such element exists, ‘nullT’ is returned; if ‘nullT’
     is not defined, an error is thrown.

‘T atOrAfter(T item)’
     returns the least element in the set that is greater than or equal
     to ‘item’.  If no such element exists, ‘nullT’ is returned; if
     ‘nullT’ is not defined, an error is thrown.

‘T before(T item)’
     returns the greatest element in the set that is strictly less than
     ‘item’.  If no such element exists, ‘nullT’ is returned; if ‘nullT’
     is not defined, an error is thrown.

‘T atOrBefore(T item)’
     returns the greatest element in the set that is less than or equal
     to ‘item’.  If no such element exists, ‘nullT’ is returned; if
     ‘nullT’ is not defined, an error is thrown.

   As an example, the following code iterates over the elements of a
sorted set ‘sset’ between ‘a’ (inclusive) and ‘b’ (exclusive):
for (var x = sset.atOrAfter(a);
     !sset.isNullT(x) && x < b;
     x = sset.after(x)) {
  <statements using x>
}
Of course, this code assumes that ‘nullT’ is defined for ‘sset’.

   Note that if you want to iterate over most or all of the elements in
a sorted set, it is usually more efficient to use a range-based ‘for’
loop and skip any elements that are not in the desired range:
for (var x : sset) {
  if (x < a) continue;
  if (x >= b) break;
  <statements using x>
}


File: asymptote.info,  Node: Using Sets as Maps,  Next: Queues,  Prev: Sets,  Up: Collections (containers)

6.20.6 Using Sets as Maps
-------------------------

In mathematics, a map is sometimes defined as a set of ordered pairs.
In the Asymptote programming language, this definition can be taken
literally: a ‘Set_T’ of ‘Pair_K_V’ objects can be used as a map from ‘K’
to ‘V’, provided that the equivalence, hash, and comparison functions
are configured to look only at the first element of the pair.  This can
be useful in particular for sorted sets, which have functionality such
as ‘atOrAfter’ and ‘after’ that is not currently available in the ‘Map’
API.

   We illustrate this approach by implementing a piecewise linear
function (in this case we can use the builtin ‘pair’ type rather than a
‘Pair_K_V’).

from collections.btreegeneral(T=pair) access
    SortedSet_T as SortedSet_pair,
    BTreeSet_T as BTreeSet_pair;

bool keysLT(pair a, pair b) { return a.x < b.x; }
bool isnan(pair a) { return isnan(a.y); }

struct PLFunction {
  SortedSet_pair pairs = BTreeSet_pair(
    lessThan=keysLT,
    nullT=(nan,nan),
    isNullT = isnan
  );
  void operator [=] (real x, real y) {
    pairs.push((x,y));
  }
  real operator [] (real x) {
    pair p = (x, 0);
    pair left = pairs.atOrBefore(p);
    pair right = pairs.atOrAfter(p);
    if (isnan(left.y) || isnan(right.y)) {
      return nan;
    }
    if (left.x == right.x) {
      return left.y;
    }
    real t = (x - left.x) / (right.x - left.x);
    return interp(left.y, right.y, t);
  }
}

PLFunction f;
f[0] = 17;
f[1] = 18;
f[2001] = 19;

for (real x : new real[] {-1, 0, 0.5, 1, 2, 2001, 2002}) {
  write('f[' + (string)x + '] = ' + (string)f[x]);
}

outputs
f[-1] = nan
f[0] = 17
f[0.5] = 17.5
f[1] = 18
f[2] = 18.0005
f[2001] = 19
f[2002] = nan


File: asymptote.info,  Node: Queues,  Next: Wrapping arrays,  Prev: Using Sets as Maps,  Up: Collections (containers)

6.20.7 Queues
-------------

Two standard datastructures are stacks and queues.  A stack is a
last-in-first-out (LIFO) structure; in Asymptote, arrays can be used as
stacks via the ‘push’ and ‘pop’ methods.  A queue is a
first-in-first-out (FIFO) structure; the ‘collections.queue(T)’ module
provides a queue implementation.

‘Queue_T makeQueue(T[] initialData)’
     returns a queue initialized with a copy of the array ‘initialData’.
     The parameter ‘initialData’ is required in order to specify the
     type ‘T’ in case there are multiple ‘Queue_T’ types in the current
     scope.  If no initial data is needed, use ‘makeQueue(new T[])’ to
     create an empty array as a parameter.

   The ‘Queue_T’ structure has the following methods:
‘void push(T item)’
     adds ‘item’ to the end of the queue.

‘T pop()’
     removes and returns the first item in the queue.  If the queue is
     empty, an error is thrown.

‘T peek()’
     returns the first item in the queue without removing it.  If the
     queue is empty, an error is thrown.

‘int size()’
     returns the number of items in the queue.

‘Iter_T operator iter()’
     returns an iterator over the items in the queue, in the order they
     will be returned by ‘pop()’.

     If an item is pushed or popped, then any iterators created before
     that change are undermined and may behave unpredictably.  The
     current implementation attempts to throw errors if an iterator is
     used after being undermined, but this behavior is not guaranteed
     and might be removed in future versions for performance reasons.

   In addition to these methods, there is also an autounraveled implicit
cast from ‘Queue_T’ to ‘Iterable_T’.  This cast allows a ‘Queue_T’ to be
passed as a parameter to utilities like ‘zip’ or ‘enumerate’.


File: asymptote.info,  Node: Wrapping arrays,  Prev: Queues,  Up: Collections (containers)

6.20.8 Wrapping arrays
----------------------

‘Asymptote’ arrays have a builtin ‘==’ operator that returns an array of
bools rather than a single bool.  This is useful in a number of
contexts, but it also prevents them from being used in pairs or as
values in hashmaps.  The ‘collections.wraparray(T)’ templated module
provides a wrapper for arrays that has a more conventional ‘==’ operator
that returns a single bool.  It also provides a ‘hash()’ method that, if
initialized, allows the array to be used as a key in a hashmap.

   The main type in ‘collections.wraparray(T)’ is ‘Array_T’.  While
‘Array_T’ is not a drop-in replacement for ‘T[]’, it is as close as any
structure can be.  For convenience, the function ‘Array_T wrap(T[]
data)’ can be used to convert an array to an ‘Array_T’.  Additionally,
‘Array_T’ has autounraveled implicit casts to and from ‘T[]’, so that it
can be used in most places where ‘T[]’ is expected (and vice versa).
Note that these casts do _not_ make copies, so if you modify the
original array, the changes will be reflected in the ‘Array_T’ and vice
versa.

   The methods and fields of ‘Array_T’ are identical to those of ‘T[]’,
with the following exceptions:
‘int length()’
     is a method (requires parentheses afterwards) that returns the
     length of the array.
‘void cyclic(bool b)’
     sets the ‘cyclic’ field of the wrapped array.
‘bool cyclic()’
     returns the ‘cyclic’ field of the wrapped array.
‘int[] keys()’
     is a method (requires parentheses afterwards) that returns the
     ‘keys’ field of the wrapped array.
‘T[] data’
     is a field that holds the wrapped array.

   Additionally, the iteration syntax ‘for (T x : a)’ is supported if
‘a’ is an ‘Array_T’, as is element access ‘a[i]’.  However, slices are
not supported.

   To make an ‘Array_T’ hashable, pass a second argument to the ‘wrap’
function that computes an integer hash of a single element.  For
instance, the following constructs hashable arrays of strings and
integers:
from collections.wraparray(T=string) access Array_T as Array_string;
from collections.wraparray(T=int) access Array_T as Array_int;
string[] s = {'hello', 'world'};
Array_string ss = wrap(s, new int(string s) { return s.hash(); });
int[] a = {1, 2, 3};
Array_int aa = wrap(a, new int(int i) { return i; });
   The resulting hash function works by converting all the elements of
the array to integers using the provided function, and then hashing the
resulting ‘int[]’, *note Hash functions::.

   Important note: If you use an ‘Array_T’ as a key in a hashmap, you
must ensure that the array is not modified while it is a key in the
hashmap.


File: asymptote.info,  Node: LaTeX usage,  Next: User modules,  Prev: Programming,  Up: Top

7 LaTeX usage
*************

‘Asymptote’ comes with a convenient LaTeX style file ‘asymptote.sty’
(v1.36 or later required) that makes LaTeX ‘Asymptote’-aware.  Entering
‘Asymptote’ code directly into the LaTeX source file, at the point where
it is needed, keeps figures organized and avoids the need to invent new
file names for each figure.  Simply add the line
‘\usepackage{asymptote}’ at the beginning of your file and enclose your
‘Asymptote’ code within a ‘\begin{asy}...\end{asy}’ environment.  As
with the LaTeX ‘comment’ environment, the ‘\end{asy}’ command must
appear on a line by itself, with no trailing commands/comments.  A blank
line is not allowed after ‘\begin{asy}’.

   The sample LaTeX file below, named ‘latexusage.tex’, can be run as
follows:
latex latexusage
asy latexusage-*.asy
latex latexusage

or
pdflatex latexusage
asy latexusage-*.asy
pdflatex latexusage
To switch between using inline Asymptote code with ‘latex’ and
‘pdflatex’ you may first need to remove the files ‘latexusage-*.tex’.

   An even better method for processing a LaTeX file with embedded
‘Asymptote’ code is to use the ‘latexmk’ utility from
     <http://mirror.ctan.org/support/latexmk/>
after putting the contents of
<https://raw.githubusercontent.com/vectorgraphics/asymptote/HEAD/doc/latexmkrc>
in a file ‘latexmkrc’ in the same directory.  The command
latexmk -pdf latexusage
will then call ‘Asymptote’ automatically, recompiling only the figures
that have changed.  Since each figure is compiled in a separate system
process, this method also tends to use less memory.  To store the
figures in a separate directory named ‘asy’, one can define
\def\asydir{asy}
   in ‘latexusage.tex’.  External ‘Asymptote’ code can be included with
\asyinclude[<options>]{<filename.asy>}
so that ‘latexmk’ will recognize when the code is changed.  Note that
‘latexmk’ requires ‘perl’, available from <https://www.perl.org/>.

   One can specify ‘width’, ‘height’, ‘keepAspect’, ‘viewportwidth’,
‘viewportheight’, ‘attach’, and ‘inline’.  ‘keyval’-style options to the
‘asy’ and ‘asyinclude’ environments.  Three-dimensional PRC files may
either be embedded within the page (the default) or attached as
annotated (but printable) attachments, using the ‘attach’ option and the
‘attachfile2’ (or older ‘attachfile’) LaTeX package.  The ‘inline’
option generates inline LaTeX code instead of EPS or PDF files.  This
makes 2D LaTeX symbols visible to the ‘\begin{asy}...\end{asy}’
environment.  In this mode, Asymptote correctly aligns 2D LaTeX symbols
defined outside of ‘\begin{asy}...\end{asy}’, but treats their size as
zero; an optional second string can be given to ‘Label’ to provide an
estimate of the unknown label size.

   Note that if the ‘latex’ TeX engine is used with the ‘inline’ option,
labels might not show up in DVI viewers that cannot handle raw
‘PostScript’ code.  One can use ‘dvips’/‘dvipdf’ to produce
‘PostScript’/PDF output (we recommend using the modified version of
‘dvipdf’ in the ‘Asymptote’ patches directory, which accepts the ‘dvips
-z’ hyperdvi option).

   Here now is ‘latexusage.tex’:
\documentclass[12pt]{article}

% Use this form to include EPS (latex) or PDF (pdflatex) files:
%\usepackage{asymptote}

% Use this form with latex or pdflatex to include inline LaTeX code by default:
\usepackage[inline]{asymptote}

% Use this form with latex or pdflatex to create PDF attachments by default:
%\usepackage[attach]{asymptote}

% Enable this line to support the attach option:
%\usepackage[dvips]{attachfile2}

\begin{document}

% Optional subdirectory for latex files (no spaces):
\def\asylatexdir{}
% Optional subdirectory for asy files (no spaces):
\def\asydir{}

\begin{asydef}
// Global Asymptote definitions can be put here.
settings.prc=true;
import three;
usepackage("bm");
texpreamble("\def\V#1{\bm{#1}}");
// One can globally override the default toolbar settings here:
// settings.toolbar=true;
\end{asydef}

Here is a venn diagram produced with Asymptote, drawn to width 4cm:

\def\A{A}
\def\B{\V{B}}

%\begin{figure}
\begin{center}
\begin{asy}
size(4cm,0);
pen colour1=red;
pen colour2=green;

pair z0=(0,0);
pair z1=(-1,0);
pair z2=(1,0);
real r=1.5;
path c1=circle(z1,r);
path c2=circle(z2,r);
fill(c1,colour1);
fill(c2,colour2);

picture intersection=new picture;
fill(intersection,c1,colour1+colour2);
clip(intersection,c2);

add(intersection);

draw(c1);
draw(c2);

//draw("$\A$",box,z1);              // Requires [inline] package option.
//draw(Label("$\B$","$B$"),box,z2); // Requires [inline] package option.
draw("$A$",box,z1);            
draw("$\V{B}$",box,z2);

pair z=(0,-2);
real m=3;
margin BigMargin=Margin(0,m*dot(unit(z1-z),unit(z0-z)));

draw(Label("$A\cap B$",0),conj(z)--z0,Arrow,BigMargin);
draw(Label("$A\cup B$",0),z--z0,Arrow,BigMargin);
draw(z--z1,Arrow,Margin(0,m));
draw(z--z2,Arrow,Margin(0,m));

shipout(bbox(0.25cm));
\end{asy}
%\caption{Venn diagram}\label{venn}
\end{center}
%\end{figure}

Each graph is drawn in its own environment. One can specify the width
and height to \LaTeX\ explicitly. This 3D example can be viewed
interactively either with Adobe Reader or Asymptote's fast OpenGL-based
renderer. To support {\tt latexmk}, 3D figures should specify
\verb+inline=true+. It is sometimes desirable to embed 3D files as annotated
attachments; this requires the \verb+attach=true+ option as well as the
\verb+attachfile2+ \LaTeX\ package.
\begin{center}
\begin{asy}[height=4cm,inline=true,attach=false,viewportwidth=\linewidth]
currentprojection=orthographic(5,4,2);
draw(unitcube,blue);
label("$V-E+F=2$",(0,1,0.5),3Y,blue+fontsize(17pt));
\end{asy}
\end{center}

One can also scale the figure to the full line width:
\begin{center}
\begin{asy}[width=\the\linewidth,inline=true]
pair z0=(0,0);
pair z1=(2,0);
pair z2=(5,0);
pair zf=z1+0.75*(z2-z1);

draw(z1--z2);
dot(z1,red+0.15cm);
dot(z2,darkgreen+0.3cm);
label("$m$",z1,1.2N,red);
label("$M$",z2,1.5N,darkgreen);
label("$\hat{\ }$",zf,0.2*S,fontsize(24pt)+blue);

pair s=-0.2*I;
draw("$x$",z0+s--z1+s,N,red,Arrows,Bars,PenMargins);
s=-0.5*I;
draw("$\bar{x}$",z0+s--zf+s,blue,Arrows,Bars,PenMargins);
s=-0.95*I;
draw("$X$",z0+s--z2+s,darkgreen,Arrows,Bars,PenMargins);
\end{asy}
\end{center}
\end{document}
[./latexusage]


File: asymptote.info,  Node: User modules,  Next: Specialty modules,  Prev: LaTeX usage,  Up: Top

8 User modules
**************

These are the most commonly used base modules for creating graphics and
visualizations.

* Menu:

* graph::                       2D linear & logarithmic graphs
* three::                       3D vector graphics
* graph3::                      3D linear & logarithmic graphs
* palette::                     Color density images and palettes
* colormap::                    Colormaps from matplotlib
* texcolors::                   CMYK TeX colors
* x11colors::                   RGB ‘X11’ colors
* fontsize::                    Nonstandard font sizes
* contour::                     Contour lines
* geometry::                    Geometry routines
* grid3::                       3D grids
* interpolate::                 Interpolation routines
* markers::                     Custom path marker routines
* math::                        Extend ‘Asymptote’'s math capabilities
* patterns::                    Custom fill and draw patterns
* roundedpath::                 Round the sharp corners of paths
* slopefield::                  Slope fields
* smoothcontour3::              Smooth implicit surfaces
* solids::                      3D solid geometry
* stats::                       Statistics routines and histograms
* tube::                        3D rotation minimizing tubes


File: asymptote.info,  Node: graph,  Next: three,  Up: User modules

8.1 ‘graph’
===========

This module implements two-dimensional linear and logarithmic graphs,
including automatic scale and tick selection (with the ability to
override manually).  A graph is a ‘guide’ (that can be drawn with the
draw command, with an optional legend) constructed with one of the
following routines:

   • guide graph(picture pic=currentpicture, real f(real), real a, real b,
                 int n=ngraph, real T(real)=identity,
                 interpolate join=operator --);
     guide[] graph(picture pic=currentpicture, real f(real), real a, real b,
                  int n=ngraph, real T(real)=identity, bool3 cond(real),
                  interpolate join=operator --);

     return a graph using the scaling information for picture ‘pic’
     (*note automatic scaling::) of the function ‘f’ on the interval
     [‘T’(‘a’),‘T’(‘b’)], sampling at ‘n’ points evenly spaced in
     [‘a’,‘b’], optionally restricted by the bool3 function ‘cond’ on
     [‘a’,‘b’].  If ‘cond’ is:
        • ‘true’, the point is added to the existing guide;
        • ‘default’, the point is added to a new guide;
        • ‘false’, the point is omitted and a new guide is begun.
     The points are connected using the interpolation specified by
     ‘join’:

        • ‘operator --’ (linear interpolation; the abbreviation
          ‘Straight’ is also accepted);

        • ‘operator ..’ (piecewise Bezier cubic spline interpolation;
          the abbreviation ‘Spline’ is also accepted);

        • ‘linear’ (linear interpolation),
        • ‘Hermite’ (standard cubic spline interpolation using boundary
          condition ‘notaknot’, ‘natural’, ‘periodic’, ‘clamped(real
          slopea, real slopeb)’), or ‘monotonic’.  The abbreviation
          ‘Hermite’ is equivalent to ‘Hermite(notaknot)’ for nonperiodic
          data and ‘Hermite(periodic)’ for periodic data).

   • guide graph(picture pic=currentpicture, real x(real), real y(real),
                 real a, real b, int n=ngraph, real T(real)=identity,
                 interpolate join=operator --);
     guide[] graph(picture pic=currentpicture, real x(real), real y(real),
                   real a, real b, int n=ngraph, real T(real)=identity,
                   bool3 cond(real), interpolate join=operator --);

     returns a graph using the scaling information for picture ‘pic’ of
     the parametrized function (‘x’(t),‘y’(t)) for t in the interval
     [‘T’(‘a’),‘T’(‘b’)], sampling at ‘n’ points evenly spaced in
     [‘a’,‘b’], optionally restricted by the bool3 function ‘cond’ on
     [‘a’,‘b’], using the given interpolation type.

   • guide graph(picture pic=currentpicture, pair z(real), real a, real b,
                 int n=ngraph, real T(real)=identity,
                 interpolate join=operator --);
     guide[] graph(picture pic=currentpicture, pair z(real), real a, real b,
                   int n=ngraph, real T(real)=identity, bool3 cond(real),
                   interpolate join=operator --);

     returns a graph using the scaling information for picture ‘pic’ of
     the parametrized function ‘z’(t) for t in the interval
     [‘T’(‘a’),‘T’(‘b’)], sampling at ‘n’ points evenly spaced in
     [‘a’,‘b’], optionally restricted by the bool3 function ‘cond’ on
     [‘a’,‘b’], using the given interpolation type.

   • guide graph(picture pic=currentpicture, pair[] z,
                 interpolate join=operator --);
     guide[] graph(picture pic=currentpicture, pair[] z, bool3[] cond,
                   interpolate join=operator --);

     returns a graph using the scaling information for picture ‘pic’ of
     the elements of the array ‘z’, optionally restricted to those
     indices for which the elements of the boolean array ‘cond’ are
     ‘true’, using the given interpolation type.

   • guide graph(picture pic=currentpicture, real[] x, real[] y,
                 interpolate join=operator --);
     guide[] graph(picture pic=currentpicture, real[] x, real[] y,
                   bool3[] cond, interpolate join=operator --);

     returns a graph using the scaling information for picture ‘pic’ of
     the elements of the arrays (‘x’,‘y’), optionally restricted to
     those indices for which the elements of the boolean array ‘cond’
     are ‘true’, using the given interpolation type.

   • guide polargraph(picture pic=currentpicture, real f(real), real a,
                      real b, int n=ngraph, interpolate join=operator --);

     returns a polar-coordinate graph using the scaling information for
     picture ‘pic’ of the function ‘f’ on the interval [‘a’,‘b’],
     sampling at ‘n’ evenly spaced points, with the given interpolation
     type.

   • guide polargraph(picture pic=currentpicture, real[] r, real[] theta,
                      interpolate join=operator--);
     returns a polar-coordinate graph using the scaling information for
     picture ‘pic’ of the elements of the arrays (‘r’,‘theta’), using
     the given interpolation type.

   An axis can be drawn on a picture with one of the following commands:

   • void xaxis(picture pic=currentpicture, Label L="", axis axis=YZero,
                real xmin=-infinity, real xmax=infinity, pen p=currentpen,
                ticks ticks=NoTicks, arrowbar arrow=None, bool above=false);

     Draw an x axis on picture ‘pic’ from x=‘xmin’ to x=‘xmax’ using pen
     ‘p’, optionally labelling it with Label ‘L’.  The relative label
     location along the axis (a real number from [0,1]) defaults to 1
     (*note Label::), so that the label is drawn at the end of the axis.
     An infinite value of ‘xmin’ or ‘xmax’ specifies that the
     corresponding axis limit will be automatically determined from the
     picture limits.  The optional ‘arrow’ argument takes the same
     values as in the ‘draw’ command (*note arrows::).  The axis is
     drawn before any existing objects in ‘pic’ unless ‘above=true’.
     The axis placement is determined by one of the following ‘axis’
     types:

     ‘YZero(bool extend=true)’
          Request an x axis at y=0 (or y=1 on a logarithmic axis)
          extending to the full dimensions of the picture, unless
          ‘extend’=false.

     ‘YEquals(real Y, bool extend=true)’
          Request an x axis at y=‘Y’ extending to the full dimensions of
          the picture, unless ‘extend’=false.

     ‘Bottom(bool extend=false)’
          Request a bottom axis.

     ‘Top(bool extend=false)’
          Request a top axis.

     ‘BottomTop(bool extend=false)’
          Request a bottom and top axis.

     Custom axis types can be created by following the examples in the
     module ‘graph.asy’.  One can easily override the default values for
     the standard axis types:
     import graph;
     
     YZero=new axis(bool extend=true) {
       return new void(picture pic, axisT axis) {
         real y=pic.scale.x.scale.logarithmic ? 1 : 0;
         axis.value=I*pic.scale.y.T(y);
         axis.position=1;
         axis.side=right;
         axis.align=2.5E;
         axis.value2=Infinity;
         axis.extend=extend;
       };
     };
     YZero=YZero();
     

     The default tick option is ‘NoTicks’.  The options ‘LeftTicks’,
     ‘RightTicks’, or ‘Ticks’ can be used to draw ticks on the left,
     right, or both sides of the path, relative to the direction in
     which the path is drawn.  These tick routines accept a number of
     optional arguments:
     ticks LeftTicks(Label format="", ticklabel ticklabel=null,
                     bool beginlabel=true, bool endlabel=true,
                     int N=0, int n=0, real Step=0, real step=0,
                     bool begin=true, bool end=true, tickmodifier modify=None,
                     real Size=0, real size=0, bool extend=false,
                     pen pTick=nullpen, pen ptick=nullpen);

     If any of these parameters are omitted, reasonable defaults will be
     chosen:
     ‘Label format’
          override the default tick label format (‘defaultformat’,
          initially "$%.4g$"), rotation, pen, and alignment (for
          example, ‘LeftSide’, ‘Center’, or ‘RightSide’) relative to the
          axis.  To enable LaTeX math mode fonts, the format string
          should begin and end with ‘$’ *note format::.  If the format
          string is ‘trailingzero’, trailing zeros will be added to the
          tick labels; if the format string is ‘"%"’, the tick label
          will be suppressed;
     ‘ticklabel’
          is a function ‘string(real x)’ returning the label (by
          default, format(format.s,x)) for each major tick value ‘x’;
     ‘bool beginlabel’
          include the first label;
     ‘bool endlabel’
          include the last label;
     ‘int N’
          when automatic scaling is enabled (the default; *note
          automatic scaling::), divide a linear axis evenly into this
          many intervals, separated by major ticks; for a logarithmic
          axis, this is the number of decades between labelled ticks;
     ‘int n’
          divide each interval into this many subintervals, separated by
          minor ticks;
     ‘real Step’
          the tick value spacing between major ticks (if ‘N’=‘0’);
     ‘real step’
          the tick value spacing between minor ticks (if ‘n’=‘0’);
     ‘bool begin’
          include the first major tick;
     ‘bool end’
          include the last major tick;
     ‘tickmodifier modify;’
          an optional function that takes and returns a ‘tickvalue’
          structure having real[] members ‘major’ and ‘minor’ consisting
          of the tick values (to allow modification of the automatically
          generated tick values);
     ‘real Size’
          the size of the major ticks (in ‘PostScript’ coordinates);
     ‘real size’
          the size of the minor ticks (in ‘PostScript’ coordinates);
     ‘bool extend;’
          extend the ticks between two axes (useful for drawing a grid
          on the graph);
     ‘pen pTick’
          an optional pen used to draw the major ticks;
     ‘pen ptick’
          an optional pen used to draw the minor ticks.

     For convenience, the predefined tickmodifiers ‘OmitTick(...real[]
     x)’, ‘OmitTickInterval(real a, real b)’, and
     ‘OmitTickIntervals(real[] a, real[] b)’ can be used to remove
     specific auto-generated ticks and their labels.  The
     ‘OmitFormat(string s=defaultformat, ... real[] x)’ ticklabel can be
     used to remove specific tick labels but not the corresponding
     ticks.  The tickmodifier ‘NoZero’ is an abbreviation for
     ‘OmitTick(0)’ and the ticklabel ‘NoZeroFormat’ is an abbreviation
     for ‘OmitFormat(0)’.

     It is also possible to specify custom tick locations with
     ‘LeftTicks’, ‘RightTicks’, and ‘Ticks’ by passing explicit real
     arrays ‘Ticks’ and (optionally) ‘ticks’ containing the locations of
     the major and minor ticks, respectively:
     ticks LeftTicks(Label format="", ticklabel ticklabel=null,
                     bool beginlabel=true, bool endlabel=true,
                     real[] Ticks, real[] ticks=new real[],
                     real Size=0, real size=0, bool extend=false,
                     pen pTick=nullpen, pen ptick=nullpen)

   • void yaxis(picture pic=currentpicture, Label L="", axis axis=XZero,
                real ymin=-infinity, real ymax=infinity, pen p=currentpen,
                ticks ticks=NoTicks, arrowbar arrow=None, bool above=false,
                bool autorotate=true);

     Draw a y axis on picture ‘pic’ from y=‘ymin’ to y=‘ymax’ using pen
     ‘p’, optionally labelling it with a Label ‘L’ that is autorotated
     unless ‘autorotate=false’.  The relative location of the label (a
     real number from [0,1]) defaults to 1 (*note Label::).  An infinite
     value of ‘ymin’ or ‘ymax’ specifies that the corresponding axis
     limit will be automatically determined from the picture limits.
     The optional ‘arrow’ argument takes the same values as in the
     ‘draw’ command (*note arrows::).  The axis is drawn before any
     existing objects in ‘pic’ unless ‘above=true’.  The tick type is
     specified by ‘ticks’ and the axis placement is determined by one of
     the following ‘axis’ types:

     ‘XZero(bool extend=true)’
          Request a y axis at x=0 (or x=1 on a logarithmic axis)
          extending to the full dimensions of the picture, unless
          ‘extend’=false.

     ‘XEquals(real X, bool extend=true)’
          Request a y axis at x=‘X’ extending to the full dimensions of
          the picture, unless ‘extend’=false.

     ‘Left(bool extend=false)’
          Request a left axis.

     ‘Right(bool extend=false)’
          Request a right axis.

     ‘LeftRight(bool extend=false)’
          Request a left and right axis.

   • For convenience, the functions
     void xequals(picture pic=currentpicture, Label L="", real x,
                  bool extend=false, real ymin=-infinity, real ymax=infinity,
                  pen p=currentpen, ticks ticks=NoTicks, bool above=true,
                  arrowbar arrow=None);
     and
     void yequals(picture pic=currentpicture, Label L="", real y,
                  bool extend=false, real xmin=-infinity, real xmax=infinity,
                  pen p=currentpen, ticks ticks=NoTicks, bool above=true,
                  arrowbar arrow=None);
     can be respectively used to call ‘yaxis’ and ‘xaxis’ with the
     appropriate axis types ‘XEquals(x,extend)’ and ‘YEquals(y,extend)’.
     This is the recommended way of drawing vertical or horizontal lines
     and axes at arbitrary locations.

   • void axes(picture pic=currentpicture, Label xlabel="", Label ylabel="",
               bool extend=true,
               pair min=(-infinity,-infinity), pair max=(infinity,infinity),
               pen p=currentpen, arrowbar arrow=None, bool above=false);
     This convenience routine draws both x and y axes on picture ‘pic’
     from ‘min’ to ‘max’, with optional labels ‘xlabel’ and ‘ylabel’ and
     any arrows specified by ‘arrow’.  The axes are drawn on top of
     existing objects in ‘pic’ only if ‘above=true’.

   • void axis(picture pic=currentpicture, Label L="", path g,
               pen p=currentpen, ticks ticks, ticklocate locate,
               arrowbar arrow=None, int[] divisor=new int[],
               bool above=false, bool opposite=false);

     This routine can be used to draw on picture ‘pic’ a general axis
     based on an arbitrary path ‘g’, using pen ‘p’.  One can optionally
     label the axis with Label ‘L’ and add an arrow ‘arrow’.  The tick
     type is given by ‘ticks’.  The optional integer array ‘divisor’
     specifies what tick divisors to try in the attempt to produce
     uncrowded tick labels.  A ‘true’ value for the flag ‘opposite’
     identifies an unlabelled secondary axis (typically drawn opposite a
     primary axis).  The axis is drawn before any existing objects in
     ‘pic’ unless ‘above=true’.  The tick locator ‘ticklocate’ is
     constructed by the routine
     ticklocate ticklocate(real a, real b, autoscaleT S=defaultS,
                           real tickmin=-infinity, real tickmax=infinity,
                           real time(real)=null, pair dir(real)=zero);
     where ‘a’ and ‘b’ specify the respective tick values at
     ‘point(g,0)’ and ‘point(g,length(g))’, ‘S’ specifies the
     autoscaling transformation, the function ‘real time(real v)’
     returns the time corresponding to the value ‘v’, and ‘pair dir(real
     t)’ returns the absolute tick direction as a function of ‘t’ (zero
     means draw the tick perpendicular to the axis).

   • These routines are useful for manually putting ticks and labels on
     axes (if the variable ‘Label’ is given as the ‘Label’ argument, the
     ‘format’ argument will be used to format a string based on the tick
     location):
     void xtick(picture pic=currentpicture, Label L="", explicit pair z,
                pair dir=N, string format="",
                real size=Ticksize, pen p=currentpen);
     void xtick(picture pic=currentpicture, Label L="", real x,
                pair dir=N, string format="",
                real size=Ticksize, pen p=currentpen);
     void ytick(picture pic=currentpicture, Label L="", explicit pair z,
                pair dir=E, string format="",
                real size=Ticksize, pen p=currentpen);
     void ytick(picture pic=currentpicture, Label L="", real y,
                pair dir=E, string format="",
                real size=Ticksize, pen p=currentpen);
     void tick(picture pic=currentpicture, pair z,
               pair dir, real size=Ticksize, pen p=currentpen);
     void labelx(picture pic=currentpicture, Label L="", explicit pair z,
                 align align=S, string format="", pen p=currentpen);
     void labelx(picture pic=currentpicture, Label L="", real x,
                 align align=S, string format="", pen p=currentpen);
     void labelx(picture pic=currentpicture, Label L,
                 string format="", explicit pen p=currentpen);
     void labely(picture pic=currentpicture, Label L="", explicit pair z,
                 align align=W, string format="", pen p=currentpen);
     void labely(picture pic=currentpicture, Label L="", real y,
                 align align=W, string format="", pen p=currentpen);
     void labely(picture pic=currentpicture, Label L,
                 string format="", explicit pen p=currentpen);

   Here are some simple examples of two-dimensional graphs:

  1. This example draws a textbook-style graph of y= exp(x), with the y
     axis starting at y=0:
     import graph;
     size(150,0);
     
     real f(real x) {return exp(x);}
     pair F(real x) {return (x,f(x));}
     
     draw(graph(f,-4,2,operator ..),red);
     
     xaxis("$x$");
     yaxis("$y$",0);
     
     labely(1,E);
     label("$e^x$",F(1),SE);
     

                                [./exp]

  2. The next example draws a scientific-style graph with a legend.  The
     position of the legend can be adjusted either explicitly or by
     using the graphical user interface (*note GUI::).  If an
     ‘UnFill(real xmargin=0, real ymargin=xmargin)’ or ‘Fill(pen)’
     option is specified to ‘add’, the legend will obscure any
     underlying objects.  Here we illustrate how to clip the portion of
     the picture covered by a label:

     import graph;
     
     size(400,200,IgnoreAspect);
     
     real Sin(real t) {return sin(2pi*t);}
     real Cos(real t) {return cos(2pi*t);}
     
     draw(graph(Sin,0,1),red,"$\sin(2\pi x)$");
     draw(graph(Cos,0,1),blue,"$\cos(2\pi x)$");
     
     xaxis("$x$",BottomTop,LeftTicks);
     yaxis("$y$",LeftRight,RightTicks(trailingzero));
     
     label("LABEL",point(0),UnFill(1mm));
     
     add(legend(),point(E),20E,UnFill);

                           [./lineargraph0]

     To specify a fixed size for the graph proper, use ‘attach’:
     import graph;
     
     size(250,200,IgnoreAspect);
     
     real Sin(real t) {return sin(2pi*t);}
     real Cos(real t) {return cos(2pi*t);}
     
     draw(graph(Sin,0,1),red,"$\sin(2\pi x)$");
     draw(graph(Cos,0,1),blue,"$\cos(2\pi x)$");
     
     xaxis("$x$",BottomTop,LeftTicks);
     yaxis("$y$",LeftRight,RightTicks(trailingzero));
     
     label("LABEL",point(0),UnFill(1mm));
     
     attach(legend(),truepoint(E),20E,UnFill);

     A legend can have multiple entries per line:
     import graph;
     size(8cm,6cm,IgnoreAspect);
     
     typedef real realfcn(real);
     realfcn F(real p) {
       return new real(real x) {return sin(p*x);};
     }
     
     for(int i=1; i < 5; ++i)
       draw(graph(F(i*pi),0,1),Pen(i),
            "$\sin("+(i == 1 ? "" : (string) i)+"\pi x)$");
     xaxis("$x$",BottomTop,LeftTicks);
     yaxis("$y$",LeftRight,RightTicks(trailingzero));
     
     attach(legend(2),(point(S).x,truepoint(S).y),10S,UnFill);

                              [./legend]

  3. This example draws a graph of one array versus another (both of the
     same size) using custom tick locations and a smaller font size for
     the tick labels on the y axis.
     import graph;
     
     size(200,150,IgnoreAspect);
     
     real[] x={0,1,2,3};
     real[] y=x^2;
     
     draw(graph(x,y),red);
     
     xaxis("$x$",BottomTop,LeftTicks);
     yaxis("$y$",LeftRight,
           RightTicks(Label(fontsize(8pt)),new real[]{0,4,9}));

                             [./datagraph]

  4. This example shows how to graph columns of data read from a file.
     import graph;
     
     size(200,150,IgnoreAspect);
     
     file in=input("filegraph.dat").line();
     real[][] a=in;
     a=transpose(a);
     
     real[] x=a[0];
     real[] y=a[1];
     
     draw(graph(x,y),red);
     
     xaxis("$x$",BottomTop,LeftTicks);
     yaxis("$y$",LeftRight,RightTicks);

                             [./filegraph]

  5. The next example draws two graphs of an array of coordinate pairs,
     using frame alignment and data markers.  In the left-hand graph,
     the markers, constructed with
     marker marker(path g, markroutine markroutine=marknodes,
                   pen p=currentpen, filltype filltype=NoFill,
                   bool above=true);
     using the path ‘unitcircle’ (*note filltype::), are drawn below
     each node.  Any frame can be converted to a marker, using
     marker marker(frame f, markroutine markroutine=marknodes,
                   bool above=true);
     In the right-hand graph, the unit n-sided regular polygon
     ‘polygon(int n)’ and the unit n-point cyclic cross ‘cross(int n,
     bool round=true, real r=0)’ (where ‘r’ is an optional "inner"
     radius) are used to build a custom marker frame.  Here
     ‘markuniform(bool centered=false, int n, bool rotated=false)’ adds
     this frame at ‘n’ uniformly spaced points along the arclength of
     the path, optionally rotated by the angle of the local tangent to
     the path (if centered is true, the frames will be centered within
     ‘n’ evenly spaced arclength intervals).  Alternatively, one can use
     markroutine ‘marknodes’ to request that the marks be placed at each
     Bezier node of the path, or markroutine ‘markuniform(pair z(real
     t), real a, real b, int n)’ to place marks at points ‘z(t)’ for n
     evenly spaced values of ‘t’ in ‘[a,b]’.

     These markers are predefined:
     marker[] Mark={
       marker(scale(circlescale)*unitcircle),
       marker(polygon(3)),marker(polygon(4)),
       marker(polygon(5)),marker(invert*polygon(3)),
       marker(cross(4)),marker(cross(6)),marker(diamond),marker(plus);
     };
     
     marker[] MarkFill={
       marker(scale(circlescale)*unitcircle,Fill),marker(polygon(3),Fill),
       marker(polygon(4),Fill),marker(polygon(5),Fill),
       marker(invert*polygon(3),Fill),marker(diamond,Fill)
     };

     The example also illustrates the ‘errorbar’ routines:

     void errorbars(picture pic=currentpicture, pair[] z, pair[] dp,
                    pair[] dm={}, bool[] cond={}, pen p=currentpen,
                    real size=0);
     
     void errorbars(picture pic=currentpicture, real[] x, real[] y,
                    real[] dpx, real[] dpy, real[] dmx={}, real[] dmy={},
                    bool[] cond={}, pen p=currentpen, real size=0);

     Here, the positive and negative extents of the error are given by
     the absolute values of the elements of the pair array ‘dp’ and the
     optional pair array ‘dm’.  If ‘dm’ is not specified, the positive
     and negative extents of the error are assumed to be equal.
     import graph;
     
     picture pic;
     real xsize=200, ysize=140;
     size(pic,xsize,ysize,IgnoreAspect);
     
     pair[] f={(5,5),(50,20),(90,90)};
     pair[] df={(0,0),(5,7),(0,5)};
     
     errorbars(pic,f,df,red);
     draw(pic,graph(pic,f),"legend",
          marker(scale(0.8mm)*unitcircle,red,FillDraw(blue),above=false));
     
     scale(pic,true);
     
     xaxis(pic,"$x$",BottomTop,LeftTicks);
     yaxis(pic,"$y$",LeftRight,RightTicks);
     add(pic,legend(pic),point(pic,NW),20SE,UnFill);
     
     picture pic2;
     size(pic2,xsize,ysize,IgnoreAspect);
     
     frame mark;
     filldraw(mark,scale(0.8mm)*polygon(6),green,green);
     draw(mark,scale(0.8mm)*cross(6),blue);
     
     draw(pic2,graph(pic2,f),marker(mark,markuniform(5)));
     
     scale(pic2,true);
     
     xaxis(pic2,"$x$",BottomTop,LeftTicks);
     yaxis(pic2,"$y$",LeftRight,RightTicks);
     
     yequals(pic2,55.0,red+Dotted);
     xequals(pic2,70.0,red+Dotted);
     
     // Fit pic to W of origin:
     add(pic.fit(),(0,0),W);
     
     // Fit pic2 to E of (5mm,0):
     add(pic2.fit(),(5mm,0),E);
     

                             [./errorbars]

  6. A custom mark routine can be also be specified:
     import graph;
     
     size(200,100,IgnoreAspect);
     
     markroutine marks() {
       return new void(picture pic=currentpicture, frame f, path g) {
         path p=scale(1mm)*unitcircle;
         for(int i=0; i <= length(g); ++i) {
           pair z=point(g,i);
           frame f;
           if(i % 4 == 0) {
             fill(f,p);
             add(pic,f,z);
           } else {
             if(z.y > 50) {
               pic.add(new void(frame F, transform t) {
                   path q=shift(t*z)*p;
                   unfill(F,q);
                   draw(F,q);
                 });
             } else {
               draw(f,p);
               add(pic,f,z);
             }
           }
         }
       };
     }
     
     pair[] f={(5,5),(40,20),(55,51),(90,30)};
     
     draw(graph(f),marker(marks()));
     
     scale(true);
     
     xaxis("$x$",BottomTop,LeftTicks);
     yaxis("$y$",LeftRight,RightTicks);

                           [./graphmarkers]

  7. This example shows how to label an axis with arbitrary strings.
     import graph;
     
     size(400,150,IgnoreAspect);
     
     real[] x=sequence(12);
     real[] y=sin(2pi*x/12);
     
     scale(false);
     
     string[] month={"Jan","Feb","Mar","Apr","May","Jun",
                     "Jul","Aug","Sep","Oct","Nov","Dec"};
     
     draw(graph(x,y),red,MarkFill[0]);
     
     xaxis(BottomTop,LeftTicks(new string(real x) {
           return month[round(x % 12)];}));
     yaxis("$y$",LeftRight,RightTicks(4));

                             [./monthaxis]

  8. The next example draws a graph of a parametrized curve.  The calls
     to
     xlimits(picture pic=currentpicture, real min=-infinity,
             real max=infinity, bool crop=NoCrop);
     and the analogous function ‘ylimits’ can be uncommented to set the
     respective axes limits for picture ‘pic’ to the specified ‘min’ and
     ‘max’ values.  Alternatively, the function
     void limits(picture pic=currentpicture, pair min, pair max, bool crop=NoCrop);
     can be used to limit the axes to the box having opposite vertices
     at the given pairs).  Existing objects in picture ‘pic’ will be
     cropped to lie within the given limits if ‘crop’=‘Crop’.  The
     function ‘crop(picture pic)’ can be used to crop a graph to the
     current graph limits.
     import graph;
     
     size(0,200);
     
     real x(real t) {return cos(2pi*t);}
     real y(real t) {return sin(2pi*t);}
     
     draw(graph(x,y,0,1));
     
     //limits((0,-1),(1,0),Crop);
     
     xaxis("$x$",BottomTop,LeftTicks);
     yaxis("$y$",LeftRight,RightTicks(trailingzero));
     
     

                          [./parametricgraph]

     The function
     guide graphwithderiv(pair f(real), pair fprime(real), real a, real b,
                          int n=ngraph#10);
     can be used to construct the graph of the parametric function ‘f’
     on ‘[a,b]’ with the control points of the ‘n’ Bezier segments
     determined by the specified derivative ‘fprime’:
     unitsize(2cm);
     import graph;
     pair F(real t) {
       return (1.3*t,-4.5*t^2+3.0*t+1.0);
     }
     pair Fprime(real t) {
       return (1.3,-9.0*t+3.0);
     }
     path g=graphwithderiv(F,Fprime,0,0.9,4);
     dot(g,red);
     draw(g,arrow=Arrow(TeXHead));

                          [./graphwithderiv]

     The next example illustrates how one can extract a common axis
     scaling factor.
     import graph;
     
     axiscoverage=0.9;
     size(200,IgnoreAspect);
     
     real[] x={-1e-11,1e-11};
     real[] y={0,1e6};
     
     real xscale=round(log10(max(x)));
     real yscale=round(log10(max(y)))-1;
     
     draw(graph(x*10^(-xscale),y*10^(-yscale)),red);
     
     xaxis("$x/10^{"+(string) xscale+"}$",BottomTop,LeftTicks);
     yaxis("$y/10^{"+(string) yscale+"}$",LeftRight,RightTicks(trailingzero));

                            [./scaledgraph]

     Axis scaling can be requested and/or automatic selection of the
     axis limits can be inhibited with one of these ‘scale’ routines:
     void scale(picture pic=currentpicture, scaleT x, scaleT y);
     
     void scale(picture pic=currentpicture, bool xautoscale=true,
                bool yautoscale=xautoscale, bool zautoscale=yautoscale);

     This sets the scalings for picture ‘pic’.  The ‘graph’ routines
     accept an optional ‘picture’ argument for determining the
     appropriate scalings to use; if none is given, it uses those set
     for ‘currentpicture’.

     Two frequently used scaling routines ‘Linear’ and ‘Log’ are
     predefined in ‘graph’.

     All picture coordinates (including those in paths and those given
     to the ‘label’ and ‘limits’ functions) are always treated as linear
     (post-scaled) coordinates.  Use
     pair Scale(picture pic=currentpicture, pair z);
     to convert a graph coordinate into a scaled picture coordinate.

     The x and y components can be individually scaled using the
     analogous routines
     real ScaleX(picture pic=currentpicture, real x);
     real ScaleY(picture pic=currentpicture, real y);

     The predefined scaling routines can be given two optional boolean
     arguments: ‘automin=false’ and ‘automax=automin’.  These default to
     ‘false’ but can be respectively set to ‘true’ to enable automatic
     selection of "nice" axis minimum and maximum values.  The ‘Linear’
     scaling can also take as optional final arguments a multiplicative
     scaling factor and intercept (for a depth axis, ‘Linear(-1)’
     requests axis reversal).

     For example, to draw a log/log graph of a function, use
     ‘scale(Log,Log)’:
     import graph;
     
     size(200,200,IgnoreAspect);
     
     real f(real t) {return 1/t;}
     
     scale(Log,Log);
     
     draw(graph(f,0.1,10));
     
     //limits((1,0.1),(10,0.5),Crop);
     
     dot(Label("(3,5)",align=S),Scale((3,5)));
     
     xaxis("$x$",BottomTop,LeftTicks);
     yaxis("$y$",LeftRight,RightTicks);
     

                             [./loggraph]

     By extending the ticks, one can easily produce a logarithmic grid:
     import graph;
     size(200,200,IgnoreAspect);
     
     real f(real t) {return 1/t;}
     
     scale(Log,Log);
     draw(graph(f,0.1,10),red);
     pen thin=linewidth(0.5*linewidth());
     xaxis("$x$",BottomTop,LeftTicks(begin=false,end=false,extend=true,
                                     ptick=thin));
     yaxis("$y$",LeftRight,RightTicks(begin=false,end=false,extend=true,
                                      ptick=thin));
     


                              [./loggrid]

     One can also specify custom tick locations and formats for
     logarithmic axes:
     import graph;
     
     size(300,175,IgnoreAspect);
     scale(Log,Log);
     draw(graph(identity,5,20));
     xlimits(5,20);
     ylimits(1,100);
     xaxis("$M/M_\odot$",BottomTop,LeftTicks(DefaultFormat,
                                             new real[] {6,10,12,14,16,18}));
     yaxis("$\nu_{\rm upp}$ [Hz]",LeftRight,RightTicks(DefaultFormat));
     

                             [./logticks]

     It is easy to draw logarithmic graphs with respect to other bases:
     import graph;
     size(200,IgnoreAspect);
     
     // Base-2 logarithmic scale on y-axis:
     
     real log2(real x) {static real log2=log(2); return log(x)/log2;}
     real pow2(real x) {return 2^x;}
     
     scaleT yscale=scaleT(log2,pow2,logarithmic=true);
     scale(Linear,yscale);
     
     real f(real x) {return 1+x^2;}
     
     draw(graph(f,-4,4));
     
     yaxis("$y$",ymin=1,ymax=f(5),RightTicks(Label(Fill(white))),EndArrow);
     xaxis("$x$",xmin=-5,xmax=5,LeftTicks,EndArrow);

                             [./log2graph]

     Here is an example of "broken" linear x and logarithmic y axes that
     omit the segments [3,8] and [100,1000], respectively.  In the case
     of a logarithmic axis, the break endpoints are automatically
     rounded to the nearest integral power of the base.
     import graph;
     
     size(200,150,IgnoreAspect);
     
     // Break the x axis at 3; restart at 8:
     real a=3, b=8;
     
     // Break the y axis at 100; restart at 1000:
     real c=100, d=1000;
     
     scale(Broken(a,b),BrokenLog(c,d));
     
     real[] x={1,2,4,6,10};
     real[] y=x^4;
     
     draw(graph(x,y),red,MarkFill[0]);
     
     xaxis("$x$",BottomTop,LeftTicks(Break(a,b)));
     yaxis("$y$",LeftRight,RightTicks(Break(c,d)));
     
     label(rotate(90)*Break,(a,point(S).y));
     label(rotate(90)*Break,(a,point(N).y));
     label(Break,(point(W).x,ScaleY(c)));
     label(Break,(point(E).x,ScaleY(c)));
     

                            [./brokenaxis]

  9. ‘Asymptote’ can draw secondary axes with the routines
     picture secondaryX(picture primary=currentpicture, void f(picture));
     picture secondaryY(picture primary=currentpicture, void f(picture));

     In this example, ‘secondaryY’ is used to draw a secondary linear y
     axis against a primary logarithmic y axis:
     import graph;
     texpreamble("\def\Arg{\mathop {\rm Arg}\nolimits}");
     
     size(10cm,5cm,IgnoreAspect);
     
     real ampl(real x) {return 2.5/sqrt(1+x^2);}
     real phas(real x) {return -atan(x)/pi;}
     
     scale(Log,Log);
     draw(graph(ampl,0.01,10));
     ylimits(0.001,100);
     
     xaxis("$\omega\tau_0$",BottomTop,LeftTicks);
     yaxis("$|G(\omega\tau_0)|$",Left,RightTicks);
     
     picture q=secondaryY(new void(picture pic) {
         scale(pic,Log,Linear);
         draw(pic,graph(pic,phas,0.01,10),red);
         ylimits(pic,-1.0,1.5);
         yaxis(pic,"$\Arg G/\pi$",Right,red,
               LeftTicks("$% #.1f$",
                         begin=false,end=false));
         yequals(pic,1,Dotted);
       });
     label(q,"(1,0)",Scale(q,(1,0)),red);
     add(q);
     

                               [./Bode]

     A secondary logarithmic y axis can be drawn like this:
     import graph;
     
     size(9cm,6cm,IgnoreAspect);
     string data="secondaryaxis.csv";
     
     file in=input(data).line().csv();
     
     string[] titlelabel=in;
     string[] columnlabel=in;
     
     real[][] a=in;
     a=transpose(a);
     real[] t=a[0], susceptible=a[1], infectious=a[2], dead=a[3], larvae=a[4];
     real[] susceptibleM=a[5], exposed=a[6], infectiousM=a[7];
     
     scale(true);
     
     draw(graph(t,susceptible,t >= 10 & t <= 15));
     draw(graph(t,dead,t >= 10 & t <= 15),dashed);
     
     xaxis("Time ($\tau$)",BottomTop,LeftTicks);
     yaxis(Left,RightTicks);
     
     picture secondary=secondaryY(new void(picture pic) {
         scale(pic,Linear(true),Log(true));
         draw(pic,graph(pic,t,infectious,t >= 10 & t <= 15),red);
         yaxis(pic,Right,red,LeftTicks(begin=false,end=false));
       });
     
     add(secondary);
     label(shift(5mm*N)*"Proportion of crows",point(NW),E);

                           [./secondaryaxis]

  10. Here is a histogram example, which uses the ‘stats’ module.
     import graph;
     import stats;
     
     size(400,200,IgnoreAspect);
     
     int n=10000;
     real[] a=new real[n];
     for(int i=0; i < n; ++i) a[i]=Gaussrand();
     
     draw(graph(Gaussian,min(a),max(a)),blue);
     
     // Optionally calculate "optimal" number of bins a la Shimazaki and Shinomoto.
     int N=bins(a);
     
     histogram(a,min(a),max(a),N,normalize=true,low=0,lightred,black,bars=true);
     
     xaxis("$x$",BottomTop,LeftTicks);
     yaxis("$dP/dx$",LeftRight,RightTicks(trailingzero));

                             [./histogram]

  11. Here is an example of reading column data in from a file and a
     least-squares fit, using the ‘stats’ module.
     size(400,200,IgnoreAspect);
     
     import graph;
     import stats;
     
     file fin=input("leastsquares.dat").line();
     
     real[][] a=fin;
     a=transpose(a);
     
     real[] t=a[0], rho=a[1];
     
     // Read in parameters from the keyboard:
     //real first=getreal("first");
     //real step=getreal("step");
     //real last=getreal("last");
     
     real first=100;
     real step=50;
     real last=700;
     
     // Remove negative or zero values of rho:
     t=rho > 0 ? t : null;
     rho=rho > 0 ? rho : null;
     
     scale(Log(true),Linear(true));
     
     int n=step > 0 ? ceil((last-first)/step) : 0;
     
     real[] T,xi,dxi;
     
     for(int i=0; i <= n; ++i) {
       real first=first+i*step;
       real[] logrho=(t >= first & t <= last) ? log(rho) : null;
       real[] logt=(t >= first & t <= last) ? -log(t) : null;
     
       if(logt.length < 2) break;
     
       // Fit to the line logt=L.m*logrho+L.b:
       linefit L=leastsquares(logt,logrho);
     
       T.push(first);
       xi.push(L.m);
       dxi.push(L.dm);
     }
     
     draw(graph(T,xi),blue);
     errorbars(T,xi,dxi,red);
     
     crop();
     
     ylimits(0);
     
     xaxis("$T$",BottomTop,LeftTicks);
     yaxis("$\xi$",LeftRight,RightTicks);

                           [./leastsquares]

  12. Here is an example that illustrates the general ‘axis’ routine.
     import graph;
     size(0,100);
     
     path g=ellipse((0,0),1,2);
     
     scale(true);
     
     axis(Label("C",align=10W),g,LeftTicks(endlabel=false,8,end=false),
          ticklocate(0,360,new real(real v) {
              path h=(0,0)--max(abs(max(g)),abs(min(g)))*dir(v);
              return intersect(g,h)[0];}));

                            [./generalaxis]

  13. To draw a vector field of ‘n’ arrows evenly spaced along the
     arclength of a path, use the routine
     picture vectorfield(path vector(real), path g, int n, bool truesize=false,
                         pen p=currentpen, arrowbar arrow=Arrow);
     as illustrated in this simple example of a flow field:
     import graph;
     defaultpen(1.0);
     
     size(0,150,IgnoreAspect);
     
     real arrowsize=4mm;
     real arrowlength=2arrowsize;
     
     typedef path vector(real);
     
     // Return a vector interpolated linearly between a and b.
     vector vector(pair a, pair b) {
       return new path(real x) {
         return (0,0)--arrowlength*interp(a,b,x);
       };
     }
     
     real f(real x) {return 1/x;}
     
     real epsilon=0.5;
     path g=graph(f,epsilon,1/epsilon);
     
     int n=3;
     draw(g);
     xaxis("$x$");
     yaxis("$y$");
     
     add(vectorfield(vector(W,W),g,n,true));
     add(vectorfield(vector(NE,NW),(0,0)--(point(E).x,0),n,true));
     add(vectorfield(vector(NE,NE),(0,0)--(0,point(N).y),n,true));
     

                               [./flow]

  14. To draw a vector field of ‘nx’\times‘ny’ arrows in ‘box(a,b)’, use
     the routine
     picture vectorfield(path vector(pair), pair a, pair b,
                         int nx=nmesh, int ny=nx, bool truesize=false,
                         real maxlength=truesize ? 0 : maxlength(a,b,nx,ny),
                         bool cond(pair z)=null, pen p=currentpen,
                         arrowbar arrow=Arrow, margin margin=PenMargin)
     as illustrated in this example:
     import graph;
     size(100);
     
     pair a=(0,0);
     pair b=(2pi,2pi);
     
     path vector(pair z) {return (sin(z.x),cos(z.y));}
     
     add(vectorfield(vector,a,b));

                            [./vectorfield]

  15. The following scientific graphs, which illustrate many features of
     ‘Asymptote’'s graphics routines, were generated from the examples
     ‘diatom.asy’ and ‘westnile.asy’, using the comma-separated data in
     ‘diatom.csv’ and ‘westnile.csv’.


                              [./diatom]

                             [./westnile]


File: asymptote.info,  Node: three,  Next: graph3,  Prev: graph,  Up: User modules

8.2 ‘three’
===========

This module fully extends the notion of guides and paths in ‘Asymptote’
to three dimensions.  It introduces the new types guide3, path3, and
surface.  Guides in three dimensions are specified with the same syntax
as in two dimensions except that triples ‘(x,y,z)’ are used in place of
pairs ‘(x,y)’ for the nodes and direction specifiers.  This
generalization of John Hobby's spline algorithm is shape-invariant under
three-dimensional rotation, scaling, and shifting, and reduces in the
planar case to the two-dimensional algorithm used in ‘Asymptote’,
‘MetaPost’, and ‘MetaFont’ [see J. C. Bowman, Proceedings in Applied
Mathematics and Mechanics, 7:1, 2010021-2010022 (2007)].

   For example, a unit circle in the XY plane may be filled and drawn
like this:
import three;

size(100);

path3 g=(1,0,0)..(0,1,0)..(-1,0,0)..(0,-1,0)..cycle;
draw(g);
draw(O--Z,red+dashed,Arrow3);
draw(((-1,-1,0)--(1,-1,0)--(1,1,0)--(-1,1,0)--cycle));
dot(g,red);

                            [./unitcircle3]
and then distorted into a saddle:
import three;

size(100,0);
path3 g=(1,0,0)..(0,1,1)..(-1,0,0)..(0,-1,1)..cycle;
draw(g);
draw(((-1,-1,0)--(1,-1,0)--(1,1,0)--(-1,1,0)--cycle));
dot(g,red);

                              [./saddle]

   Module ‘three’ provides constructors for converting two-dimensional
paths to three-dimensional ones, and vice-versa:
path3 path3(path p, triple plane(pair)=XYplane);
path path(path3 p, pair P(triple)=xypart);

   A Bezier surface, the natural two-dimensional generalization of
Bezier curves, is defined in ‘three_surface.asy’ as a structure
containing an array of Bezier patches.  Surfaces may drawn with one of
the routines
void draw(picture pic=currentpicture, surface s, int nu=1, int nv=1,
          material surfacepen=currentpen, pen meshpen=nullpen,
          light light=currentlight, light meshlight=nolight, string name="",
          render render=defaultrender);
void draw(picture pic=currentpicture, surface s, int nu=1, int nv=1,
          material[] surfacepen, pen meshpen,
          light light=currentlight, light meshlight=nolight, string name="",
          render render=defaultrender);
void draw(picture pic=currentpicture, surface s, int nu=1, int nv=1,
          material[] surfacepen, pen[] meshpen=nullpens,
          light light=currentlight, light meshlight=nolight, string name="",
          render render=defaultrender);

   The parameters ‘nu’ and ‘nv’ specify the number of subdivisions for
drawing optional mesh lines for each Bezier patch.  The optional ‘name’
parameter is used as a prefix for naming the surface patches in the PRC
model tree.  Here material is a structure defined in ‘three_light.asy’:
struct material {
  pen[] p; // diffusepen,emissivepen,specularpen
  real opacity;
  real shininess;
  real metallic;
  real fresnel0;
}
These material properties are used to implement physically based
rendering (PBR) using light properties defined in ‘plain_prethree.asy’
and ‘three_light.asy’:
struct light {
  real[][] diffuse;
  real[][] specular;
  pen background=nullpen; // Background color of the canvas.
  real specularfactor;
  triple[] position; // Only directional lights are currently implemented.
}

light Viewport=light(specularfactor=3,(0.25,-0.25,1));

light White=light(new pen[] {rgb(0.38,0.38,0.45),rgb(0.6,0.6,0.67),
                             rgb(0.5,0.5,0.57)},specularfactor=3,
  new triple[] {(-2,-1.5,-0.5),(2,1.1,-2.5),(-0.5,0,2)});

light Headlamp=light(gray(0.8),specular=gray(0.7),
                     specularfactor=3,dir(42,48));

currentlight=Headlamp;

light nolight;
   The ‘currentlight.background’ (or ‘background’ member of the
specified ‘light’) can be used to set the background color for 2D (or
3D) images.  The default background is white for ‘HTML’ images and
transparent for all other formats.  One can request a completely
transparent background for 3D ‘WebGL’ images with
‘currentlight.background=black+opacity(0.0);’

   ‘render’

   A function ‘render()’ may be assigned to the optional ‘render’
parameter allows one to pass specialized rendering options to the
surface drawing routines, via arguments such as:
  bool tessellate;   // use tessellated mesh to store straight patches
  real margin;       // shrink amount for rendered OpenGL viewport, in bp.
  bool partnames;    // assign part name indices to compound objects
  bool defaultnames; // assign default names to unnamed objects
  interaction interaction; // billboard interaction mode
   along with the rendering parameters for the legacy PRC format
described in ‘three.asy’.

   Asymptote also supports image-based lighting with the setting
‘settings.ibl=true’.  This uses pre-rendered EXR images from the
directory specified by ‘-imageDir’ (which defaults to ‘ibl’) or, for
‘WebGL’ rendering, the URL specified by ‘-imageURL’ (which defaults to
<https://vectorgraphics.gitlab.io/asymptote/ibl>).  Additional rendered
images can be generated on an ‘NVIDIA’ GPU using the ‘reflect’ program
in the ‘cudareflect’ subdirectory of the ‘Asymptote’ source directory.

   Sample Bezier surfaces are contained in the example files
‘BezierSurface.asy’, ‘teapot.asy’, ‘teapotIBL.asy’, and
‘parametricsurface.asy’.

   The structure ‘render’ contains specialized rendering options
documented at the beginning of module ‘three’.

   The example ‘elevation.asy’ illustrates how to draw a surface with
patch-dependent colors.  The examples ‘sphericalharmonic.asy’,
‘vertexshading.asy’ and ‘smoothelevation.asy’ illustrate
vertex-dependent colors, which are supported by ‘Asymptote’'s native
‘OpenGL’/‘WebGL’ renderers, the three-dimensional vector output format
‘V3D’, and two-dimensional vector output (‘settings.render=0’).  Since
the legacy PRC output format does not support vertex shading of Bezier
surfaces, PRC patches are shaded with the mean of the four vertex
colors.

   A surface can be constructed from a cyclic ‘path3’ with the
constructor
surface surface(path3 external, triple[] internal=new triple[],
                pen[] colors=new pen[], bool3 planar=default);
and then filled:
draw(surface(unitsquare3,new triple[] {X,Y,Z,O}),red);
draw(surface(O--X{Y}..Y{-X}--cycle,new triple[] {Z}),red);
draw(surface(path3(polygon(5))),red,nolight);
draw(surface(unitcircle3),red,nolight);
draw(surface(unitcircle3,new pen[] {red,green,blue,black}),nolight);
The first example draws a Bezier patch and the second example draws a
Bezier triangle.  The third and fourth examples are planar surfaces.
The last example constructs a patch with vertex-specific colors.  A
three-dimensional planar surface in the plane ‘plane’ can be constructed
from a two-dimensional cyclic path ‘g’ with the constructor
surface surface(path p, triple plane(pair)=XYplane);
and then filled:
draw(surface((0,0)--E+2N--2E--E+N..0.2E..cycle),red);
Planar Bezier surfaces patches are constructed using Orest Shardt's
‘bezulate’ routine, which decomposes (possibly nonsimply connected)
regions bounded (according to the ‘zerowinding’ fill rule) by simple
cyclic paths (intersecting only at the endpoints) into subregions
bounded by cyclic paths of length ‘4’ or less.

   A more efficient routine also exists for drawing tessellations
composed of many 3D triangles, with specified vertices, and optional
normals or vertex colors:
void draw(picture pic=currentpicture, triple[] v, int[][] vi,
          triple[] n={}, int[][] ni=vi, material m=currentpen, pen[] p={},
          int[][] pi=vi, light light=currentlight);
   Here, the triple array ‘v’ lists the (typically distinct) vertices,
while the array ‘vi’ contains integer arrays of length 3 containing the
indices of the elements in ‘v’ that form the vertices of each triangle.
Similarly, the arguments ‘n’ and ‘ni’ contain optional normal data and
‘p’ and ‘pi’ contain optional pen vertex data.  If more than one normal
or pen is specified for a vertex, the last one is used.  An example of
this tessellation facility is given in ‘triangles.asy’.

   Arbitrary thick three-dimensional curves and line caps (which the
‘OpenGL’ standard does not require implementations to provide) are
constructed with
tube tube(path3 p, real width, render render=defaultrender);
this returns a tube structure representing a tube of diameter ‘width’
centered approximately on ‘g’.  The tube structure consists of a surface
‘s’ and the actual tube center, path3 ‘center’.  Drawing thick lines as
tubes can be slow to render, especially with the ‘Adobe Reader’
renderer.  The setting ‘thick=false’ can be used to disable this feature
and force all lines to be drawn with ‘linewidth(0)’ (one pixel wide,
regardless of the resolution).  By default, mesh and contour lines in
three-dimensions are always drawn thin, unless an explicit line width is
given in the pen parameter or the setting ‘thin’ is set to ‘false’.  The
pens ‘thin()’ and ‘thick()’ defined in ‘plain_pens.asy’ can also be used
to override these defaults for specific draw commands.

There are six choices for viewing 3D ‘Asymptote’ output:

  1. Use the native ‘Asymptote’ adaptive ‘OpenGL’-based renderer (with
     the command-line option ‘-V’ and the default settings
     ‘outformat=""’ and ‘render=-1’).  On ‘UNIX’ systems with graphics
     support for multisampling, the sample width can be controlled with
     the setting ‘multisample’.  The ratio of physical to logical screen
     pixels can be specified with the setting ‘devicepixelratio’.  An
     initial screen position can be specified with the pair setting
     ‘position’, where negative values are interpreted as relative to
     the corresponding maximum screen dimension.  The default settings
     import settings;
     leftbutton=new string[] {"rotate","zoom","shift","pan"};
     middlebutton=new string[] {""};
     rightbutton=new string[] {"zoom","rotateX","rotateY","rotateZ"};
     wheelup=new string[] {"zoomin"};
     wheeldown=new string[] {"zoomout"};
     bind the mouse buttons as follows:
        • Left: rotate
        • Shift Left: zoom
        • Ctrl Left: shift viewport
        • Alt Left: pan
        • Wheel Up: zoom in
        • Wheel Down: zoom out
        • Right: zoom
        • Shift Right: rotate about the X axis
        • Ctrl Right: rotate about the Y axis
        • Alt Right: rotate about the Z axis

     The keyboard bindings are:
        • h: home
        • f: toggle fitscreen
        • x: spin about the X axis
        • y: spin about the Y axis
        • z: spin about the Z axis
        • s: stop spinning
        • m: rendering mode (solid/patch/mesh)
        • e: export
        • c: show camera parameters
        • p: play animation
        • r: reverse animation
        • : step animation
        • +: expand
        • =: expand
        • >: expand
        • -: shrink
        • _: shrink
        • <: shrink
        • q: exit
        • Ctrl-q: exit

  2. Generate ‘WebGL’ interactive vector graphics output with the the
     command-line option and ‘-f html’ (or the setting
     ‘outformat="html"’).  The resulting 3D HTML file can then be viewed
     directly in any modern desktop or mobile browser, or even embedded
     within another web page:
     <iframe src="logo3.html" width="561" height="321" frameborder="0">
     </iframe>

     Normally, ‘WebGL’ files generated by ‘Asymptote’ are dynamically
     remeshed to fit the browser window dimensions.  However, the
     setting ‘absolute=true’ can be used to force the image to be
     rendered at its designed size (accounting for multiple device
     pixels per ‘css’ pixel).

     For specialized applications, the setting ‘keys=true’ can be used
     to generate an identifying key immediately before the ‘WebGL’ code
     for each generated object.  The default key, the ‘"line:column"’ of
     the associated function call of the top-level source code, can be
     overwritten by adding ‘KEY="x"’ as the first argument of the
     function call, where ‘x’ represents user-supplied text.

     The interactive ‘WebGL’ files produced by ‘Asymptote’ use the
     default mouse and (many of the same) key bindings as the ‘OpenGL’
     renderer.  Zooming via the mouse wheel of a ‘WebGL’ image embedded
     within another page is disabled until the image is activated by a
     click or touch event and will remain enabled until the ‘ESC’ key is
     pressed.

     By default, viewing the 3D HTML files generated by Asymptote
     requires network access to download the ‘AsyGL’ rendering library,
     which is normally cached by the browser for future use.  However,
     the setting ‘offline=true’ can be used to embed this small (about
     48kB) library within a stand-alone HTML file that can be viewed
     offline.

  3. Render the scene to a specified rasterized format ‘outformat’ at
     the resolution of ‘n’ pixels per ‘bp’, as specified by the setting
     ‘render=n’.  A negative value of ‘n’ is interpreted as ‘|2n|’ for
     EPS and PDF formats and ‘|n|’ for other formats.  The default value
     of ‘render’ is -1.  By default, the scene is internally rendered at
     twice the specified resolution; this can be disabled by setting
     ‘antialias=1’.  High resolution rendering is done by tiling the
     image.  If your graphics card allows it, the rendering can be made
     more efficient by increasing the maximum tile size ‘maxtile’ to
     your screen dimensions (indicated by ‘maxtile=(0,0)’.  If your
     video card generates unwanted black stripes in the output, try
     setting the horizontal and vertical components of ‘maxtiles’ to
     something less than your screen dimensions.  The tile size is also
     limited by the setting ‘maxviewport’, which restricts the maximum
     width and height of the viewport.  Some graphics drivers support
     batch mode (‘-noV’) rendering in an iconified window; this can be
     enabled with the setting ‘iconify=true’.

  4. Embed the 3D legacy PRC format in a PDF file and view the resulting
     PDF file with version ‘9.0’ or later of ‘Adobe Reader’.  This
     requires ‘settings.outformat="pdf"’ and ‘settings.prc=true’, which
     can be specified by the command-line options ‘-f pdf’ and ‘-f prc’,
     put in the ‘Asymptote’ configuration file (*note configuration
     file::), or specified in the script before module ‘three’ (or
     ‘graph3’) is imported.  The ‘media9’ LaTeX package is also required
     (*note embed::).  The example ‘100d.asy’ illustrates how one can
     generate a list of predefined views (see ‘100d.views’).  A
     stationary preview image with a resolution of ‘n’ pixels per ‘bp’
     can be embedded with the setting ‘render=n’; this allows the file
     to be viewed with other ‘PDF’ viewers.  Alternatively, the file
     ‘externalprc.tex’ illustrates how the resulting PRC and rendered
     image files can be extracted and processed in a separate LaTeX
     file.  However, see *note LaTeX usage:: for an easier way to embed
     three-dimensional ‘Asymptote’ pictures within LaTeX.  For
     specialized applications where only the raw PRC file is required,
     specify ‘settings.outformat="prc"’.  The PRC specification is
     available from
     <https://web.archive.org/web/20081204104459/http://livedocs.adobe.com/acrobat_sdk/9/Acrobat9_HTMLHelp/API_References/PRCReference/PRC_Format_Specification/>

  5. Output a V3D portable compressed vector graphics file using
     ‘settings.outformat="v3d"’, which can be viewed with an external
     viewer or converted to an alternate 3D format using the Python
     ‘pyv3d’ library.  V3D content can be automatically embedded within
     a PDF file using the options ‘settings.outformat="pdf"’ and
     ‘settings.v3d=true’.  Alternatively, a V3D file ‘file.v3d’ may be
     manually embedded within a PDF file using the ‘media9’ LaTeX
     package:
     \includemedia[noplaybutton,width=100pt,height=200pt]{}{file.v3d}%
     An online ‘Javascript’-based V3D-aware ‘PDF’ viewer is available at
     <https://github.com/vectorgraphics/pdfv3dReader>.

     The V3D specification and the ‘pyv3d’ library are available at
     <https://github.com/vectorgraphics/v3d>.  A V3D file ‘file.v3d’ may
     be imported and viewed by ‘Asymptote’ either by specifying
     ‘file.v3d’ on the command line
     asy -V file.v3d
     or using the ‘v3d’ module and ‘importv3d’ function in interactive
     mode (or within an ‘Asymptote’ file):
     import v3d;
     importv3d("file.v3d");

  6. Project the scene to a two-dimensional vector (EPS or PDF) format
     with ‘render=0’.  Only limited support for hidden surface removal,
     lighting, and transparency is available with this approach (*note
     PostScript3D::).

   Automatic picture sizing in three dimensions is accomplished with
double deferred drawing.  The maximal desired dimensions of the scene in
each of the three dimensions can optionally be specified with the
routine
void size3(picture pic=currentpicture, real x, real y=x, real z=y,
          bool keepAspect=pic.keepAspect);
A simplex linear programming problem is then solved to produce a 3D
version of a frame (actually implemented as a 3D picture).  The result
is then fit with another application of deferred drawing to the viewport
dimensions corresponding to the usual two-dimensional picture ‘size’
parameters.  The global pair ‘viewportmargin’ may be used to add
horizontal and vertical margins to the viewport dimensions.
Alternatively, a minimum ‘viewportsize’ may be specified.  A 3D picture
‘pic’ can be explicitly fit to a 3D frame by calling
frame pic.fit3(projection P=currentprojection);
and then added to picture ‘dest’ about ‘position’ with
void add(picture dest=currentpicture, frame src, triple position=(0,0,0));

   For convenience, the ‘three’ module defines ‘O=(0,0,0)’, ‘X=(1,0,0)’,
‘Y=(0,1,0)’, and ‘Z=(0,0,1)’, along with a unitcircle in the XY plane:
path3 unitcircle3=X..Y..-X..-Y..cycle;

   A general (approximate) circle can be drawn perpendicular to the
direction ‘normal’ with the routine
path3 circle(triple c, real r, triple normal=Z);

   A circular arc centered at ‘c’ with radius ‘r’ from
‘c+r*dir(theta1,phi1)’ to ‘c+r*dir(theta2,phi2)’, drawing
counterclockwise relative to the normal vector
‘cross(dir(theta1,phi1),dir(theta2,phi2))’ if ‘theta2 > theta1’ or if
‘theta2 == theta1’ and ‘phi2 >= phi1’, can be constructed with
path3 arc(triple c, real r, real theta1, real phi1, real theta2, real phi2,
          triple normal=O);
   The normal must be explicitly specified if ‘c’ and the endpoints are
colinear.  If ‘r’ < 0, the complementary arc of radius ‘|r|’ is
constructed.  For convenience, an arc centered at ‘c’ from triple ‘v1’
to ‘v2’ (assuming ‘|v2-c|=|v1-c|’) in the direction CCW
(counter-clockwise) or CW (clockwise) may also be constructed with
path3 arc(triple c, triple v1, triple v2, triple normal=O,
          bool direction=CCW);
When high accuracy is needed, the routines ‘Circle’ and ‘Arc’ defined in
‘graph3’ may be used instead.  See *note GaussianSurface:: for an
example of a three-dimensional circular arc.

   The representation ‘O--O+u--O+u+v--O+v--cycle’ of the plane passing
through point ‘O’ with normal ‘cross(u,v)’ is returned by
path3 plane(triple u, triple v, triple O=O);
   A three-dimensional box with opposite vertices at triples ‘v1’ and
‘v2’ may be drawn with the function
path3[] box(triple v1, triple v2);
For example, a unit box is predefined as
path3[] unitbox=box(O,(1,1,1));
   ‘Asymptote’ also provides optimized definitions for the
three-dimensional paths ‘unitsquare3’ and ‘unitcircle3’, along with the
surfaces ‘unitdisk’, ‘unitplane’, ‘unitcube’, ‘unitcylinder’,
‘unitcone’, ‘unitsolidcone’, ‘unitfrustum(real t1, real t2)’,
‘unitsphere’, and ‘unithemisphere’.

These projections to two dimensions are predefined:
‘oblique’
‘oblique(real angle)’
     The point ‘(x,y,z)’ is projected to ‘(x-0.5z,y-0.5z)’.  If an
     optional real argument is given, the negative z axis is drawn at
     this angle in degrees.  The projection ‘obliqueZ’ is a synonym for
     ‘oblique’.

‘obliqueX’
‘obliqueX(real angle)’
     The point ‘(x,y,z)’ is projected to ‘(y-0.5x,z-0.5x)’.  If an
     optional real argument is given, the negative x axis is drawn at
     this angle in degrees.

‘obliqueY’
‘obliqueY(real angle)’
     The point ‘(x,y,z)’ is projected to ‘(x+0.5y,z+0.5y)’.  If an
     optional real argument is given, the positive y axis is drawn at
     this angle in degrees.

‘orthographic(triple camera, triple up=Z, triple target=O,
             real zoom=1, pair viewportshift=0, bool showtarget=true,
             bool center=true)’
     This projects from three to two dimensions using the view as seen
     at a point infinitely far away in the direction ‘unit(camera)’,
     orienting the camera so that, if possible, the vector ‘up’ points
     upwards.  Parallel lines are projected to parallel lines.  The
     bounding volume is expanded to include ‘target’ if
     ‘showtarget=true’.  If ‘center=true’, the target will be adjusted
     to the center of the bounding volume.

‘orthographic(real x, real y, real z, triple up=Z, triple target=O,
             real zoom=1, pair viewportshift=0, bool showtarget=true,
             bool center=true)’
     This is equivalent to
     orthographic((x,y,z),up,target,zoom,viewportshift,showtarget,center)

     The routine
     triple camera(real alpha, real beta);
     can be used to compute the camera position with the x axis below
     the horizontal at angle ‘alpha’, the y axis below the horizontal at
     angle ‘beta’, and the z axis up.

‘perspective(triple camera, triple up=Z, triple target=O,
            real zoom=1, real angle=0, pair viewportshift=0,
            bool showtarget=true, bool autoadjust=true,
            bool center=autoadjust)’
     This projects from three to two dimensions, taking account of
     perspective, as seen from the location ‘camera’ looking at
     ‘target’, orienting the camera so that, if possible, the vector
     ‘up’ points upwards.  If ‘autoadjust=true’, the camera will
     automatically be adjusted to lie outside the bounding volume for
     all possible interactive rotations about ‘target’.  If
     ‘center=true’, the target will be adjusted to the center of the
     bounding volume.

‘perspective(real x, real y, real z, triple up=Z, triple target=O,
            real zoom=1, real angle=0, pair viewportshift=0,
            bool showtarget=true, bool autoadjust=true,
            bool center=autoadjust)’
     This is equivalent to
     perspective((x,y,z),up,target,zoom,angle,viewportshift,showtarget,
                 autoadjust,center)

The default projection, ‘currentprojection’, is initially set to
‘perspective(5,4,2)’.

   We also define standard orthographic views used in technical drawing:
projection LeftView=orthographic(-X,showtarget=true);
projection RightView=orthographic(X,showtarget=true);
projection FrontView=orthographic(-Y,showtarget=true);
projection BackView=orthographic(Y,showtarget=true);
projection BottomView=orthographic(-Z,showtarget=true);
projection TopView=orthographic(Z,showtarget=true);
The function
void addViews(picture dest=currentpicture, picture src,
              projection[][] views=SixViewsUS,
              bool group=true, filltype filltype=NoFill);
adds to picture ‘dest’ an array of views of picture ‘src’ using the
layout projection[][] ‘views’.  The default layout ‘SixViewsUS’ aligns
the projection ‘FrontView’ below ‘TopView’ and above ‘BottomView’, to
the right of ‘LeftView’ and left of ‘RightView’ and ‘BackView’.  The
predefined layouts are:
projection[][] ThreeViewsUS={{TopView},
                             {FrontView,RightView}};

projection[][] SixViewsUS={{null,TopView},
                           {LeftView,FrontView,RightView,BackView},
                           {null,BottomView}};

projection[][] ThreeViewsFR={{RightView,FrontView},
                             {null,TopView}};

projection[][] SixViewsFR={{null,BottomView},
                           {RightView,FrontView,LeftView,BackView},
                           {null,TopView}};

projection[][] ThreeViews={{FrontView,TopView,RightView}};

projection[][] SixViews={{FrontView,TopView,RightView},
                         {BackView,BottomView,LeftView}};

   A triple or path3 can be projected to a pair or path, with
‘project(triple, projection P=currentprojection)’ or ‘project(path3,
projection P=currentprojection)’.

   It is occasionally useful to be able to invert a projection, sending
a pair ‘z’ onto the plane perpendicular to ‘normal’ and passing through
‘point’:
triple invert(pair z, triple normal, triple point,
              projection P=currentprojection);
A pair ‘z’ on the projection plane can be inverted to a triple with the
routine
triple invert(pair z, projection P=currentprojection);
A pair direction ‘dir’ on the projection plane can be inverted to a
triple direction relative to a point ‘v’ with the routine
triple invert(pair dir, triple v, projection P=currentprojection).

   Three-dimensional objects may be transformed with one of the
following built-in transform3 types (the identity transformation is
‘identity4’):

‘shift(triple v)’
     translates by the triple ‘v’;
‘xscale3(real x)’
     scales by ‘x’ in the x direction;
‘yscale3(real y)’
     scales by ‘y’ in the y direction;
‘zscale3(real z)’
     scales by ‘z’ in the z direction;
‘scale3(real s)’
     scales by ‘s’ in the x, y, and z directions;
‘scale(real x, real y, real z)’
     scales by ‘x’ in the x direction, by ‘y’ in the y direction, and by
     ‘z’ in the z direction;
‘rotate(real angle, triple v)’
     rotates by ‘angle’ in degrees about the axis ‘O--v’;
‘rotate(real angle, triple u, triple v)’
     rotates by ‘angle’ in degrees about the axis ‘u--v’;
‘reflect(triple u, triple v, triple w)’
     reflects about the plane through ‘u’, ‘v’, and ‘w’.

   When not multiplied on the left by a transform3, three-dimensional
TeX Labels are drawn as Bezier surfaces directly on the projection
plane:
void label(picture pic=currentpicture, Label L, triple position,
           align align=NoAlign, pen p=currentpen,
           light light=nolight, string name="",
           render render=defaultrender, interaction interaction=
           settings.autobillboard ? Billboard : Embedded)
The optional ‘name’ parameter is used as a prefix for naming the label
patches in the PRC model tree.  The default interaction is ‘Billboard’,
which means that labels are rotated interactively so that they always
face the camera.  The interaction ‘Embedded’ means that the label
interacts as a normal ‘3D’ surface, as illustrated in the example
‘billboard.asy’.  Alternatively, a label can be transformed from the
‘XY’ plane by an explicit transform3 or mapped to a specified
two-dimensional plane with the predefined transform3 types ‘XY’, ‘YZ’,
‘ZX’, ‘YX’, ‘ZY’, ‘ZX’.  There are also modified versions of these
transforms that take an optional argument ‘projection
P=currentprojection’ that rotate and/or flip the label so that it is
more readable from the initial viewpoint.

   A transform3 that projects in the direction ‘dir’ onto the plane with
normal ‘n’ through point ‘O’ is returned by
transform3 planeproject(triple n, triple O=O, triple dir=n);
One can use
triple normal(path3 p);
to find the unit normal vector to a planar three-dimensional path ‘p’.
As illustrated in the example ‘planeproject.asy’, a transform3 that
projects in the direction ‘dir’ onto the plane defined by a planar path
‘p’ is returned by
transform3 planeproject(path3 p, triple dir=normal(p));

   The functions
surface extrude(path p, triple axis=Z);
surface extrude(Label L, triple axis=Z);
return the surface obtained by extruding path ‘p’ or Label ‘L’ along
‘axis’.

   Three-dimensional versions of the path functions ‘length’, ‘size’,
‘point’, ‘dir’, ‘accel’, ‘radius’, ‘precontrol’, ‘postcontrol’,
‘arclength’, ‘arctime’, ‘reverse’, ‘subpath’, ‘intersect’,
‘intersections’, ‘intersectionpoint’, ‘intersectionpoints’, ‘min’,
‘max’, ‘cyclic’, and ‘straight’ are also defined.

   The routine
real[] intersect(path3 p, surface s, real fuzz=-1);
returns a real array of length 3 containing the intersection times, if
any, of a path ‘p’ with a surface ‘s’.  The routine
real[][] intersections(path3 p, surface s, real fuzz=-1);
returns all (unless there are infinitely many) intersection times of a
path ‘p’ with a surface ‘s’ as a sorted array of real arrays of length
3, and
triple[] intersectionpoints(path3 p, surface s, real fuzz=-1);
returns the corresponding intersection points.  Here, the computations
are performed to the absolute error specified by ‘fuzz’, or if ‘fuzz <
0’, to machine precision.  The routine
real orient(triple a, triple b, triple c, triple d);
is a numerically robust computation of ‘dot(cross(a-d,b-d),c-d)’, which
is the determinant
|a.x a.y a.z 1|
|b.x b.y b.z 1|
|c.x c.y c.z 1|
|d.x d.y d.z 1|
   The result is negative (positive) if ‘a’, ‘b’, ‘c’ appear in
counterclockwise (clockwise) order when viewed from ‘d’ or zero if all
four points are coplanar.

   The routine
real insphere(triple a, triple b, triple c, triple d, triple e);
returns a positive (negative) value if ‘e’ lies inside (outside) the
sphere passing through points ‘a,b,c,d’ oriented so that
‘dot(cross(a-d,b-d),c-d)’ is positive, or zero if all five points are
cospherical.  The value returned is the determinant
|a.x a.y a.z a.x^2+a.y^2+a.z^2 1|
|b.x b.y b.z b.x^2+b.y^2+b.z^2 1|
|c.x c.y c.z c.x^2+c.y^2+c.z^2 1|
|d.x d.y d.z d.x^2+d.y^2+d.z^2 1|
|e.x e.y e.z e.x^2+e.y^2+e.z^2 1|

   Here is an example showing all five guide3 connectors:
import graph3;

size(200);

currentprojection=orthographic(500,-500,500);

triple[] z=new triple[10];

z[0]=(0,100,0); z[1]=(50,0,0); z[2]=(180,0,0);

for(int n=3; n <= 9; ++n)
  z[n]=z[n-3]+(200,0,0);

path3 p=z[0]..z[1]---z[2]::{Y}z[3]
&z[3]..z[4]--z[5]::{Y}z[6]
&z[6]::z[7]---z[8]..{Y}z[9];

draw(p,grey+linewidth(4mm),currentlight);

xaxis3(Label(XY()*"$x$",align=-3Y),red,above=true);
yaxis3(Label(XY()*"$y$",align=-3X),red,above=true);

                               [./join3]

   Three-dimensional versions of bars or arrows can be drawn with one of
the specifiers ‘None’, ‘Blank’, ‘BeginBar3’, ‘EndBar3’ (or equivalently
‘Bar3’), ‘Bars3’, ‘BeginArrow3’, ‘MidArrow3’, ‘EndArrow3’ (or
equivalently ‘Arrow3’), ‘Arrows3’, ‘BeginArcArrow3’, ‘EndArcArrow3’ (or
equivalently ‘ArcArrow3’), ‘MidArcArrow3’, and ‘ArcArrows3’.
Three-dimensional bars accept the optional arguments ‘(real size=0,
triple dir=O)’.  If ‘size=O’, the default bar length is used; if
‘dir=O’, the bar is drawn perpendicular to the path and the initial
viewing direction.  The predefined three-dimensional arrowhead styles
are ‘DefaultHead3’, ‘HookHead3’, ‘TeXHead3’.  Versions of the
two-dimensional arrowheads lifted to three-dimensional space and aligned
according to the initial viewpoint (or an optionally specified ‘normal’
vector) are also defined: ‘DefaultHead2(triple normal=O)’,
‘HookHead2(triple normal=O)’, ‘TeXHead2(triple normal=O)’.  These are
illustrated in the example ‘arrows3.asy’.

   Module ‘three’ also defines the three-dimensional margins
‘NoMargin3’, ‘BeginMargin3’, ‘EndMargin3’, ‘Margin3’, ‘Margins3’,
‘BeginPenMargin2’, ‘EndPenMargin2’, ‘PenMargin2’, ‘PenMargins2’,
‘BeginPenMargin3’, ‘EndPenMargin3’, ‘PenMargin3’, ‘PenMargins3’,
‘BeginDotMargin3’, ‘EndDotMargin3’, ‘DotMargin3’, ‘DotMargins3’,
‘Margin3’, and ‘TrueMargin3’.

   The routine
void pixel(picture pic=currentpicture, triple v, pen p=currentpen,
           real width=1);
can be used to draw on picture ‘pic’ a pixel of width ‘width’ at
position ‘v’ using pen ‘p’.

   Further three-dimensional examples are provided in the files
‘near_earth.asy’, ‘conicurv.asy’, and (in the ‘animations’ subdirectory)
‘cube.asy’.

   Limited support for projected vector graphics (effectively
three-dimensional nonrendered ‘PostScript’) is available with the
setting ‘render=0’.  This currently only works for piecewise planar
surfaces, such as those produced by the parametric ‘surface’ routines in
the ‘graph3’ module.  Surfaces produced by the ‘solids’ module will also
be properly rendered if the parameter ‘nslices’ is sufficiently large.

   In the module ‘bsp’, hidden surface removal of planar pictures is
implemented using a binary space partition and picture clipping.  A
planar path is first converted to a structure ‘face’ derived from
‘picture’.  A ‘face’ may be given to a two-dimensional drawing routine
in place of any ‘picture’ argument.  An array of such faces may then be
drawn, removing hidden surfaces:
void add(picture pic=currentpicture, face[] faces,
         projection P=currentprojection);
   Labels may be projected to two dimensions, using projection ‘P’, onto
the plane passing through point ‘O’ with normal ‘cross(u,v)’ by
multiplying it on the left by the transform
transform transform(triple u, triple v, triple O=O,
                    projection P=currentprojection);

   Here is an example that shows how a binary space partition may be
used to draw a two-dimensional vector graphics projection of three
orthogonal intersecting planes:
size(6cm,0);
import bsp;

real u=2.5;
real v=1;

currentprojection=oblique;

path3 y=plane((2u,0,0),(0,2v,0),(-u,-v,0));
path3 l=rotate(90,Z)*rotate(90,Y)*y;
path3 g=rotate(90,X)*rotate(90,Y)*y;

face[] faces;
filldraw(faces.push(y),project(y),yellow);
filldraw(faces.push(l),project(l),lightgrey);
filldraw(faces.push(g),project(g),green);

add(faces);

                              [./planes]


File: asymptote.info,  Node: graph3,  Next: palette,  Prev: three,  Up: User modules

8.3 ‘graph3’
============

This module implements three-dimensional versions of the functions in
‘graph.asy’.  To draw an x axis in three dimensions, use the routine
void xaxis3(picture pic=currentpicture, Label L="", axis axis=YZZero,
            real xmin=-infinity, real xmax=infinity, pen p=currentpen,
            ticks3 ticks=NoTicks3, arrowbar3 arrow=None,
            margin3 margin=NoMargin3, bool above=false,
            projection P=currentprojection);
Analogous routines ‘yaxis’ and ‘zaxis’ can be used to draw y and z axes
in three dimensions.  There is also a routine for drawing all three
axis:
void axes3(picture pic=currentpicture,
           Label xlabel="", Label ylabel="", Label zlabel="",
           bool extend=false,
           triple min=(-infinity,-infinity,-infinity),
           triple max=(infinity,infinity,infinity),
           pen p=currentpen, pen py=p, pen pz=p,
           arrowbar3 arrow=None, margin3 margin=NoMargin3,
           projection P=currentprojection);

The predefined three-dimensional axis types are
axis YZEquals(real y, real z, triple align=O, bool extend=false);
axis XZEquals(real x, real z, triple align=O, bool extend=false);
axis XYEquals(real x, real y, triple align=O, bool extend=false);
axis YZZero(triple align=O, bool extend=false);
axis XZZero(triple align=O, bool extend=false);
axis XYZero(triple align=O, bool extend=false);
axis Bounds(int type=Both, int type2=Both, triple align=O, bool extend=false);
The optional ‘align’ parameter to these routines can be used to specify
the default axis and tick label alignments.  The ‘Bounds’ axis accepts
two type parameters, each of which must be one of ‘Min’, ‘Max’, or
‘Both’.  These parameters specify which of the four possible
three-dimensional bounding box edges should be drawn.

   The three-dimensional tick options are ‘NoTicks3’, ‘InTicks’,
‘OutTicks’, and ‘InOutTicks’.  These specify the tick directions for the
‘Bounds’ axis type; other axis types inherit the direction that would be
used for the ‘Bounds(Min,Min)’ axis.

   Here is an example of a helix and bounding box axes with ticks and
axis labels, using orthographic projection:
import graph3;

size(0,200);
size3(200,IgnoreAspect);

currentprojection=orthographic(4,6,3);

real x(real t) {return cos(2pi*t);}
real y(real t) {return sin(2pi*t);}
real z(real t) {return t;}

path3 p=graph(x,y,z,0,2.7,operator ..);

draw(p,Arrow3);

scale(true);

xaxis3(XZ()*"$x$",Bounds,red,InTicks(Label,2,2));
yaxis3(YZ()*"$y$",Bounds,red,InTicks(beginlabel=false,Label,2,2));
zaxis3(XZ()*"$z$",Bounds,red,InTicks);

                               [./helix]

   The next example illustrates three-dimensional x, y, and z axes,
without autoscaling of the axis limits:
import graph3;

size(0,200);
size3(200,IgnoreAspect);

currentprojection=perspective(dir(75,20));

scale(Linear,Linear,Log);

xaxis3("$x$",0,1,red,OutTicks(2,2));
yaxis3("$y$",0,1,red,OutTicks(2,2));
zaxis3("$z$",1,30,red,OutTicks(beginlabel=false));

                               [./axis3]

   One can also place ticks along a general three-dimensional axis:
import graph3;

size(0,100);

path3 g=yscale3(2)*unitcircle3;
currentprojection=perspective(10,10,10);

axis(Label("C",position=0,align=15X),g,InTicks(endlabel=false,8,end=false),
     ticklocate(0,360,new real(real v) {
         path3 h=O--max(abs(max(g)),abs(min(g)))*dir(90,v);
         return intersect(g,h)[0];},
       new triple(real t) {return cross(dir(g,t),Z);}));


                           [./generalaxis3]

   Surface plots of matrices and functions over the region ‘box(a,b)’ in
the XY plane are also implemented:
surface surface(real[][] f, pair a, pair b, bool[][] cond={});
surface surface(real[][] f, pair a, pair b, splinetype xsplinetype,
                splinetype ysplinetype=xsplinetype, bool[][] cond={});
surface surface(real[][] f, real[] x, real[] y,
                splinetype xsplinetype=null, splinetype ysplinetype=xsplinetype,
                bool[][] cond={})
surface surface(triple[][] f, bool[][] cond={});
surface surface(real f(pair z), pair a, pair b, int nx=nmesh, int ny=nx,
                bool cond(pair z)=null);
surface surface(real f(pair z), pair a, pair b, int nx=nmesh, int ny=nx,
                splinetype xsplinetype, splinetype ysplinetype=xsplinetype,
                bool cond(pair z)=null);
surface surface(triple f(pair z), real[] u, real[] v,
                splinetype[] usplinetype, splinetype[] vsplinetype=Spline,
                bool cond(pair z)=null);
surface surface(triple f(pair z), pair a, pair b, int nu=nmesh, int nv=nu,
                bool cond(pair z)=null);
surface surface(triple f(pair z), pair a, pair b, int nu=nmesh, int nv=nu,
                splinetype[] usplinetype, splinetype[] vsplinetype=Spline,
                bool cond(pair z)=null);
The final two versions draw parametric surfaces for a function f(u,v)
over the parameter space ‘box(a,b)’, as illustrated in the example
‘parametricsurface.asy’.  An optional splinetype ‘Spline’ may be
specified.  The boolean array or function ‘cond’ can be used to control
which surface mesh cells are actually drawn (by default all mesh cells
over ‘box(a,b)’ are drawn).

   One can also construct the surface generated by rotating a path ‘g’
between ‘angle1’ to ‘angle2’ (in degrees) sampled ‘n’ times about the
line ‘c--c+axis’:
surface surface(triple c, path3 g, triple axis, int n=nslice,
                real angle1=0, real angle2=360, pen color(int i, real j)=null);
The optional argument ‘color(int i, real j)’ can be used to override the
surface color at the point obtained by rotating vertex ‘i’ by angle ‘j’
(in degrees).

Surface lighting is illustrated in the example files
‘parametricsurface.asy’ and ‘sinc.asy’.  Lighting can be disabled by
setting ‘light=nolight’, as in this example of a Gaussian surface:
import graph3;

size(200,0);

currentprojection=perspective(10,8,4);

real f(pair z) {return 0.5+exp(-abs(z)^2);}

draw((-1,-1,0)--(1,-1,0)--(1,1,0)--(-1,1,0)--cycle);

draw(arc(0.12Z,0.2,90,60,90,25),ArcArrow3);

surface s=surface(f,(-1,-1),(1,1),nx=5,Spline);

xaxis3(Label("$x$"),red,Arrow3);
yaxis3(Label("$y$"),red,Arrow3);
zaxis3(XYZero(extend=true),red,Arrow3);

draw(s,lightgray,meshpen=black+thick(),nolight,render(merge=true));

label("$O$",O,-Z+Y,red);

                          [./GaussianSurface]
A mesh can be drawn without surface filling by specifying ‘nullpen’ for
the surfacepen.

   A vector field of ‘nu’\times‘nv’ arrows on a parametric surface ‘f’
over ‘box(a,b)’ can be drawn with the routine
picture vectorfield(path3 vector(pair v), triple f(pair z), pair a, pair b,
                    int nu=nmesh, int nv=nu, bool truesize=false,
                    real maxlength=truesize ? 0 : maxlength(f,a,b,nu,nv),
                    bool cond(pair z)=null, pen p=currentpen,
                    arrowbar3 arrow=Arrow3, margin3 margin=PenMargin3)
   as illustrated in the examples ‘vectorfield3.asy’ and
‘vectorfieldsphere.asy’.


File: asymptote.info,  Node: palette,  Next: colormap,  Prev: graph3,  Up: User modules

8.4 ‘palette’
=============

‘Asymptote’ can also generate color density images and palettes.  The
following palettes are predefined in ‘palette.asy’:

‘pen[] Grayscale(int NColors=256)’
     a grayscale palette;

‘pen[] Rainbow(int NColors=32766)’
     a rainbow spectrum;

‘pen[] BWRainbow(int NColors=32761)’
     a rainbow spectrum tapering off to black/white at the ends;

‘pen[] BWRainbow2(int NColors=32761)’
     a double rainbow palette tapering off to black/white at the ends,
     with a linearly scaled intensity.

‘pen[] Wheel(int NColors=32766)’
     a full color wheel palette;

‘pen[] Gradient(int NColors=256, ... pen[] p)’
     a palette varying linearly over the specified array of pens, using
     NColors in each interpolation interval;

   The function ‘cmyk(pen[] Palette)’ may be used to convert any of
these palettes to the CMYK colorspace.

   A color density plot using palette ‘palette’ can be generated from a
function ‘f’(x,y) and added to a picture ‘pic’:
bounds image(picture pic=currentpicture, real f(real, real),
             range range=Full, pair initial, pair final,
             int nx=ngraph, int ny=nx, pen[] palette, int divs=0,
             bool antialias=false)
   The function ‘f’ will be sampled at ‘nx’ and ‘ny’ evenly spaced
points over a rectangle defined by the points ‘initial’ and ‘final’,
respecting the current graphical scaling of ‘pic’.  The color space is
scaled according to the z axis scaling (*note automatic scaling::).  If
‘divs’ > 1, the palette is quantized to ‘divs’-1 values.  A ‘bounds’
structure for the function values is returned:
struct bounds {
  real min;
  real max;
  // Possible tick intervals:
  int[] divisor;
}
This information can be used for generating an optional palette bar.
The palette color space corresponds to a range of values specified by
the argument ‘range’, which can be ‘Full’, ‘Automatic’, or an explicit
range ‘Range(real min, real max)’.  (The type ‘range’ is defined in
‘palette.asy’.)  Here ‘Full’ specifies a range varying from the minimum
to maximum values of the function over the sampling interval, while
‘Automatic’ selects "nice" limits.  The examples ‘fillcontour.asy’ and
‘imagecontour.asy’ illustrate how level sets (contour lines) can be
drawn on a color density plot (*note contour::).

   A color density plot can also be generated from an explicit real[][]
array ‘data’:
bounds image(picture pic=currentpicture, real[][] f, range range=Full,
             pair initial, pair final, pen[] palette, int divs=0,
             bool transpose=(initial.x < final.x && initial.y < final.y),
             bool copy=true, bool antialias=false);
If the initial point is to the left and below the final point, by
default the array indices are interpreted according to the Cartesian
convention (first index: x, second index: y) rather than the usual
matrix convention (first index: -y, second index: x).

   To construct an image from an array of irregularly spaced points and
an array of values ‘f’ at these points, use one of the routines
bounds image(picture pic=currentpicture, pair[] z, real[] f,
             range range=Full, pen[] palette)
bounds image(picture pic=currentpicture, real[] x, real[] y, real[] f,
             range range=Full, pen[] palette)

   An optionally labelled palette bar may be generated with the routine
void palette(picture pic=currentpicture, Label L="", bounds bounds,
             pair initial, pair final, axis axis=Right, pen[] palette,
             pen p=currentpen, paletteticks ticks=PaletteTicks,
             bool copy=true, bool antialias=false);
   The color space of ‘palette’ is taken to be over bounds ‘bounds’ with
scaling given by the z scaling of ‘pic’.  The palette orientation is
specified by ‘axis’, which may be one of ‘Right’, ‘Left’, ‘Top’, or
‘Bottom’.  The bar is drawn over the rectangle from ‘initial’ to
‘final’.  The argument ‘paletteticks’ is a special tick type (*note
ticks::) that takes the following arguments:
paletteticks PaletteTicks(Label format="", ticklabel ticklabel=null,
                          bool beginlabel=true, bool endlabel=true,
                          int N=0, int n=0, real Step=0, real step=0,
                          pen pTick=nullpen, pen ptick=nullpen);

   The image and palette bar can be fit to a frame and added and
optionally aligned to a picture at the desired location:

size(12cm,12cm);

import graph;
import palette;

int n=256;
real ninv=2pi/n;
real[][] v=new real[n][n];

for(int i=0; i < n; ++i)
  for(int j=0; j < n; ++j)
    v[i][j]=sin(i*ninv)*cos(j*ninv);

pen[] Palette=BWRainbow();

picture bar;

bounds range=image(v,(0,0),(1,1),Palette);
palette(bar,"$A$",range,(0,0),(0.5cm,8cm),Right,Palette,
        PaletteTicks("$%+#.1f$"));
add(bar.fit(),point(E),30E);

                               [./image]

   Here is an example that uses logarithmic scaling of the function
values:

import graph;
import palette;

size(10cm,10cm,IgnoreAspect);

real f(real x, real y) {
  return 0.9*pow10(2*sin(x/5+2*y^0.25)) + 0.1*(1+cos(10*log(y)));
}

scale(Linear,Log,Log);

pen[] Palette=BWRainbow();

bounds range=image(f,Automatic,(0,1),(100,100),nx=200,Palette);

xaxis("$x$",BottomTop,LeftTicks,above=true);
yaxis("$y$",LeftRight,RightTicks,above=true);

palette("$f(x,y)$",range,(0,200),(100,250),Top,Palette,
        PaletteTicks(ptick=linewidth(0.5*linewidth())));



                             [./logimage]

   One can also draw an image directly from a two-dimensional pen array
or a function ‘pen f(int, int)’:
void image(picture pic=currentpicture, pen[][] data,
           pair initial, pair final,
           bool transpose=(initial.x < final.x && initial.y < final.y),
           bool copy=true, bool antialias=false);
void image(picture pic=currentpicture, pen f(int, int), int width, int height,
           pair initial, pair final,
           bool transpose=(initial.x < final.x && initial.y < final.y),
           bool antialias=false);
as illustrated in the following examples:

size(200);

import palette;

int n=256;
real ninv=2pi/n;
pen[][] v=new pen[n][n];

for(int i=0; i < n; ++i)
  for(int j=0; j < n; ++j)
    v[i][j]=rgb(0.5*(1+sin(i*ninv)),0.5*(1+cos(j*ninv)),0);

image(v,(0,0),(1,1));


                             [./penimage]

import palette;

size(200);

real fracpart(real x) {return (x-floor(x));}

pair pws(pair z) {
  pair w=(z+exp(pi*I/5)/0.9)/(1+z/0.9*exp(-pi*I/5));
  return exp(w)*(w^3-0.5*I);
}

int N=512;

pair a=(-1,-1);
pair b=(0.5,0.5);
real dx=(b-a).x/N;
real dy=(b-a).y/N;

pen f(int u, int v) {
  pair z=a+(u*dx,v*dy);
  pair w=pws(z);
  real phase=degrees(w,warn=false);
  real modulus=w == 0 ? 0: fracpart(log(abs(w)));
  return hsv(phase,1,sqrt(modulus));
}

image(f,N,N,(0,0),(300,300),antialias=true);

                         [./penfunctionimage]

   For convenience, the module ‘palette’ also defines functions that may
be used to construct a pen array from a given function and palette:
pen[] palette(real[] f, pen[] palette);
pen[][] palette(real[][] f, pen[] palette);


File: asymptote.info,  Node: colormap,  Next: texcolors,  Prev: palette,  Up: User modules

8.5 ‘colormap’
==============

This module implements a list of colormaps converted from the Python
library matplotlib.  The colormaps can be used to generate palettes with
the ‘palette()’ method.  For example:
import colormap;
pen[] Palette = wistia.palette();
   There are two types of palettes: segmented palettes (which accept
parameters for number of colors and gamma) and listed palettes (which
return a fixed set of colors).  See
<https://matplotlib.org/tutorials/colors/colormaps.html> for the
complete list of available colormaps.


File: asymptote.info,  Node: texcolors,  Next: x11colors,  Prev: colormap,  Up: User modules

8.6 ‘texcolors’
===============

This module defines the standard 68 CMYK TeX colors as named pens.
These can be imported with:
import texcolors;
   The colors include names like ‘GreenYellow’, ‘Goldenrod’, ‘Melon’,
‘Mahogany’, and many others defined in the standard TeX color set.


File: asymptote.info,  Node: x11colors,  Next: fontsize,  Prev: texcolors,  Up: User modules

8.7 ‘x11colors’
===============

This module defines the standard 140 RGB ‘X11’ colors as named pens.
These can be imported with:
import x11colors;
   The colors include names like ‘AliceBlue’, ‘Aquamarine’, ‘CadetBlue’,
‘Chartreuse’, and many others from the standard ‘X11’ color set.

   Note that there is some overlap between the TeX and ‘X11’ color
standards, and the definitions of some colors (such as ‘Green’) actually
disagree.


File: asymptote.info,  Node: fontsize,  Next: contour,  Prev: x11colors,  Up: User modules

8.8 ‘fontsize’
==============

This module provides support for nonstandard font sizes by loading the
‘type1cm’ TeX package.  Import this module at the beginning of your file
to use arbitrary font sizes with ‘fontsize()’:
import fontsize;
defaultpen(fontsize(10pt));


File: asymptote.info,  Node: contour,  Next: geometry,  Prev: fontsize,  Up: User modules

8.9 ‘contour’
=============

This module draws contour lines.  To construct contours corresponding to
the values in a real array ‘c’ for a function ‘f’ on ‘box(a,b)’, use the
routine
guide[][] contour(real f(real, real), pair a, pair b,
                  real[] c, int nx=ngraph, int ny=nx,
                  interpolate join=operator --, int subsample=1);
The integers ‘nx’ and ‘ny’ define the resolution.  The default
resolution, ‘ngraph x ngraph’ (here ‘ngraph’ defaults to ‘100’) can be
increased for greater accuracy.  The default interpolation operator is
‘operator --’ (linear).  Spline interpolation (‘operator ..’) may
produce smoother contours but it can also lead to overshooting.  The
‘subsample’ parameter indicates the number of interior points that
should be used to sample contours within each ‘1 x 1’ box; the default
value of ‘1’ is usually sufficient.

   To construct contours for an array of data values on a uniform
two-dimensional lattice on ‘box(a,b)’, use
guide[][] contour(real[][] f, pair a, pair b, real[] c,
                  interpolate join=operator --, int subsample=1);

   To construct contours for an array of data values on a nonoverlapping
regular mesh specified by the two-dimensional array ‘z’,
guide[][] contour(pair[][] z, real[][] f, real[] c,
                  interpolate join=operator --, int subsample=1);

To construct contours for an array of values ‘f’ specified at
irregularly positioned points ‘z’, use the routine
guide[][] contour(pair[] z, real[] f, real[] c, interpolate join=operator --);
The contours themselves can be drawn with one of the routines
void draw(picture pic=currentpicture, Label[] L=new Label[],
          guide[][] g, pen p=currentpen);

void draw(picture pic=currentpicture, Label[] L=new Label[],
          guide[][] g, pen[] p);

   The following simple example draws the contour at value ‘1’ for the
function z=x^2+y^2, which is a unit circle:
import contour;
size(75);

real f(real a, real b) {return a^2+b^2;}
draw(contour(f,(-1,-1),(1,1),new real[] {1}));

                            [./onecontour]

   The next example draws and labels multiple contours for the function
z=x^2-y^2 with the resolution ‘100 x 100’, using a dashed pen for
negative contours and a solid pen for positive (and zero) contours:
import contour;

size(200);

real f(real x, real y) {return x^2-y^2;}
int n=10;
real[] c=new real[n];
for(int i=0; i < n; ++i) c[i]=(i-n/2)/n;

pen[] p=sequence(new pen(int i) {
    return (c[i] >= 0 ? solid : dashed)+fontsize(6pt);
  },c.length);

Label[] Labels=sequence(new Label(int i) {
    return Label(c[i] != 0 ? (string) c[i] : "",Relative(unitrand()),(0,0),
                 UnFill(1bp));
  },c.length);

draw(Labels,contour(f,(-1,-1),(1,1),c),p);

                           [./multicontour]

   The next examples illustrates how contour lines can be drawn on color
density images, with and without palette quantization:
import graph;
import palette;
import contour;

size(10cm,10cm);

pair a=(0,0);
pair b=(2pi,2pi);

real f(real x, real y) {return cos(x)*sin(y);}

int N=200;
int Divs=10;
int divs=1;
int n=Divs*divs;

defaultpen(1bp);
pen Tickpen=black;
pen tickpen=gray+0.5*linewidth(currentpen);
pen[] Palette=quantize(BWRainbow(),n);

bounds range=image(f,Automatic,a,b,3N,Palette,n);

// Major contours
real[] Cvals=uniform(range.min,range.max,Divs);
draw(contour(f,a,b,Cvals,N,operator --),Tickpen+squarecap+beveljoin);

// Minor contours (if divs > 1)
real[] cvals;
for(int i=0; i < Cvals.length-1; ++i)
  cvals.append(uniform(Cvals[i],Cvals[i+1],divs)[1:divs]);
draw(contour(f,a,b,cvals,N,operator --),tickpen);

xaxis("$x$",BottomTop,LeftTicks,above=true);
yaxis("$y$",LeftRight,RightTicks,above=true);

palette("$f(x,y)$",range,point(SE)+(0.5,0),point(NE)+(1,0),Right,Palette,
        PaletteTicks("$%+#0.1f$",N=Divs,n=divs,Tickpen,tickpen));

                            [./fillcontour]

import graph;
import palette;
import contour;

size(10cm,10cm);

pair a=(0,0);
pair b=(2pi,2pi);

real f(real x, real y) {return cos(x)*sin(y);}

int N=200;
int Divs=10;
int divs=1;

defaultpen(1bp);
pen Tickpen=black;
pen tickpen=gray+0.5*linewidth(currentpen);
pen[] Palette=BWRainbow();

bounds range=image(f,Automatic,a,b,N,Palette);

// Major contours
real[] Cvals=uniform(range.min,range.max,Divs);
draw(contour(f,a,b,Cvals,N,operator --),Tickpen+squarecap+beveljoin);

// Minor contours (if divs > 1)
real[] cvals;
for(int i=0; i < Cvals.length-1; ++i)
  cvals.append(uniform(Cvals[i],Cvals[i+1],divs)[1:divs]);
draw(contour(f,a,b,cvals,N,operator --),tickpen+squarecap+beveljoin);

xaxis("$x$",BottomTop,LeftTicks,above=true);
yaxis("$y$",LeftRight,RightTicks,above=true);

palette("$f(x,y)$",range,point(SE)+(0.5,0),point(NE)+(1,0),Right,Palette,
        PaletteTicks("$%+#0.1f$",N=Divs,n=divs,Tickpen,tickpen));

                           [./imagecontour]

   Finally, here is an example that illustrates the construction of
contours from irregularly spaced data:
import contour;

size(200);

int n=100;

real f(real a, real b) {return a^2+b^2;}

srand(1);

real r() {return 1.1*(rand()/randMax*2-1);}

pair[] points=new pair[n];
real[] values=new real[n];

for(int i=0; i < n; ++i) {
  points[i]=(r(),r());
  values[i]=f(points[i].x,points[i].y);
}

draw(contour(points,values,new real[]{0.25,0.5,1},operator ..),blue);

                         [./irregularcontour]

   In the above example, the contours of irregularly spaced data are
constructed by first creating a triangular mesh from an array ‘z’ of
pairs:

int[][] triangulate(pair[] z);

size(200);
int np=100;
pair[] points;

real r() {return 1.2*(rand()/randMax*2-1);}

for(int i=0; i < np; ++i)
  points.push((r(),r()));

int[][] trn=triangulate(points);

for(int i=0; i < trn.length; ++i) {
  draw(points[trn[i][0]]--points[trn[i][1]]);
  draw(points[trn[i][1]]--points[trn[i][2]]);
  draw(points[trn[i][2]]--points[trn[i][0]]);
}

for(int i=0; i < np; ++i)
  dot(points[i],red);

                            [./triangulate]

   The example ‘Gouraudcontour.asy’ illustrates how to produce color
density images over such irregular triangular meshes.  ‘Asymptote’ uses
a robust version of Paul Bourke's Delaunay triangulation algorithm based
on the public-domain exact arithmetic predicates written by Jonathan
Shewchuk.


File: asymptote.info,  Node: geometry,  Next: grid3,  Prev: contour,  Up: User modules

8.10 ‘geometry’
===============

This module, written by Philippe Ivaldi, provides an extensive set of
geometry routines, including ‘perpendicular’ symbols and a ‘triangle’
structure.  Link to the documentation for the ‘geometry’ module are
posted here: <https://asymptote.sourceforge.io/links.html>, including an
extensive set of examples,
<https://web.archive.org/web/20201130113133/http://www.piprime.fr/files/asymptote/geometry/>,
and an index:
     <https://web.archive.org/web/20201130113133/http://www.piprime.fr/files/asymptote/geometry/modules/geometry.asy.index.type.html>


File: asymptote.info,  Node: grid3,  Next: interpolate,  Prev: geometry,  Up: User modules

8.11 ‘grid3’
============

This module, contributed by Philippe Ivaldi, can be used for drawing 3D
grids.  Here is an example (further examples can be found in ‘grid3.asy’
and at
<https://web.archive.org/web/20201130113133/http://www.piprime.fr/files/asymptote/grid3/>):
import grid3;

size(8cm,0,IgnoreAspect);
currentprojection=orthographic(0.5,1,0.5);

scale(Linear, Linear, Log);

limits((-2,-2,1),(0,2,100));

grid3(XYZgrid);

xaxis3(Label("$x$",position=EndPoint,align=S),Bounds(Min,Min),
       OutTicks());
yaxis3(Label("$y$",position=EndPoint,align=S),Bounds(Min,Min),OutTicks());
zaxis3(Label("$z$",position=EndPoint,align=(-1,0.5)),Bounds(Min,Min),
       OutTicks(beginlabel=false));


                             [./grid3xyz]


File: asymptote.info,  Node: interpolate,  Next: markers,  Prev: grid3,  Up: User modules

8.12 ‘interpolate’
==================

This module implements Lagrange, Hermite, and standard cubic spline
interpolation in ‘Asymptote’, as illustrated in the example
‘interpolate1.asy’.


File: asymptote.info,  Node: markers,  Next: math,  Prev: interpolate,  Up: User modules

8.13 ‘markers’
==============

This module implements specialized routines for marking paths and
angles.  The principal mark routine provided by this module is
markroutine markinterval(int n=1, frame f, bool rotated=false);
which centers ‘n’ copies of frame ‘f’ within uniformly space intervals
in arclength along the path, optionally rotated by the angle of the
local tangent.

   The ‘marker’ (*note marker::) routine can be used to construct new
markers from these predefined frames:

frame stickframe(int n=1, real size=0, pair space=0, real angle=0,
                 pair offset=0, pen p=currentpen);
frame circlebarframe(int n=1, real barsize=0,
                     real radius=0,real angle=0,
                     pair offset=0, pen p=currentpen,
                     filltype filltype=NoFill, bool above=false);
frame crossframe(int n=3, real size=0, pair space=0,
                 real angle=0, pair offset=0, pen p=currentpen);
frame tildeframe(int n=1, real size=0, pair space=0,
                 real angle=0, pair offset=0, pen p=currentpen);

   For convenience, this module also constructs the markers
‘StickIntervalMarker’, ‘CrossIntervalMarker’, ‘CircleBarIntervalMarker’,
and ‘TildeIntervalMarker’ from the above frames.  The example
‘markers1.asy’ illustrates the use of these markers:


                             [./markers1]

   This module also provides a routine for marking an angle AOB:
void markangle(picture pic=currentpicture, Label L="",
               int n=1, real radius=0, real space=0,
               pair A, pair O, pair B, arrowbar arrow=None,
               pen p=currentpen, margin margin=NoMargin,
               marker marker=nomarker);
as illustrated in the example ‘markers2.asy’.


                             [./markers2]


File: asymptote.info,  Node: math,  Next: patterns,  Prev: markers,  Up: User modules

8.14 ‘math’
===========

This module extends ‘Asymptote’'s mathematical capabilities with useful
functions such as

‘void drawline(picture pic=currentpicture’
     draw the visible portion of the (infinite) line going through ‘P’
     and ‘Q’, without altering the size of picture ‘pic’, using pen ‘p’.

‘real intersect(triple P, triple Q, triple n, triple Z);’
     returns the intersection time of the extension of the line segment
     ‘PQ’ with the plane perpendicular to ‘n’ and passing through ‘Z’.

‘triple intersectionpoint(triple n0, triple P0, triple n1, triple P1);’
     Return any point on the intersection of the two planes with normals
     ‘n0’ and ‘n1’ passing through points ‘P0’ and ‘P1’, respectively.
     If the planes are parallel, return ‘(infinity,infinity,infinity)’.

‘pair[] quarticroots(real a, real b, real c, real d, real e);’
     returns the four complex roots of the quartic equation
     ax^4+bx^3+cx^2+dx+e=0.

‘real time(path g, real x, int n=0, real fuzz=-1)’
     returns the ‘n’th intersection time of path ‘g’ with the vertical
     line through x.

‘real time(path g, explicit pair z, int n=0, real fuzz=-1)’
     returns the ‘n’th intersection time of path ‘g’ with the horizontal
     line through ‘(0,z.y)’.

‘real value(path g, real x, int n=0, real fuzz=-1)’
     returns the ‘n’th ‘y’ value of ‘g’ at ‘x’.

‘real value(path g, explicit pair z, int n=0, real fuzz=-1)’
     returns the ‘n’th ‘x’ value of ‘g’ at ‘y=z.y’.

‘real slope(path g, real x, int n=0, real fuzz=-1)’
     returns the ‘n’th slope of ‘g’ at ‘x’.

‘real slope(path g, explicit pair z, int n=0, real fuzz=-1)’
     returns the ‘n’th slope of ‘g’ at ‘y=z.y’.

     int[][] segment(bool[] b) returns the indices of consecutive
     true-element segments of bool[] ‘b’.

‘real[] partialsum(real[] a)’
     returns the partial sums of a real array ‘a’.

‘real[] partialsum(real[] a, real[] dx)’
     returns the partial ‘dx’-weighted sums of a real array ‘a’.

‘bool increasing(real[] a, bool strict=false)’
     returns, if ‘strict=false’, whether ‘i > j’ implies ‘a[i] >= a[j]’,
     or if ‘strict=true’, whether ‘i > j’ implies implies ‘a[i] > a[j]’.

‘int unique(real[] a, real x)’
     if the sorted array ‘a’ does not contain ‘x’, insert it
     sequentially, returning the index of ‘x’ in the resulting array.

‘bool lexorder(pair a, pair b)’
     returns the strict lexicographical partial order of ‘a’ and ‘b’.

‘bool lexorder(triple a, triple b)’
     returns the strict lexicographical partial order of ‘a’ and ‘b’.


File: asymptote.info,  Node: patterns,  Next: roundedpath,  Prev: math,  Up: User modules

8.15 ‘patterns’
===============

This module implements ‘PostScript’ tiling patterns and includes several
convenient pattern generation routines.


File: asymptote.info,  Node: roundedpath,  Next: slopefield,  Prev: patterns,  Up: User modules

8.16 ‘roundedpath’
==================

This module, contributed by Stefan Knorr, is useful for rounding the
sharp corners of paths, as illustrated in the example file
‘roundpath.asy’.


File: asymptote.info,  Node: slopefield,  Next: smoothcontour3,  Prev: roundedpath,  Up: User modules

8.17 ‘slopefield’
=================

To draw a slope field for the differential equation dy/dx=f(x,y) (or
dy/dx=f(x)), use:
picture slopefield(real f(real,real), pair a, pair b,
                   int nx=nmesh, int ny=nx,
                   real tickfactor=0.5, pen p=currentpen,
                   arrowbar arrow=None);
Here, the points ‘a’ and ‘b’ are the lower left and upper right corners
of the rectangle in which the slope field is to be drawn, ‘nx’ and ‘ny’
are the respective number of ticks in the x and y directions,
‘tickfactor’ is the fraction of the minimum cell dimension to use for
drawing ticks, and ‘p’ is the pen to use for drawing the slope fields.
The return value is a picture that can be added to ‘currentpicture’ via
the ‘add(picture)’ command.

   The function
path curve(pair c, real f(real,real), pair a, pair b);
takes a point (‘c’) and a slope field-defining function ‘f’ and returns,
as a path, the curve passing through that point.  The points ‘a’ and ‘b’
represent the rectangular boundaries over which the curve is
interpolated.

   Both ‘slopefield’ and ‘curve’ alternatively accept a function ‘real
f(real)’ that depends on x only, as seen in this example:

import slopefield;

size(200);

real func(real x) {return 2x;}
add(slopefield(func,(-3,-3),(3,3),20));
draw(curve((0,0),func,(-3,-3),(3,3)),red);

                            [./slopefield1]


File: asymptote.info,  Node: smoothcontour3,  Next: solids,  Prev: slopefield,  Up: User modules

8.18 ‘smoothcontour3’
=====================

This module, written by Charles Staats, draws implicitly defined
surfaces with smooth appearance.  The purpose of this module is similar
to that of ‘contour3’: given a real-valued function f(x,y,z), construct
the surface described by the equation f(x,y,z) = 0.  The
‘smoothcontour3’ module generally produces nicer results than
‘contour3’, but takes longer to compile.  Additionally, the algorithm
assumes that the function and the surface are both smooth; if they are
not, then ‘contour3’ may be a better choice.

   To construct the null surface of a function ‘f(triple)’ or
‘ff(real,real,real)’ over ‘box(a,b)’, use the routine
surface implicitsurface(real f(triple)=null,
                        real ff(real,real,real)=null,
                        triple a,
                        triple b,
                        int n=nmesh,
                        bool keyword overlapedges=false,
                        int keyword nx=n,
                        int keyword ny=n,
                        int keyword nz=n,
                        int keyword maxdepth=8,
                        bool usetriangles=true);
The optional parameter ‘overlapedges’ attempts to compensate for an
artifact that can cause the renderer to "see through" the boundary
between patches.  Although it defaults to ‘false’, it should usually be
set to ‘true’.  The example ‘genustwo.asy’ illustrates the use of this
function.  Additional examples, together with a more in-depth
explanation of the module's usage and pitfalls, are available at
<https://github.com/charlesstaats/smoothcontour3>.


File: asymptote.info,  Node: solids,  Next: stats,  Prev: smoothcontour3,  Up: User modules

8.19 ‘solids’
=============

This solid geometry module defines a structure ‘revolution’ that can be
used to fill and draw surfaces of revolution.  The following example
uses it to display the outline of a circular cylinder of radius 1 with
axis ‘O--1.5unit(Y+Z)’ with perspective projection:
import solids;

size(0,100);

revolution r=cylinder(O,1,1.5,Y+Z);
draw(r,heavygreen);

                         [./cylinderskeleton]

   Further illustrations are provided in the example files
‘cylinder.asy’, ‘cones.asy’, ‘hyperboloid.asy’, and ‘torus.asy’.

   The structure ‘skeleton’ contains the three-dimensional wireframe
used to visualize a volume of revolution:
struct skeleton {
  struct curve {
    path3[] front;
    path3[] back;
  }
  // transverse skeleton (perpendicular to axis of revolution)
  curve transverse;
  // longitudinal skeleton (parallel to axis of revolution)
  curve longitudinal;
}


File: asymptote.info,  Node: stats,  Next: tube,  Prev: solids,  Up: User modules

8.20 ‘stats’
============

This module implements a Gaussian random number generator and a
collection of statistics routines, including ‘histogram’ and
‘leastsquares’.


File: asymptote.info,  Node: tube,  Prev: stats,  Up: User modules

8.21 ‘tube’
===========

This module extends the ‘tube’ surfaces constructed in
‘three_arrows.asy’ to arbitrary cross sections, colors, and spine
transformations.  The routine
surface tube(path3 g, coloredpath section,
             transform T(real)=new transform(real t) {return identity();},
             real corner=1, real relstep=0);
draws a tube along ‘g’ with cross section ‘section’, after applying the
transformation ‘T(t)’ at ‘point(g,t)’.  The parameter ‘corner’ controls
the number of elementary tubes at the angular points of ‘g’.  A nonzero
value of ‘relstep’ specifies a fixed relative time step (in the sense of
‘relpoint(g,t)’) to use in constructing elementary tubes along ‘g’.  The
type ‘coloredpath’ is a generalization of ‘path’ to which a ‘path’ can
be cast:
struct coloredpath
{
  path p;
  pen[] pens(real);
  int colortype=coloredSegments;
}
Here ‘p’ defines the cross section and the method ‘pens(real t)’ returns
an array of pens (interpreted as a cyclic array) used for shading the
tube patches at ‘relpoint(g,t)’.  If ‘colortype=coloredSegments’, the
tube patches are filled as if each segment of the section was colored
with the pen returned by ‘pens(t)’, whereas if ‘colortype=coloredNodes’,
the tube components are vertex shaded as if the nodes of the section
were colored.

   A ‘coloredpath’ can be constructed with one of the routines:
coloredpath coloredpath(path p, pen[] pens(real),
                        int colortype=coloredSegments);
coloredpath coloredpath(path p, pen[] pens=new pen[] {currentpen},
                        int colortype=coloredSegments);
coloredpath coloredpath(path p, pen pen(real));
In the second case, the pens are independent of the relative time.  In
the third case, the array of pens contains only one pen, which depends
of the relative time.

   The casting of ‘path’ to ‘coloredpath’ allows the use of a ‘path’
instead of a ‘coloredpath’; in this case the shading behavior is the
default shading behavior for a surface.

   An example of ‘tube’ is provided in the file ‘trefoilknot.asy’.
Further examples can be found at
<https://web.archive.org/web/20201130113133/http://www.piprime.fr/files/asymptote/tube>.


File: asymptote.info,  Node: Specialty modules,  Next: Developer modules,  Prev: User modules,  Up: Top

9 Specialty modules
*******************

These modules are for specialized applications.

* Menu:

* animation::                   Embedded PDF and MPEG movies
* animate::                     Portable clickable PDF movies with controls
* annotate::                    Annotate your PDF files
* babel::                       Interface to LaTeX ‘babel’ package
* binarytree::                  Binary tree drawing module
* bsp::                         Binary space partition for hidden surface removal
* CAD::                         2D CAD pen and measurement functions (DIN 15)
* contour3::                    Contour surfaces
* drawtree::                    Tree drawing module
* embed::                       Embedding movies, sounds, and 3D objects
* external::                    Portable movie embedding
* feynman::                     Feynman diagrams
* flowchart::                   Flowchart drawing routines
* labelpath::                   Drawing curved labels
* labelpath3::                  Drawing curved labels in 3D
* lmfit::                       Least-squares fitting
* MetaPost::                    ‘MetaPost’ compatibility routines
* obj::                         3D obj files
* ode::                         Ordinary differential equations
* pstoedit::                    ‘PostScript’ to ‘Asymptote’ conversion
* rational::                    Rational number arithmetic
* rationalSimplex::             Simplex method for rational numbers
* simplex::                     Linear programming: simplex method
* size10::                      Named font sizes for 10pt LaTeX documents
* size11::                      Named font sizes for 11pt LaTeX documents
* slide::                       Making presentations with ‘Asymptote’
* syzygy::                      Syzygy and braid drawing module
* tree::                        Dynamic binary search tree
* trembling::                   Wavy lines


File: asymptote.info,  Node: animation,  Next: animate,  Up: Specialty modules

9.1 ‘animation’
===============

This module allows one to generate animations, as illustrated by the
files ‘wheel.asy’, ‘wavepacket.asy’, and ‘cube.asy’ in the ‘animations’
subdirectory of the examples directory.  These animations use the
‘ImageMagick’ ‘magick’ program to merge multiple images into a GIF or
MPEG movie.

   The related ‘animate’ module, derived from the ‘animation’ module,
generates higher-quality portable clickable PDF movies, with optional
controls.  This requires installing the module
     <http://mirror.ctan.org/macros/latex/contrib/animate/animate.sty>
(version 2007/11/30 or later) in a new directory ‘animate’ in the local
LaTeX directory (for example, in
‘/usr/local/share/texmf/tex/latex/animate’).  On ‘UNIX’ systems, one
must then execute the command ‘texhash’.

   The example ‘pdfmovie.asy’ in the ‘animations’ directory, along with
the slide presentations ‘slidemovies.asy’ and ‘intro’, illustrate the
use of embedded PDF movies.  The examples ‘inlinemovie.tex’ and
‘inlinemovie3.tex’ show how to generate and embed PDF movies directly
within a LaTeX file (*note LaTeX usage::).  The member function
string pdf(fit fit=NoBox, real delay=animationdelay, string options="",
           bool keep=settings.keep, bool multipage=true);
of the ‘animate’ structure accepts any of the ‘animate.sty’ options, as
described here:
     <http://mirror.ctan.org/macros/latex/contrib/animate/doc/animate.pdf>


File: asymptote.info,  Node: animate,  Next: annotate,  Prev: animation,  Up: Specialty modules

9.2 ‘animate’
=============

This module loads the TeX ‘animate’ package before importing the
‘animation’ module to generate portable clickable PDF movies with
optional controls.


File: asymptote.info,  Node: annotate,  Next: babel,  Prev: animate,  Up: Specialty modules

9.3 ‘annotate’
==============

This module supports PDF annotations for viewing with ‘Adobe Reader’,
via the function
void annotate(picture pic=currentpicture, string title, string text,
              pair position);
Annotations are illustrated in the example file ‘annotation.asy’.
Currently, annotations are only implemented for the ‘latex’ (default)
and ‘tex’ TeX engines.


File: asymptote.info,  Node: babel,  Next: binarytree,  Prev: annotate,  Up: Specialty modules

9.4 ‘babel’
===========

This module implements the LaTeX ‘babel’ package in ‘Asymptote’.  For
example:
import babel;
babel("german");


File: asymptote.info,  Node: binarytree,  Next: bsp,  Prev: babel,  Up: Specialty modules

9.5 ‘binarytree’
================

This module can be used to draw an arbitrary binary tree and includes an
input routine for the special case of a binary search tree, as
illustrated in the example ‘binarytreetest.asy’:
import binarytree;

picture pic,pic2;

binarytree bt=binarytree(1,2,4,nil,5,nil,nil,0,nil,nil,3,6,nil,nil,7);
draw(pic,bt,condensed=false);

binarytree st=searchtree(10,5,2,1,3,4,7,6,8,9,15,13,12,11,14,17,16,18,19);
draw(pic2,st,blue,condensed=true);

add(pic.fit(),(0,0),10N);
add(pic2.fit(),(0,0),10S);


                          [./binarytreetest]


File: asymptote.info,  Node: bsp,  Next: CAD,  Prev: binarytree,  Up: Specialty modules

9.6 ‘bsp’
=========

This module implements hidden surface removal of planar pictures using a
binary space partition (BSP) and picture clipping.  It is used for
projecting three-dimensional scenes to two dimensions with proper
occlusion.


File: asymptote.info,  Node: CAD,  Next: contour3,  Prev: bsp,  Up: Specialty modules

9.7 ‘CAD’
=========

This module, contributed by Mark Henning, provides basic pen definitions
and measurement functions for simple 2D CAD drawings according to DIN
15.  It is documented separately, in the file ‘CAD.pdf’.


File: asymptote.info,  Node: contour3,  Next: drawtree,  Prev: CAD,  Up: Specialty modules

9.8 ‘contour3’
==============

This module draws surfaces described as the null space of real-valued
functions of (x,y,z) or ‘real[][][]’ matrices.  Its usage is illustrated
in the example file ‘magnetic.asy’.


File: asymptote.info,  Node: drawtree,  Next: embed,  Prev: contour3,  Up: Specialty modules

9.9 ‘drawtree’
==============

This is a simple tree drawing module used by the example ‘treetest.asy’.


File: asymptote.info,  Node: embed,  Next: external,  Prev: drawtree,  Up: Specialty modules

9.10 ‘embed’
============

This module provides an interface to the LaTeX package (included with
‘MikTeX’)
     <http://mirror.ctan.org/macros/latex/contrib/media9>
for embedding movies, sounds, and 3D objects into a PDF document.

   A more portable method for embedding movie files, which should work
on any platform and does not require the ‘media9’ package, is provided
by using the ‘external’ module instead of ‘embed’.

   Examples of the above two interfaces is provided in the file
‘embeddedmovie.asy’ in the ‘animations’ subdirectory of the examples
directory and in ‘externalmovie.asy’.  For a higher quality embedded
movie generated directly by ‘Asymptote’, use the ‘animate’ module along
with the ‘animate.sty’ package to embed a portable PDF animation (*note
animate::).

   An example of embedding ‘U3D’ code is provided in the file
‘embeddedu3d’.


File: asymptote.info,  Node: external,  Next: feynman,  Prev: embed,  Up: Specialty modules

9.11 ‘external’
===============

This module provides a portable method for embedding movie files in PDF
documents without requiring the ‘media9’ package.  It is an alternative
to the ‘embed’ module for better cross-platform compatibility.


File: asymptote.info,  Node: feynman,  Next: flowchart,  Prev: external,  Up: Specialty modules

9.12 ‘feynman’
==============

This module, contributed by Martin Wiebusch, is useful for drawing
Feynman diagrams, as illustrated by the examples ‘eetomumu.asy’ and
‘fermi.asy’.


File: asymptote.info,  Node: flowchart,  Next: labelpath,  Prev: feynman,  Up: Specialty modules

9.13 ‘flowchart’
================

This module provides routines for drawing flowcharts.  The primary
structure is a ‘block’, which represents a single block on the
flowchart.  The following eight functions return a position on the
appropriate edge of the block, given picture transform ‘t’:

pair block.top(transform t=identity());
pair block.left(transform t=identity());
pair block.right(transform t=identity());
pair block.bottom(transform t=identity());
pair block.topleft(transform t=identity());
pair block.topright(transform t=identity());
pair block.bottomleft(transform t=identity());
pair block.bottomright(transform t=identity());

To obtain an arbitrary position along the boundary of the block in user
coordinates, use:
pair block.position(real x, transform t=identity());

The center of the block in user coordinates is stored in ‘block.center’
and the block size in ‘PostScript’ coordinates is given by ‘block.size’.

A frame containing the block is returned by
frame block.draw(pen p=currentpen);

   The following block generation routines accept a Label, string, or
frame for their object argument:

“rectangular block with an optional header (and padding ‘dx’ around header and body):”
     block rectangle(object header, object body, pair center=(0,0),
                     pen headerpen=mediumgray, pen bodypen=invisible,
                     pen drawpen=currentpen,
                     real dx=3, real minheaderwidth=minblockwidth,
                     real minheaderheight=minblockwidth,
                     real minbodywidth=minblockheight,
                     real minbodyheight=minblockheight);
     block rectangle(object body, pair center=(0,0),
                     pen fillpen=invisible, pen drawpen=currentpen,
                     real dx=3, real minwidth=minblockwidth,
                     real minheight=minblockheight);

“parallelogram block:”
     block parallelogram(object body, pair center=(0,0),
                         pen fillpen=invisible, pen drawpen=currentpen,
                         real dx=3, real slope=2,
                         real minwidth=minblockwidth,
                         real minheight=minblockheight);

“diamond-shaped block:”
     block diamond(object body, pair center=(0,0),
                   pen fillpen=invisible, pen drawpen=currentpen,
                   real ds=5, real dw=1,
                   real height=20, real minwidth=minblockwidth,
                   real minheight=minblockheight);

“circular block:”
     block circle(object body, pair center=(0,0), pen fillpen=invisible,
                  pen drawpen=currentpen, real dr=3,
                  real mindiameter=mincirclediameter);

“rectangular block with rounded corners:”
     block roundrectangle(object body, pair center=(0,0),
                          pen fillpen=invisible, pen drawpen=currentpen,
                          real ds=5, real dw=0, real minwidth=minblockwidth,
                          real minheight=minblockheight);

“rectangular block with beveled edges:”
     block bevel(object body, pair center=(0,0), pen fillpen=invisible,
                 pen drawpen=currentpen, real dh=5, real dw=5,
                 real minwidth=minblockwidth, real minheight=minblockheight);

   To draw paths joining the pairs in ‘point’ with right-angled lines,
use the routine:
path path(pair point[], ... flowdir dir[]);
The entries in ‘dir’ identify whether successive segments between the
pairs specified by ‘point’ should be drawn in the ‘Horizontal’ or
‘Vertical’ direction.

   Here is a simple flowchart example (see also the example
‘controlsystem.asy’):

size(0,300);

import flowchart;

block block1=rectangle(Label("Example",magenta),
                       pack(Label("Start:",heavygreen),"",Label("$A:=0$",blue),
                            "$B:=1$"),(-0.5,3),palegreen,paleblue,red);
block block2=diamond(Label("Choice?",blue),(0,2),palegreen,red);
block block3=roundrectangle("Do something",(-1,1));
block block4=bevel("Don't do something",(1,1));
block block5=circle("End",(0,0));

draw(block1);
draw(block2);
draw(block3);
draw(block4);
draw(block5);

add(new void(picture pic, transform t) {
    blockconnector operator --=blockconnector(pic,t);
    //    draw(pic,block1.right(t)--block2.top(t));
    block1--Right--Down--Arrow--block2;
    block2--Label("Yes",0.5,NW)--Left--Down--Arrow--block3;
    block2--Right--Label("No",0.5,NE)--Down--Arrow--block4;
    block4--Down--Left--Arrow--block5;
    block3--Down--Right--Arrow--block5;
  });

                           [./flowchartdemo]


File: asymptote.info,  Node: labelpath,  Next: labelpath3,  Prev: flowchart,  Up: Specialty modules

9.14 ‘labelpath’
================

This module uses the ‘PSTricks’ ‘pstextpath’ macro to fit labels along a
path (properly kerned, as illustrated in the example file
‘curvedlabel.asy’), using the command
void labelpath(picture pic=currentpicture, Label L, path g,
               string justify=Centered, pen p=currentpen);
Here ‘justify’ is one of ‘LeftJustified’, ‘Centered’, or
‘RightJustified’.  The x component of a shift transform applied to the
Label is interpreted as a shift along the curve, whereas the y component
is interpreted as a shift away from the curve.  All other Label
transforms are ignored.  This module requires the ‘latex’ tex engine and
inherits the limitations of the ‘PSTricks’ ‘\pstextpath’ macro.


File: asymptote.info,  Node: labelpath3,  Next: lmfit,  Prev: labelpath,  Up: Specialty modules

9.15 ‘labelpath3’
=================

This module, contributed by Jens Schwaiger, implements a 3D version of
‘labelpath’ that does not require the ‘PSTricks’ package.  An example is
provided in ‘curvedlabel3.asy’.


File: asymptote.info,  Node: lmfit,  Next: MetaPost,  Prev: labelpath3,  Up: Specialty modules

9.16 ‘lmfit’
============

This module provides least-squares fitting routines for data analysis.
It can be used to fit mathematical models to experimental data.


File: asymptote.info,  Node: MetaPost,  Next: obj,  Prev: lmfit,  Up: Specialty modules

9.17 ‘MetaPost’
===============

This module provides some useful routines to help ‘MetaPost’ users
migrate old ‘MetaPost’ code to ‘Asymptote’.  Further contributions here
are welcome.

   Unlike ‘MetaPost’, ‘Asymptote’ does not implicitly solve linear
equations and therefore does not have the notion of a ‘whatever’
unknown.  The routine ‘extension’ (*note extension::) provides a useful
replacement for a common use of ‘whatever’: finding the intersection
point of the lines through ‘P’, ‘Q’ and ‘p’, ‘q’.  For less common
occurrences of ‘whatever’, one can use the built-in explicit linear
equation solver ‘solve’ instead.


File: asymptote.info,  Node: obj,  Next: ode,  Prev: MetaPost,  Up: Specialty modules

9.18 ‘obj’
==========

This module allows one to construct surfaces from simple obj files, as
illustrated in the example files ‘galleon.asy’ and ‘triceratops.asy’.


File: asymptote.info,  Node: ode,  Next: pstoedit,  Prev: obj,  Up: Specialty modules

9.19 ‘ode’
==========

The ‘ode’ module, illustrated in the example ‘odetest.asy’, implements a
number of explicit numerical integration schemes for ordinary
differential equations.


File: asymptote.info,  Node: pstoedit,  Next: rational,  Prev: ode,  Up: Specialty modules

9.20 ‘pstoedit’
===============

This module provides support for converting ‘PostScript’ files to
‘Asymptote’ code using the ‘pstoedit’ utility with the Asymptote
backend.


File: asymptote.info,  Node: rational,  Next: rationalSimplex,  Prev: pstoedit,  Up: Specialty modules

9.21 ‘rational’
===============

This module implements rational number arithmetic for exact computations
without floating-point errors.  It provides a ‘rational’ type with
support for basic arithmetic operations.


File: asymptote.info,  Node: rationalSimplex,  Next: simplex,  Prev: rational,  Up: Specialty modules

9.22 ‘rationalSimplex’
======================

This module provides the simplex method for linear programming using
rational arithmetic.  It is an exact version of the ‘simplex’ module
that avoids floating-point precision issues.


File: asymptote.info,  Node: simplex,  Next: size10,  Prev: rationalSimplex,  Up: Specialty modules

9.23 ‘simplex’
==============

This module solves the general linear programming problem using the
simplex method.


File: asymptote.info,  Node: size10,  Next: size11,  Prev: simplex,  Up: Specialty modules

9.24 ‘size10’
=============

This module provides a simple way to use the named LaTeX font sizes
corresponding to ‘\documentclass[10pt]{article}’:
import size10;
pen small=fontcommand("\small");
label("M",0,W,red+small);
pen normal=fontsize(9pt);
label("M",0,E,normal);


File: asymptote.info,  Node: size11,  Next: slide,  Prev: size10,  Up: Specialty modules

9.25 ‘size11’
=============

This module provides a simple way to use the named LaTeX font sizes
corresponding to ‘\documentclass[11pt]{article}’.  The usage is similar
to ‘size10’.


File: asymptote.info,  Node: slide,  Next: syzygy,  Prev: size11,  Up: Specialty modules

9.26 ‘slide’
============

This module provides a simple yet high-quality facility for making
presentation slides, including portable embedded PDF animations (see the
file ‘slidemovies.asy’).  A simple example is provided in
‘slidedemo.asy’.


File: asymptote.info,  Node: syzygy,  Next: tree,  Prev: slide,  Up: Specialty modules

9.27 ‘syzygy’
=============

This module automates the drawing of braids, relations, and syzygies,
along with the corresponding equations, as illustrated in the example
‘knots.asy’.


File: asymptote.info,  Node: tree,  Next: trembling,  Prev: syzygy,  Up: Specialty modules

9.28 ‘tree’
===========

This module implements an example of a dynamic binary search tree.


File: asymptote.info,  Node: trembling,  Prev: tree,  Up: Specialty modules

9.29 ‘trembling’
================

This module, written by Philippe Ivaldi and illustrated in the example
‘floatingdisk.asy’, allows one to draw wavy lines, as if drawn by hand.


File: asymptote.info,  Node: Developer modules,  Next: Options,  Prev: Specialty modules,  Up: Top

10 Developer modules
********************

These modules are for ‘Asymptote’ developers.

* Menu:

* plain::                       Default ‘Asymptote’ base file
* bezulate::                    Subdivision surface module
* simplex2::                    Two-variable simplex method
* v3d::                         V3D 3D graphics format


File: asymptote.info,  Node: plain,  Next: bezulate,  Up: Developer modules

10.1 ‘plain’
============

This is the default ‘Asymptote’ base file drawing language (such as the
‘picture’ structure).

   By default, an implicit ‘private import plain;’ occurs before
translating a file and before the first command given in interactive
mode.  This also applies when translating files for module definitions
(except when translating ‘plain’, of course).  This means that the types
and functions defined in ‘plain’ are accessible in almost all
‘Asymptote’ code.  Use the ‘-noautoplain’ command-line option to disable
this feature.


File: asymptote.info,  Node: bezulate,  Next: simplex2,  Prev: plain,  Up: Developer modules

10.2 ‘bezulate’
===============

This module provides the ‘bezulate’ routine for subdivision surface
generation.  It decomposes (possibly nonsimply connected) meshes into a
set of Bezier patches suitable for rendering.  The module is used
internally by ‘Asymptote’ for creating smooth surfaces from mesh data.


File: asymptote.info,  Node: simplex2,  Next: v3d,  Prev: bezulate,  Up: Developer modules

10.3 ‘simplex2’
===============

This module solves a special case of the two-variable linear programming
problem used by the module ‘plain’ for automatic sizing of pictures
(*note deferred drawing::).


File: asymptote.info,  Node: v3d,  Prev: simplex2,  Up: Developer modules

10.4 ‘v3d’
==========

This module provides support for the V3D 3D graphics format, which can
be viewed using the Python ‘pyv3d’ library or embedded within PDF
documents.  V3D content can be automatically generated by setting
‘settings.v3d=true’, or existing V3D files can be imported using the
‘importv3d’ function:
import v3d;
importv3d("file.v3d");


File: asymptote.info,  Node: Options,  Next: Interactive mode,  Prev: Developer modules,  Up: Top

11 Command-line options
***********************

Type ‘asy -h’ to see the full list of command-line options supported by
‘Asymptote’:
Usage: ../asy [options] [file ...]

Options (negate boolean options by replacing - with -no): 

-GPUblockSize n        Compute shader block size [8]
-GPUcompress           Compress GPU transparent fragment counts [false]
-GPUindexing           Compute indexing partial sums on GPU [true]
-GPUinterlock          Use fragment shader interlock [true]
-GPUlocalSize n        Compute shader local size [256]
-V,-View               View output; command-line only
-absolute              Use absolute WebGL dimensions [false]
-a,-align C|B|T|Z      Center, Bottom, Top, or Zero page alignment [C]
-aligndir pair         Directional page alignment (overrides align) [(0,0)]
-animating             [false]
-antialias n           Antialiasing width for rasterized output [2]
-auto3D                Automatically activate 3D scene [true]
-autobillboard         3D labels always face viewer by default [true]
-autoimport str        Module to automatically import
-autoplain             Enable automatic importing of plain [true]
-autoplay              Autoplay 3D WebGL animations [true]
-autorotate            Enable automatic PDF page rotation [false]
-axes3                 Show 3D axes in PDF output [true]
-batchMask             Mask fpu exceptions in batch mode [false]
-batchView             View output in batch mode [false]
-bw                    Convert all colors to black and white false
-cd directory          Set current directory; command-line only
-cmyk                  Convert rgb colors to cmyk false
-c,-command str        Command to autoexecute
-compact               Conserve memory at the expense of speed false
-compress              Compress images in PDF output [true]
-convertOptions str    []
-curlAfterRead         Allow libcurl after reading local files via input() false
-d,-debug              Enable debugging messages and traceback false
-device n              Set Vulkan device [-1]
-devicepixelratio n    Ratio of physical to logical pixels [0]
-digits n              Default output file precision [7]
-divisor n             Garbage collect using purge(divisor=n) [2]
-dvipsOptions str      []
-dvisvgmMultipleFiles  dvisvgm supports multiple files [true]
-dvisvgmOptions str    [--optimize]
-embed                 Embed rendered preview image [true]
-e,-environment        Show summary of environment settings; command-line only
-exitonEOF             Exit interactive mode on EOF [true]
-fitscreen             Fit rendered image to screen [true]
-framerate frames/s    Animation speed [30]
-fxaa                  Enable FXAA. Multisampling is turned off if FXAA is enabled [false]
-glOptions str         []
-globalread            Allow read from other directory true
-globalwrite           Allow write to other directory false
-gray                  Convert all colors to grayscale false
-gsOptions str         []
-h,-help               Show summary of options; command-line only
-historylines n        Retain n lines of history [1000]
-htmlviewerOptions str
                       []
-hyperrefOptions str   [setpagesize=false,unicode,pdfborder=0 0 0]
-ibl                   Enable environment map image-based lighting [false]
-iconify               [false]
-image str             Environment image name [snowyField]
-imageDir str          Environment image library directory [ibl]
-imageURL str          Environment image library URL [https://vectorgraphics.gitlab.io/asymptote/ibl]
-inlineimage           Generate inline embedded image [false]
-inlinetex             Generate inline TeX code [false]
-inpipe n              Input pipe [-1]
-interactiveMask       Mask fpu exceptions in interactive mode [true]
-interactiveView       View output in interactive mode [true]
-interactiveWrite      Write expressions entered at the prompt to stdout [true]
-interrupt             [false]
-k,-keep               Keep intermediate files [false]
-keepaux               Keep intermediate LaTeX .aux files [false]
-keys                  Generate WebGL keys false
-level n               Postscript level [3]
-licenses              Show license and third-party attribution; command-line only
-l,-listvariables      List available global functions and variables [false]
-localhistory          Use a local interactive history file [false]
-loop                  Loop 3D animations [false]
-lossy                 Use single precision for V3D reals [false]
-lsp                   Interactive mode for the Language Server Protocol [false]
-m,-mask               Mask fpu exceptions; command-line only
-maxFramesInFlight n   Maximum frames queued to the GPU [3]
-maxtile pair          Maximum rendering tile size [(1024,768)]
-maxviewport pair      Maximum viewport size [(0,0)]
-multiline             Input code over multiple lines at the prompt [false]
-multipleView          View output from multiple batch-mode files [false]
-multisample n         Multisampling width for screen images [4]
-offline               Produce offline html files [false]
-offscreen             Use offscreen rendering [false]
-O,-offset pair        PostScript offset [(0,0)]
-f,-outformat format   Convert each output file to specified format
-o,-outname name       Alternative output directory/file prefix
-outpipe n             Output pipe [-1]
-paperheight bp        Default page height [0]
-paperwidth bp         Default page width [0]
-p,-parseonly          Parse file [false]
-pdfreload             Automatically reload document in pdfviewer [false]
-pdfreloadOptions str  []
-pdfreloaddelay usec   Delay before attempting initial pdf reload [750000]
-pdfviewerOptions str  []
-position pair         Initial 3D rendering screen position [(0,0)]
-prc                   Embed 3D PRC graphics in PDF output [false]
-prerender resolution  Prerender V3D objects (0 implies vector output) [0]
-prompt str            Prompt [> ]
-prompt2 str           Continuation prompt for multiline input  [..]
-psviewerOptions str   []
-q,-quiet              Suppress welcome text and noninteractive stdout [false]
-render n              Render 3D graphics using n pixels per bp (-1=auto) [-1]
-resizestep step       Resize step [1.2]
-reverse               reverse 3D animations [false]
-rgb                   Convert cmyk colors to rgb false
-safe                  Disable system call true
-scroll n              Scroll standard output n lines at a time [0]
-shiftHoldDistance n   WebGL touch screen distance limit for shift mode [20]
-shiftWaitTime ms      WebGL touch screen shift mode delay [200]
-spinstep deg/s        Spin speed [60]
-srgb                  Render 3D images in sRGB space [false]
-svgemulation          Emulate unimplemented SVG shading [true]
-tabcompletion         Interactive prompt auto-completion [true]
-tex engine            latex|pdflatex|xelatex|lualatex|tex|pdftex|luatex|context|none [latex]
-thick                 Render thick 3D lines [true]
-thin                  Render thin 3D lines [true]
-threads               Use POSIX threads for 3D rendering [true]
-toolbar               Show 3D toolbar in PDF output [true]
-s,-translate          Show translated virtual machine code [false]
-twice                 Run LaTeX twice (to resolve references) [false]
-twosided              Use two-sided 3D lighting model for rendering [true]
-u,-user str           General purpose user string
-v3d                   Embed 3D V3D graphics in PDF output [false]
-v,-verbose            Increase verbosity level (can specify multiple times) 0
-version               Show version; command-line only
-vibrateTime ms        WebGL shift mode vibrate duration [25]
-viewportmargin pair   Horizontal and vertical 3D viewport margin [(0.5,0.5)]
-vsync                 Vertically synchronize with monitor [false]
-vulkan                Use Vulkan renderer if available [true]
-wait                  Wait for child processes to finish before exiting [false]
-warn str              Enable warning; command-line only
-webgl2                Use webgl2 if available [false]
-where                 Show where listed variables are declared [false]
-wsl                   Run asy under the Windows Subsystem for Linux [false]
-xasy                  Interactive mode for xasy false
-zoomPinchCap limit    WebGL maximum zoom pinch [100]
-zoomPinchFactor n     WebGL zoom pinch sensitivity [10]
-zoomThreshold threshold
                       Zoom remesh threshold [0.02]
-zoomfactor factor     Zoom step factor [1.05]
-zoomstep step         Mouse motion zoom step [0.1]

   All boolean options can be negated by prepending ‘no’ to the option
name.

   If no arguments are given, ‘Asymptote’ runs in interactive mode
(*note Interactive mode::).  In this case, the default output file is
‘out.eps’.

   If ‘-’ is given as the file argument, ‘Asymptote’ reads from standard
input.

   If multiple files are specified, they are treated as separate
‘Asymptote’ runs.

   If the string ‘autoimport’ is nonempty, a module with this name is
automatically imported for each run as the final step in loading module
‘plain’.

   Default option values may be entered as ‘Asymptote’ code in a
configuration file named ‘config.asy’ (or the file specified by the
environment variable ‘ASYMPTOTE_CONFIG’ or ‘-config’ option).
‘Asymptote’ will look for this file in its usual search path (*note
Search paths::).  Typically the configuration file is placed in the
‘.asy’ directory in the user's home directory (‘%USERPROFILE%\.asy’
under ‘MSDOS’).  Configuration variables are accessed using the long
form of the option names:
import settings;
outformat="pdf";
batchView=false;
interactiveView=true;
batchMask=false;
interactiveMask=true;
   Command-line options override these defaults.  Most configuration
variables may also be changed at runtime.  The advanced configuration
variables ‘dvipsOptions’, ‘hyperrefOptions’, ‘convertOptions’,
‘gsOptions’, ‘htmlviewerOptions’, ‘psviewerOptions’, ‘pdfviewerOptions’,
‘pdfreloadOptions’, ‘glOptions’, and ‘dvisvgmOptions’ allow specialized
options to be passed as a string to the respective applications or
libraries.  The default value of ‘hyperrefOptions’ is
‘setpagesize=false,unicode,pdfborder=0 0 0’.

   If you insert
import plain;
settings.autoplain=true;
at the beginning of the configuration file, it can contain arbitrary
‘Asymptote’ code.

   The default output format is EPS for the (default) ‘latex’ and ‘tex’
tex engine and PDF for the ‘pdflatex’, ‘xelatex’, ‘context’, ‘luatex’,
and ‘lualatex’ tex engines.  Alternative output formats may be produced
using the ‘-f’ option (or ‘outformat’ setting).

   To produce SVG output, you will need ‘dvisvgm’ (version 3.2.1 or
later) from <https://dvisvgm.de>, which can display SVG output (used by
the ‘xasy’ editor) for embedded EPS, PDF, PNG, and JPEG images included
with the ‘graphic()’ function.  The generated output is optimized with
the default setting ‘settings.dvisvgmOptions="--optimize"’.

   ‘Asymptote’ can also produce any output format supported by the
‘ImageMagick’ ‘magick’ program (version 7 or later).  The optional
setting ‘-render n’ requests an output resolution of ‘n’ pixels per
‘bp’.  Antialiasing is controlled by the parameter ‘antialias’, which by
default specifies a sampling width of 2 pixels.  To give other options
to ‘magick’, use the ‘convertOptions’ setting or call ‘magick convert’
manually.  This example emulates how ‘Asymptote’ produces antialiased
‘tiff’ output at one pixel per ‘bp’:
asy -o - venn | magick convert -alpha Off -density 144x144 -geometry 50%x eps:- venn.tiff

   If the option ‘-nosafe’ is given, ‘Asymptote’ runs in unsafe mode.
This enables the ‘int system(string s)’ and ‘int system(string[] s)’
calls, allowing one to execute arbitrary shell commands.  The default
mode, ‘-safe’, disables this call.

   A ‘PostScript’ offset may be specified as a pair (in ‘bp’ units) with
the ‘-O’ option:
asy -O 0,0 file
The default offset is zero.  The pair ‘aligndir’ specifies an optional
direction on the boundary of the page (mapped to the rectangle
[-1,1]\times[-1,1]) to which the picture should be aligned; the default
value ‘(0,0)’ specifies center alignment.

   The ‘-c’ (‘command’) option may be used to execute arbitrary
‘Asymptote’ code on the command line as a string.  It is not necessary
to terminate the string with a semicolon.  Multiple ‘-c’ options are
executed in the order they are given.  For example
asy -c 2+2 -c "sin(1)" -c "size(100); draw(unitsquare)"
produces the output
4
0.841470984807897
and draws a unitsquare of size ‘100’.

   The ‘-u’ (‘user’) option may be used to specify arbitrary ‘Asymptote’
settings on the command line as a string.  It is not necessary to
terminate the string with a semicolon.  Multiple ‘-u’ options are
executed in the order they are given.  Command-line code like ‘-u
x=sqrt(2)’ can be executed within a module like this:
real x;
usersetting();
write(x);

   When the ‘-l’ (‘listvariables’) option is used with file arguments,
only global functions and variables defined in the specified file(s) are
listed.

   Additional debugging output is produced with each additional ‘-v’
option:
‘-v’
     Display top-level module and final output file names.
‘-vv’
     Also display imported and included module names and final LaTeX and
     ‘dvips’ processing information.
‘-vvv’
     Also output LaTeX bidirectional pipe diagnostics.
‘-vvvv’
     Also output knot guide solver diagnostics.
‘-vvvvv’
     Also output ‘Asymptote’ traceback diagnostics.

   Type ‘asy -licenses’ to see the license and third-party attribution
information for ‘Asymptote’ and its bundled components.  Type ‘asy
-licenses=full’ or ‘asy --licenses=full’ to print the full license texts
(read from the installed ‘licenses/’ directory).


File: asymptote.info,  Node: Interactive mode,  Next: GUI,  Prev: Options,  Up: Top

12 Interactive mode
*******************

Interactive mode is entered by executing the command ‘asy’ with no file
arguments.  When the ‘-multiline’ option is disabled (the default), each
line must be a complete ‘Asymptote’ statement (unless explicitly
continued by a final backslash character ‘\’); it is not necessary to
terminate input lines with a semicolon.  If one assigns
‘settings.multiline=true’, interactive code can be entered over multiple
lines; in this mode, the automatic termination of interactive input
lines by a semicolon is inhibited.  Multiline mode is useful for cutting
and pasting ‘Asymptote’ code directly into the interactive input buffer.

   Interactive mode can be conveniently used as a calculator:
expressions entered at the interactive prompt (for which a corresponding
‘write’ function exists) are automatically evaluated and written to
‘stdout’.  If the expression is non-writable, its type signature will be
printed out instead.  In either case, the expression can be referred to
using the symbol ‘%’ in the next line input at the prompt.  For example:
> 2+3
5
> %*4
20
> 1/%
0.05
> sin(%)
0.0499791692706783
> currentpicture
<picture currentpicture>
> %.size(200,0)
>

   The ‘%’ symbol, when used as a variable, is shorthand for the
identifier ‘operator answer’, which is set by the prompt after each
written expression evaluation.

   The following special commands are supported only in interactive mode
and must be entered immediately after the prompt:

‘help’
     view the manual;
‘erase’
     erase ‘currentpicture’;
‘reset’
     reset the ‘Asymptote’ environment to its initial state, except for
     changes to the settings module (*note settings::), the current
     directory (*note cd::), and breakpoints (*note Debugger::);
‘input FILE’
     does an interactive reset, followed by the command ‘include FILE’.
     If the file name ‘FILE’ contains nonalphanumeric characters,
     enclose it with quotation marks.  A trailing semi-colon followed by
     optional ‘Asymptote’ commands may be entered on the same line.
‘quit’
     exit interactive mode (‘exit’ is a synonym; the abbreviation ‘q’ is
     also accepted unless there exists a top-level variable named ‘q’).
     A history of the most recent 1000 (this number can be changed with
     the ‘historylines’ configuration variable) previous commands will
     be retained in the file ‘.asy/history’ in the user's home directory
     (unless the command-line option ‘-localhistory’ was specified, in
     which case the history will be stored in the file ‘.asy_history’ in
     the current directory).

   Typing ‘ctrl-C’ interrupts the execution of ‘Asymptote’ code and
returns control to the interactive prompt.

   Interactive mode is implemented with the GNU ‘readline’ library, with
command history and auto-completion.  To customize the key bindings,
see: <https://tiswww.case.edu/php/chet/readline/readline.html>

   The file ‘asymptote.py’ in the ‘Asymptote’ system directory provides
an alternative way of entering ‘Asymptote’ commands interactively,
coupled with the full power of ‘Python’.  Copy this file to your ‘Python
path’ and then execute from within ‘Python 3’ the commands
from asymptote import *
g=asy()
g.size(200)
g.draw("unitcircle")
g.send("draw(unitsquare)")
g.fill("unitsquare, blue")
g.clip("unitcircle")
g.label("\"$O$\", (0,0), SW")


File: asymptote.info,  Node: GUI,  Next: Command-Line Interface,  Prev: Interactive mode,  Up: Top

13 Graphical User Interface
***************************

* Menu:

* GUI installation::            Installing ‘xasy’
* GUI usage::                   Using ‘xasy’ to edit objects

In the event that adjustments to the final figure are required, the
preliminary Graphical User Interface (GUI) ‘xasy’ included with
‘Asymptote’ allows you to move graphical objects and draw new ones.  The
modified figure can then be saved as a normal ‘Asymptote’ file.


File: asymptote.info,  Node: GUI installation,  Next: GUI usage,  Up: GUI

13.1 GUI installation
=====================

As ‘xasy’ is written in the interactive scripting language ‘Python/Qt’,
it requires ‘Python’ (<https://www.python.org>), along with the ‘Python’
packages ‘PySide6’, ‘cson’, and ‘numpy’:

pip3 install cson numpy PySide6

   Pictures are deconstructed into the SVG image format.  Since ‘Qt5’
does not support ‘SVG’ clipping, you will need the ‘rsvg-convert’
utility, which is part of the ‘librsvg2-tools’ package on ‘UNIX’ systems
and the ‘librsvg’ package on ‘macOS X’; under ‘Microsoft Windows’, it is
available as

   <https://sourceforge.net/projects/tumagcc/files/rsvg-convert-2.40.20.7z>

   Ensure that ‘rsvg-convert’ is available in ‘PATH’, or specify the
location of ‘rsvg-convert’ in ‘rsvgConverterPath’ option within ‘xasy’'s
settings.

   Deconstruction of a picture into its components is fastest when using
the LaTeX TeX engine.  The default setting ‘dvisvgmMultipleFiles=true’
speeds up deconstruction under PDF TeX engines.


File: asymptote.info,  Node: GUI usage,  Prev: GUI installation,  Up: GUI

13.2 GUI usage
==============

The arrow keys (or mouse wheel) are convenient for temporarily raising
and lowering objects within ‘xasy’, allowing an object to be selected.
Pressing the arrow keys will pan while the shift key is held and zoom
while the control key is held.  The mouse wheel will pan while the alt
or shift keys is held and zoom while the control key is held.  In
translate mode, an object can be dragged coarsely with the mouse or
positioned finely with the arrow keys while holding down the mouse
button.

   Deconstruction of compound objects (such as arrows) can be prevented
by enclosing them within the commands
void begingroup(picture pic=currentpicture);
void endgroup(picture pic=currentpicture);
   By default, the elements of a picture or frame will be grouped
together on adding them to a picture.  However, the elements of a frame
added to another frame are not grouped together by default: their
elements will be individually deconstructed (*note add::).


File: asymptote.info,  Node: Command-Line Interface,  Next: Language server protocol,  Prev: GUI,  Up: Top

14 Command-Line Interface
*************************

‘Asymptote’ code may be sent to the <https://asymptote.ualberta.ca>
server directly from the command line

   • SVG output:

     ‘curl --data-binary 'import venn;'
     'asymptote.ualberta.ca:10007?f=svg' | display -’

   • HTML output:

     ‘curl --data-binary
     @/usr/local/share/doc/asymptote/examples/Klein.asy
     'asymptote.ualberta.ca:10007' -o Klein.html’

   • V3D output:

     ‘curl --data-binary 'import teapot;'
     'asymptote.ualberta.ca:10007?f=v3d' -o teapot.v3d’

   • PDF output with rendered bitmap at 2 pixels per bp:

     ‘curl --data-binary 'import teapot;'
     'asymptote.ualberta.ca:10007?f=pdf' -o teapot.pdf’

   • PDF output with rendered bitmap at 4 pixels per bp:

     ‘curl --data-binary 'import teapot;'
     'asymptote.ualberta.ca:10007?f=pdf&render=4' -o teapot.pdf’

   • PRC output:

     ‘curl --data-binary 'import teapot;'
     'asymptote.ualberta.ca:10007?f=pdf&prc' -o teapot.pdf’

   • PRC output with rendered preview bitmap at 4 pixels per bp:

     ‘curl --data-binary 'import teapot;'
     'asymptote.ualberta.ca:10007?f=pdf&prc&render=4' -o teapot.pdf’

   The source code for the command-line interface is available at
<https://github.com/vectorgraphics/asymptote-http-server>.


File: asymptote.info,  Node: Language server protocol,  Next: PostScript to Asymptote,  Prev: Command-Line Interface,  Up: Top

15 Language server protocol
***************************

Under ‘UNIX’ and ‘macOS X’, ‘Asymptote’ supports features of the
Language Server Protocol (LSP)
(https://en.wikipedia.org/wiki/Language_Server_Protocol), including
function signature and variable matching.  Under ‘MSWindows’,
‘Asymptote’ currently supports LSP only when compiled within the
‘Windows Subsystem for Linux’.

   ‘Emacs’ users can enable the ‘Asymptote’ language server protocol by
installing ‘lsp-mode’ using the following procedure:

   • Add to the ‘.emacs’ initialization file:
     (require 'package)
     (add-to-list 'package-archives '("melpa" . "https://melpa.org/packages/") t)
     (package-initialize)

   • Launch emacs and execute
     M-x package-refresh-contents
     M-x package-install
     and select ‘lsp-mode’.

   • Add to the ‘.emacs’ initialization file:
     (require 'lsp-mode)
     (add-to-list 'lsp-language-id-configuration '(asy-mode . "asymptote"))
     
     (lsp-register-client
      (make-lsp-client :new-connection (lsp-stdio-connection '("asy" "-lsp"))
                       :activation-fn (lsp-activate-on "asymptote")
                       :major-modes '(asy-mode)
                       :server-id 'asyls
                       )
      )

   • Launch emacs and execute
     M-x lsp


File: asymptote.info,  Node: PostScript to Asymptote,  Next: Help,  Prev: Language server protocol,  Up: Top

16 ‘PostScript’ to ‘Asymptote’
******************************

The excellent ‘PostScript’ editor ‘pstoedit’ (version 3.50 or later;
available from <https://sourceforge.net/projects/pstoedit/>) can convert
‘PostScript’ to ‘Asymptote’ code.  This driver includes native clipping,
even-odd fill rule, ‘PostScript’ subpath, and full image support.  Here
is an example:

‘asy -V /usr/local/share/doc/asymptote/examples/venn.asy’
pstoedit -f asy venn.eps test.asy
asy -V test

If the line widths aren't quite correct, try giving ‘pstoedit’ the
‘-dis’ option.  If the fonts aren't typeset correctly, try giving
‘pstoedit’ the ‘-dt’ option.


File: asymptote.info,  Node: Help,  Next: Debugger,  Prev: PostScript to Asymptote,  Up: Top

17 Help
*******

A list of frequently asked questions (FAQ) is maintained at
     <https://asymptote.sourceforge.io/FAQ>
Questions on installing and using ‘Asymptote’ that are not addressed in
the FAQ should be sent to the ‘Asymptote’ forum:
     <https://sourceforge.net/p/asymptote/discussion/409349>
Including an example that illustrates what you are trying to do will
help you get useful feedback.  LaTeX problems can often be diagnosed
with the ‘-vv’ or ‘-vvv’ command-line options.  Contributions in the
form of patches or ‘Asymptote’ modules can be posted here:
     <https://sourceforge.net/p/asymptote/patches>
To receive announcements of upcoming releases, please subscribe to
‘Asymptote’ at
     <https://sourceforge.net/projects/asymptote/>
If you find a bug in ‘Asymptote’, please check (if possible) whether the
bug is still present in the latest ‘git’ developmental code (*note
Git::) before submitting a bug report.  New bugs can be reported at
     <https://github.com/vectorgraphics/asymptote/issues>
To see if the bug has already been fixed, check bugs with Status
‘Closed’ and recent lines in
     <https://asymptote.sourceforge.io/ChangeLog>

   ‘Asymptote’ can be configured with the optional GNU library
‘libsigsegv’, available from <https://www.gnu.org/software/libsigsegv/>,
which allows one to distinguish user-generated ‘Asymptote’ stack
overflows (*note stack overflow::) from true segmentation faults (due to
internal C++ programming errors; please submit the ‘Asymptote’ code that
generates such segmentation faults along with your bug report).


File: asymptote.info,  Node: Debugger,  Next: Credits,  Prev: Help,  Up: Top

18 Debugger
***********

Asymptote now includes a line-based (as opposed to code-based) debugger
that can assist the user in following flow control.  To set a break
point in file ‘file’ at line ‘line’, use the command

void stop(string file, int line, code s=quote{});
The optional argument ‘s’ may be used to conditionally set the variable
‘ignore’ in ‘plain_debugger.asy’ to ‘true’.  For example, the first 10
instances of this breakpoint will be ignored (the variable ‘int count=0’
is defined in ‘plain_debugger.asy’):
stop("test",2,quote{ignore=(++count <= 10);});

   To set a break point in file ‘file’ at the first line containing the
string ‘text’, use

void stop(string file, string text, code s=quote{});
To list all breakpoints, use:
void breakpoints();
To clear a breakpoint, use:
void clear(string file, int line);
To clear all breakpoints, use:
void clear();

   The following commands may be entered at the debugging prompt:

‘h’
     help;
‘c’
     continue execution;

‘i’
     step to the next instruction;

‘s’
     step to the next executable line;

‘n’
     step to the next executable line in the current file;

‘f’
     step to the next file;

‘r’
     return to the file associated with the most recent breakpoint;

‘t’
     toggle tracing (‘-vvvvv’) mode;

‘q’
     quit debugging and end execution;

‘x’
     exit the debugger and run to completion.

Arbitrary ‘Asymptote’ code may also be entered at the debugging prompt;
however, since the debugger is implemented with ‘eval’, currently only
top-level (global) variables can be displayed or modified.

   The debugging prompt may be entered manually with the call
void breakpoint(code s=quote{});


File: asymptote.info,  Node: Credits,  Next: General Index,  Prev: Debugger,  Up: Top

19 Acknowledgments
******************

Financial support for the development of ‘Asymptote’ was generously
provided by the Natural Sciences and Engineering Research Council of
Canada, the Pacific Institute for Mathematical Sciences, and the
University of Alberta Faculty of Science.

   We also would like to acknowledge the previous work of John D. Hobby,
author of the program ‘MetaPost’ that inspired the development of
‘Asymptote’, and Donald E. Knuth, author of TeX and ‘MetaFont’ (on which
‘MetaPost’ is based).

   The authors of ‘Asymptote’ are Andy Hammerlindl, John Bowman, and Tom
Prince.  Sean Healy designed the ‘Asymptote’ logo.  Other contributors
include Orest Shardt, Jesse Frohlich, Michail Vidiassov, Charles Staats,
Philippe Ivaldi, Olivier Guibé, Radoslav Marinov, Jeff Samuelson, Chris
Savage, Jacques Pienaar, Mark Henning, Steve Melenchuk, Martin Wiebusch,
Stefan Knorr, Supakorn "Jamie" Rassameemasmuang, Jacob Skitsko, Joseph
Chaumont, and Oliver Cheng.  Pedram Emami developed the ‘Asymptote Web
Application’ hosted at <https://asymptote.ualberta.ca>:

   <https://github.com/vectorgraphics/asymptoteWebApplication>


File: asymptote.info,  Node: General Index,  Prev: Credits,  Up: Top

General Index
*************

[index]
* Menu:

* -:                                     Arithmetic & logical.
                                                             (line   14)
* --:                                    Paths.              (line   16)
* -- <1>:                                Self & prefix operators.
                                                             (line    6)
* ---:                                   Bezier curves.      (line   85)
* -=:                                    Self & prefix operators.
                                                             (line    6)
* ::                                     Arithmetic & logical.
                                                             (line   61)
* :::                                    Bezier curves.      (line   71)
* !:                                     Arithmetic & logical.
                                                             (line   57)
* !=:                                    Structures.         (line   62)
* != <1>:                                Arithmetic & logical.
                                                             (line   37)
* ?:                                     Arithmetic & logical.
                                                             (line   61)
* .. (two dots):                         Paths.              (line   16)
* .asy:                                  Search paths.       (line   12)
* []:                                    Bracket operators.  (line    6)
* [], Map.operator []:                   The Map API.        (line   25)
* [], Map.operator [=]:                  The Map API.        (line   31)
* [=]:                                   Bracket operators.  (line    6)
* *:                                     Pens.               (line   15)
* * <1>:                                 Arithmetic & logical.
                                                             (line   16)
* **:                                    Arithmetic & logical.
                                                             (line   31)
* *=:                                    Self & prefix operators.
                                                             (line    6)
* /:                                     Arithmetic & logical.
                                                             (line   18)
* /=:                                    Self & prefix operators.
                                                             (line    6)
* &:                                     Bezier curves.      (line   85)
* & <1>:                                 Arithmetic & logical.
                                                             (line   49)
* &&:                                    Arithmetic & logical.
                                                             (line   47)
* #:                                     Arithmetic & logical.
                                                             (line   20)
* %:                                     Arithmetic & logical.
                                                             (line   25)
* % <1>:                                 Interactive mode.   (line   16)
* %=:                                    Self & prefix operators.
                                                             (line    6)
* ^:                                     Arithmetic & logical.
                                                             (line   29)
* ^ <1>:                                 Arithmetic & logical.
                                                             (line   55)
* ^^:                                    Paths.              (line   23)
* ^=:                                    Self & prefix operators.
                                                             (line    6)
* +:                                     Pens.               (line   15)
* + <1>:                                 Arithmetic & logical.
                                                             (line   13)
* ++:                                    Self & prefix operators.
                                                             (line    6)
* +=:                                    Self & prefix operators.
                                                             (line    6)
* <:                                     Arithmetic & logical.
                                                             (line   39)
* <=:                                    Arithmetic & logical.
                                                             (line   41)
* ==:                                    Structures.         (line   62)
* == <1>:                                Arithmetic & logical.
                                                             (line   36)
* >:                                     Arithmetic & logical.
                                                             (line   45)
* >=:                                    Arithmetic & logical.
                                                             (line   43)
* |:                                     Arithmetic & logical.
                                                             (line   53)
* ||:                                    Arithmetic & logical.
                                                             (line   51)
* 2D graphs:                             graph.              (line    6)
* 3D graphs:                             graph3.             (line    6)
* 3D grids:                              grid3.              (line    6)
* 3D PostScript:                         three.              (line  685)
* a4:                                    Configuring.        (line   63)
* abort:                                 Data types.         (line  364)
* abs:                                   Data types.         (line   65)
* abs <1>:                               Data types.         (line  144)
* abs <2>:                               Mathematical functions.
                                                             (line   35)
* abs2:                                  Data types.         (line   65)
* abs2 <1>:                              Data types.         (line  144)
* absolute:                              Configuring.        (line   38)
* absolute <1>:                          three.              (line  255)
* accel:                                 Paths and guides.   (line  126)
* accel <1>:                             Paths and guides.   (line  132)
* accel <2>:                             three.              (line  586)
* access:                                Import.             (line    6)
* access <1>:                            Import.             (line   45)
* acknowledgments:                       Credits.            (line    6)
* acos:                                  Mathematical functions.
                                                             (line    6)
* aCos:                                  Mathematical functions.
                                                             (line   20)
* acosh:                                 Mathematical functions.
                                                             (line    6)
* add:                                   Frames and pictures.
                                                             (line  205)
* add <1>:                               Frames and pictures.
                                                             (line  219)
* add <2>:                               three.              (line  363)
* add, Map.add:                          The Map API.        (line   68)
* add, Set.add:                          The Set API.        (line   24)
* add, Set.add <1>:                      The Set API.        (line   30)
* addViews:                              three.              (line  480)
* adjust:                                Pens.               (line  123)
* advance:                               Iterators.          (line   22)
* advance <1>:                           collections.iter(T).
                                                             (line   16)
* after, SortedSet.after:                The SortedSet API.  (line   47)
* Ai:                                    Mathematical functions.
                                                             (line   59)
* Ai_deriv:                              Mathematical functions.
                                                             (line   59)
* Airy:                                  Mathematical functions.
                                                             (line   59)
* alias:                                 Structures.         (line   62)
* alias <1>:                             Arrays.             (line  194)
* Align:                                 label.              (line   20)
* aligndir:                              Options.            (line  226)
* all:                                   Arrays.             (line  361)
* Allow:                                 Pens.               (line  416)
* and:                                   Bezier curves.      (line   56)
* AND:                                   Arithmetic & logical.
                                                             (line   68)
* angle:                                 Data types.         (line   73)
* animate:                               Configuring.        (line   15)
* animate <1>:                           Files.              (line  163)
* animate <2>:                           animation.          (line   12)
* animate <3>:                           animate.            (line    6)
* animation:                             animation.          (line    6)
* animation <1>:                         animation.          (line    6)
* annotate:                              annotate.           (line    6)
* antialias:                             three.              (line  282)
* antialias <1>:                         Options.            (line  200)
* append:                                Files.              (line   38)
* append <1>:                            Arrays.             (line   50)
* arc:                                   Paths and guides.   (line   24)
* Arc:                                   Paths and guides.   (line   37)
* arc <1>:                               three.              (line  374)
* ArcArrow:                              draw.               (line   30)
* ArcArrow3:                             three.              (line  652)
* ArcArrows:                             draw.               (line   30)
* ArcArrows3:                            three.              (line  652)
* arclength:                             Paths and guides.   (line  153)
* arclength <1>:                         three.              (line  586)
* arcpoint:                              Paths and guides.   (line  163)
* arctime:                               Paths and guides.   (line  157)
* arctime <1>:                           three.              (line  586)
* arguments:                             Default arguments.  (line    6)
* arithmetic operators:                  Arithmetic & logical.
                                                             (line    6)
* array:                                 Data types.         (line  284)
* array <1>:                             Arrays.             (line  123)
* array iteration:                       Programming.        (line   56)
* arrays:                                Arrays.             (line    6)
* arrow:                                 Drawing commands.   (line   34)
* Arrow:                                 draw.               (line   26)
* arrow <1>:                             label.              (line   77)
* arrow keys:                            Drawing in interactive mode.
                                                             (line   11)
* arrow keys <1>:                        GUI usage.          (line    6)
* Arrow3:                                three.              (line  652)
* arrowbar:                              draw.               (line   26)
* arrowhead:                             draw.               (line   50)
* arrows:                                draw.               (line   26)
* Arrows:                                draw.               (line   26)
* Arrows3:                               three.              (line  652)
* as:                                    Import.             (line   67)
* ascii:                                 Data types.         (line  309)
* ascii <1>:                             Data types.         (line  309)
* asin:                                  Mathematical functions.
                                                             (line    6)
* aSin:                                  Mathematical functions.
                                                             (line   20)
* asinh:                                 Mathematical functions.
                                                             (line    6)
* Aspect:                                Frames and pictures.
                                                             (line   45)
* assert:                                Data types.         (line  369)
* assignment:                            Programming.        (line   31)
  associative array, See Map_K_V.
* asy:                                   Data types.         (line  359)
* asy <1>:                               Import.             (line  114)
* asy-mode:                              Editing modes.      (line    6)
* asy.vim:                               Editing modes.      (line   32)
* asygl:                                 Configuring.        (line   69)
* asyinclude:                            LaTeX usage.        (line   42)
* Asymptote Web Application:             Description.        (line    6)
* ASYMPTOTE_CONFIG:                      Options.            (line  171)
* asymptote.sty:                         LaTeX usage.        (line    6)
* asymptote.xml:                         Editing modes.      (line   48)
* atan:                                  Mathematical functions.
                                                             (line    6)
* aTan:                                  Mathematical functions.
                                                             (line   20)
* atan2:                                 Mathematical functions.
                                                             (line    6)
* atanh:                                 Mathematical functions.
                                                             (line    6)
* atleast:                               Bezier curves.      (line   56)
* atOrAfter, SortedSet.atOrAfter:        The SortedSet API.  (line   52)
* atOrBefore, SortedSet.atOrBefore:      The SortedSet API.  (line   62)
* attach:                                Frames and pictures.
                                                             (line  264)
* attach <1>:                            LaTeX usage.        (line   47)
* attach <2>:                            graph.              (line  404)
* attribution:                           Options.            (line  271)
* autoadjust:                            three.              (line  449)
* autoimport:                            Options.            (line  167)
* automatic scaling:                     graph.              (line  708)
* automatic scaling <1>:                 graph.              (line  708)
* autounravel:                           Autounravel.        (line    6)
* axialshade:                            fill.               (line   43)
* axis:                                  graph.              (line  922)
* axis <1>:                              graph.              (line 1004)
* axis <2>:                              graph3.             (line   70)
* axis <3>:                              graph3.             (line   86)
* azimuth:                               Data types.         (line  154)
* babel:                                 babel.              (line    6)
* background:                            three.              (line   76)
* background <1>:                        three.              (line   97)
* background color:                      Frames and pictures.
                                                             (line  168)
* BackView:                              three.              (line  473)
* Bar:                                   draw.               (line   19)
* Bar3:                                  three.              (line  652)
* Bars:                                  draw.               (line   19)
* Bars3:                                 three.              (line  652)
* barsize:                               draw.               (line   19)
* basealign:                             Pens.               (line  181)
* baseline:                              label.              (line   97)
* batch mode:                            Drawing in batch mode.
                                                             (line    6)
* beep:                                  Data types.         (line  382)
* before, SortedSet.before:              The SortedSet API.  (line   57)
* BeginArcArrow:                         draw.               (line   30)
* BeginArcArrow3:                        three.              (line  652)
* BeginArrow:                            draw.               (line   26)
* BeginArrow3:                           three.              (line  652)
* BeginBar:                              draw.               (line   19)
* BeginBar3:                             three.              (line  652)
* BeginDotMargin:                        draw.               (line   87)
* BeginDotMargin3:                       three.              (line  668)
* BeginMargin:                           draw.               (line   68)
* BeginMargin3:                          three.              (line  668)
* BeginPenMargin:                        draw.               (line   76)
* BeginPenMargin2:                       three.              (line  668)
* BeginPenMargin3:                       three.              (line  668)
* BeginPoint:                            label.              (line   62)
* Bessel:                                Mathematical functions.
                                                             (line   59)
* bevel:                                 flowchart.          (line   72)
* beveljoin:                             Pens.               (line  149)
* Bezier curves:                         Bezier curves.      (line    6)
* Bezier patch:                          three.              (line  142)
* Bezier triangle:                       three.              (line  142)
* bezulate:                              three.              (line  160)
* bezulate <1>:                          bezulate.           (line    6)
* Bi:                                    Mathematical functions.
                                                             (line   59)
* Bi_deriv:                              Mathematical functions.
                                                             (line   59)
* Billboard:                             three.              (line  556)
* binary:                                Files.              (line   80)
* binary format:                         Files.              (line   80)
* binary operators:                      Arithmetic & logical.
                                                             (line    6)
* binarytree:                            binarytree.         (line    6)
* black stripes:                         three.              (line  282)
* Blank:                                 draw.               (line   26)
* block.bottom:                          flowchart.          (line   19)
* block.bottomleft:                      flowchart.          (line   19)
* block.bottomright:                     flowchart.          (line   19)
* block.center:                          flowchart.          (line   24)
* block.draw:                            flowchart.          (line   29)
* block.left:                            flowchart.          (line   19)
* block.position:                        flowchart.          (line   23)
* block.right:                           flowchart.          (line   19)
* block.top:                             flowchart.          (line   19)
* block.topleft:                         flowchart.          (line   19)
* block.topright:                        flowchart.          (line   19)
* bool:                                  Data types.         (line   14)
* bool3:                                 Data types.         (line   25)
* boolean operators:                     Arithmetic & logical.
                                                             (line    6)
* Bottom:                                graph.              (line  130)
* BottomTop:                             graph.              (line  136)
* BottomView:                            three.              (line  473)
* bounding box:                          Frames and pictures.
                                                             (line  168)
* Bounds:                                graph3.             (line   25)
* bounds:                                palette.            (line   43)
* box:                                   Frames and pictures.
                                                             (line  118)
* box <1>:                               three.              (line  396)
* box <2>:                               three.              (line  398)
* bp:                                    Drawing in batch mode.
                                                             (line   23)
* brace:                                 Paths and guides.   (line   51)
* bracket operators:                     Bracket operators.  (line    6)
* break:                                 Programming.        (line   52)
* breakpoints:                           Debugger.           (line   21)
* brick:                                 Pens.               (line  338)
* broken axis:                           graph.              (line  819)
* bsp:                                   bsp.                (line    6)
* BTreeMap_K_V:                          Initializing a Map. (line   15)
* BTreeSet_T:                            The SortedSet API.  (line    6)
* bug reports:                           Help.               (line   19)
* buildcycle:                            Paths and guides.   (line  270)
* Button-1:                              GUI.                (line    6)
* Button-2:                              GUI.                (line    6)
* BWRainbow:                             palette.            (line   15)
* BWRainbow2:                            palette.            (line   18)
* C string:                              Data types.         (line  217)
* CAD:                                   CAD.                (line    6)
* camera:                                three.              (line  443)
* casts:                                 Casts.              (line    6)
* cbrt:                                  Mathematical functions.
                                                             (line    6)
* cd:                                    Files.              (line   26)
* ceil:                                  Mathematical functions.
                                                             (line   26)
* Center:                                label.              (line   67)
* center:                                three.              (line  426)
* checker:                               Pens.               (line  338)
* Chinese:                               Pens.               (line  297)
* choose:                                Mathematical functions.
                                                             (line   39)
* Ci:                                    Mathematical functions.
                                                             (line   59)
* circle:                                Paths and guides.   (line   10)
* Circle:                                Paths and guides.   (line   18)
* circle <1>:                            three.              (line  370)
* circle <2>:                            flowchart.          (line   61)
* circlebarframe:                        markers.            (line   18)
* CJK:                                   Pens.               (line  297)
* clamped:                               graph.              (line   36)
* clang:                                 Compiling from UNIX source.
                                                             (line   34)
* clear:                                 Files.              (line  101)
* clear <1>:                             Debugger.           (line   23)
* clip:                                  clip.               (line    6)
* CLZ:                                   Arithmetic & logical.
                                                             (line   68)
* cm:                                    Figure size.        (line   18)
* cmd:                                   Configuring.        (line   30)
* cmyk:                                  Pens.               (line   38)
* colatitude:                            Data types.         (line  159)
* collections:                           Collections (containers).
                                                             (line    6)
* collections.btreemap:                  Maps.               (line   41)
* collections.enumerate:                 Attaching a counter to an iterator.
                                                             (line    6)
* collections.genericpair:               Pairs.              (line    6)
* collections.hashmap:                   Maps.               (line   41)
* collections.iter:                      Iterators.          (line   31)
* collections.iter <1>:                  collections.iter(T).
                                                             (line    6)
* collections.map:                       Maps.               (line   41)
* collections.zip:                       Combining (zipping) iterators.
                                                             (line   37)
* collections.zip2:                      Combining (zipping) iterators.
                                                             (line    6)
* color:                                 Pens.               (line   23)
* color <1>:                             graph3.             (line  137)
* coloredNodes:                          tube.               (line   25)
* coloredpath:                           tube.               (line   18)
* coloredSegments:                       tube.               (line   25)
* colorless:                             Pens.               (line   57)
* colormap:                              colormap.           (line    6)
* colors:                                Pens.               (line   54)
* comma:                                 Files.              (line   65)
* comma-separated-value mode:            Arrays.             (line  396)
* command-line interface:                Command-Line Interface.
                                                             (line    6)
* command-line options:                  Configuring.        (line   86)
* command-line options <1>:              Options.            (line    6)
* comment character:                     Files.              (line   16)
* compass directions:                    Labels.             (line   18)
* Compiling from UNIX source:            Compiling from UNIX source.
                                                             (line    6)
* complement:                            Arrays.             (line  160)
* concat:                                Arrays.             (line  190)
* conditional:                           Programming.        (line   31)
* conditional <1>:                       Arithmetic & logical.
                                                             (line   61)
* config:                                Configuring.        (line   69)
* config <1>:                            Options.            (line  171)
* configuration file:                    Configuring.        (line   15)
* configuration file <1>:                Options.            (line  171)
* configuring:                           Configuring.        (line    6)
* conj:                                  Data types.         (line   62)
* constructors:                          Structures.         (line   90)
* containers:                            Collections (containers).
                                                             (line    6)
* contains, Map.contains:                The Map API.        (line   22)
* contains, Set.contains:                The Set API.        (line   19)
* context:                               Options.            (line  200)
* continue:                              Programming.        (line   52)
* continue <1>:                          Debugger.           (line   31)
* contour:                               contour.            (line    6)
* contour3:                              contour3.           (line    6)
* controls:                              Bezier curves.      (line   45)
* controls <1>:                          three.              (line    6)
* controlSpecifier:                      Paths and guides.   (line  397)
* convert:                               Files.              (line  163)
* convertOptions:                        Options.            (line  186)
* Coons shading:                         fill.               (line   78)
* copy:                                  Arrays.             (line  187)
* cos:                                   Mathematical functions.
                                                             (line    6)
* Cos:                                   Mathematical functions.
                                                             (line   20)
* cosh:                                  Mathematical functions.
                                                             (line    6)
* cputime:                               Mathematical functions.
                                                             (line   97)
* crop:                                  graph.              (line  637)
* cropping graphs:                       graph.              (line  637)
* cross:                                 Data types.         (line  106)
* cross <1>:                             Data types.         (line  197)
* cross <2>:                             graph.              (line  478)
* crossframe:                            markers.            (line   22)
* crosshatch:                            Pens.               (line  355)
* csv:                                   Arrays.             (line  378)
* csv <1>:                               Arrays.             (line  396)
* CTZ:                                   Arithmetic & logical.
                                                             (line   68)
* cubicroots:                            Arrays.             (line  350)
* curl:                                  Bezier curves.      (line   67)
* curl <1>:                              three.              (line    6)
* curlSpecifier:                         Paths and guides.   (line  409)
* currentlight:                          three.              (line   76)
* currentpen:                            Pens.               (line    6)
* currentprojection:                     three.              (line  470)
* curve:                                 slopefield.         (line   20)
* custom axis types:                     graph.              (line  139)
* custom mark routine:                   graph.              (line  575)
* custom tick locations:                 graph.              (line  231)
* cut:                                   Paths and guides.   (line  251)
* cycle:                                 Figure size.        (line   29)
* cycle <1>:                             Paths.              (line   16)
* cycle <2>:                             three.              (line    6)
* cyclic:                                Paths and guides.   (line   85)
* cyclic <1>:                            Paths and guides.   (line  377)
* cyclic <2>:                            Arrays.             (line   50)
* cyclic <3>:                            three.              (line  586)
* Cyrillic:                              Pens.               (line  291)
* dashdotted:                            Pens.               (line  102)
* dashed:                                Pens.               (line  102)
* data types:                            Data types.         (line    6)
* date:                                  Data types.         (line  321)
* Debian:                                UNIX binary distributions.
                                                             (line   19)
* debugger:                              Debugger.           (line    6)
* declaration:                           Programming.        (line   31)
* deconstruct:                           GUI usage.          (line    6)
* default arguments:                     Default arguments.  (line    6)
* defaultformat:                         graph.              (line  173)
* DefaultHead:                           draw.               (line   50)
* DefaultHead3:                          three.              (line  652)
* defaultpen:                            Pens.               (line   49)
* defaultpen <1>:                        Pens.               (line  122)
* defaultpen <2>:                        Pens.               (line  127)
* defaultpen <3>:                        Pens.               (line  139)
* defaultpen <4>:                        Pens.               (line  245)
* defaultpen <5>:                        Pens.               (line  416)
* defaultpen <6>:                        Pens.               (line  440)
* defaultrender:                         three.              (line   46)
* deferred drawing:                      Deferred drawing.   (line    6)
* deferred drawing <1>:                  simplex2.           (line    6)
* degrees:                               Data types.         (line   78)
* degrees <1>:                           Mathematical functions.
                                                             (line   17)
* Degrees:                               Mathematical functions.
                                                             (line   17)
* delete:                                Files.              (line  158)
* delete <1>:                            Arrays.             (line   50)
* delete, Map.delete:                    The Map API.        (line   39)
* delete, Set.delete:                    The Set API.        (line   34)
* delete, Set.delete <1>:                The Set API.        (line   39)
* description:                           Description.        (line    6)
* developer modules:                     Developer modules.  (line    6)
* devicepixelratio:                      three.              (line  198)
* diagonal:                              Arrays.             (line  335)
* diamond:                               flowchart.          (line   54)
  dictionary, See Map_K_V.
* diffuse:                               three.              (line   76)
* diffusepen:                            three.              (line   66)
* dimension:                             Arrays.             (line  378)
* dimension <1>:                         Arrays.             (line  401)
* dir:                                   Search paths.       (line    9)
* dir <1>:                               Data types.         (line   90)
* dir <2>:                               Data types.         (line  181)
* dir <3>:                               Paths and guides.   (line  109)
* dir <4>:                               three.              (line  586)
* direction specifier:                   Bezier curves.      (line    6)
* directory:                             Files.              (line   26)
* dirSpecifier:                          Paths and guides.   (line  391)
* dirtime:                               Paths and guides.   (line  166)
* display:                               Configuring.        (line   15)
* do:                                    Programming.        (line   52)
* DOSendl:                               Files.              (line   65)
* DOSnewl:                               Files.              (line   65)
* dot:                                   draw.               (line  119)
* dot <1>:                               Data types.         (line  103)
* dot <2>:                               Data types.         (line  194)
* dot <3>:                               Arrays.             (line  290)
* dot <4>:                               Arrays.             (line  293)
* DotMargin:                             draw.               (line   83)
* DotMargin3:                            three.              (line  668)
* DotMargins:                            draw.               (line   89)
* DotMargins3:                           three.              (line  668)
* dotted:                                Pens.               (line  102)
* double deferred drawing:               three.              (line  348)
* double precision:                      Files.              (line   80)
* draw:                                  Drawing commands.   (line   34)
* draw <1>:                              draw.               (line    6)
* draw <2>:                              draw.               (line  149)
* Draw:                                  Frames and pictures.
                                                             (line  148)
* draw <3>:                              three.              (line  168)
* drawer:                                Deferred drawing.   (line   31)
* drawing commands:                      Drawing commands.   (line    6)
* drawline:                              math.               (line    9)
* drawtree:                              drawtree.           (line    6)
* dvips:                                 Configuring.        (line   69)
* dvipsOptions:                          Options.            (line  186)
* dvisvgm:                               Configuring.        (line   69)
* dvisvgm <1>:                           Options.            (line  205)
* dvisvgmMultipleFiles:                  GUI installation.   (line   24)
* dvisvgmOptions:                        Options.            (line  186)
* E:                                     Labels.             (line   18)
* E <1>:                                 Mathematical functions.
                                                             (line   59)
* Editing modes:                         Editing modes.      (line    6)
* Ei:                                    Mathematical functions.
                                                             (line   59)
* ellipse:                               Paths and guides.   (line   45)
* elliptic functions:                    Mathematical functions.
                                                             (line   59)
* else:                                  Programming.        (line   31)
* emacs:                                 Editing modes.      (line    6)
* embed:                                 embed.              (line    6)
* Embedded:                              three.              (line  556)
* emissivepen:                           three.              (line   66)
* empty:                                 Frames and pictures.
                                                             (line    7)
* empty, Map.empty:                      The Map API.        (line   19)
* empty, Set.empty:                      The Set API.        (line   16)
* EndArcArrow:                           draw.               (line   30)
* EndArcArrow3:                          three.              (line  652)
* EndArrow:                              draw.               (line   26)
* EndArrow3:                             three.              (line  652)
* EndBar:                                draw.               (line   19)
* EndBar3:                               three.              (line  652)
* EndDotMargin:                          draw.               (line   89)
* EndDotMargin3:                         three.              (line  668)
* endl:                                  Files.              (line   65)
* EndMargin:                             draw.               (line   69)
* EndMargin3:                            three.              (line  668)
* EndPenMargin:                          draw.               (line   78)
* EndPenMargin2:                         three.              (line  668)
* EndPenMargin3:                         three.              (line  668)
* EndPoint:                              label.              (line   62)
* enumerate:                             Attaching a counter to an iterator.
                                                             (line    6)
* envelope:                              label.              (line  111)
* environment variables:                 Configuring.        (line   90)
* eof:                                   Files.              (line  101)
* eof <1>:                               Arrays.             (line  375)
* eol:                                   Files.              (line  101)
* eol <1>:                               Arrays.             (line  375)
* EPS:                                   label.              (line   85)
* EPS <1>:                               Options.            (line  200)
* erase:                                 Drawing in interactive mode.
                                                             (line   11)
* erase <1>:                             Data types.         (line  257)
* erase <2>:                             Frames and pictures.
                                                             (line    7)
* erase <3>:                             Frames and pictures.
                                                             (line  272)
* erf:                                   Mathematical functions.
                                                             (line    6)
* erfc:                                  Mathematical functions.
                                                             (line    6)
* error:                                 Files.              (line   16)
* error <1>:                             Files.              (line  101)
* error bars:                            graph.              (line  529)
* errorbars:                             graph.              (line  478)
* eval:                                  Import.             (line  110)
* eval <1>:                              Import.             (line  120)
* evenodd:                               Paths.              (line   37)
* evenodd <1>:                           Pens.               (line  164)
* exit:                                  Data types.         (line  373)
* exit <1>:                              Interactive mode.   (line   54)
* exit <2>:                              Debugger.           (line   56)
* exp:                                   Mathematical functions.
                                                             (line    6)
* expi:                                  Data types.         (line   86)
* expi <1>:                              Data types.         (line  177)
* explicit:                              Casts.              (line    6)
* explicit casts:                        Casts.              (line   21)
* expm1:                                 Mathematical functions.
                                                             (line    6)
* exponential integral:                  Mathematical functions.
                                                             (line   59)
* extendcap:                             Pens.               (line  139)
* extension:                             Paths and guides.   (line  246)
* extension <1>:                         MetaPost.           (line   10)
* external:                              embed.              (line   11)
* external <1>:                          external.           (line    6)
* extract, Set.extract:                  The Set API.        (line   85)
* extrude:                               three.              (line  580)
* F:                                     Mathematical functions.
                                                             (line   59)
* fabs:                                  Mathematical functions.
                                                             (line    6)
* face:                                  three.              (line  692)
* factorial:                             Mathematical functions.
                                                             (line   39)
* Fedora:                                UNIX binary distributions.
                                                             (line   15)
* feynman:                               feynman.            (line    6)
* fft:                                   Arrays.             (line  260)
* fft <1>:                               Arrays.             (line  274)
* fft <2>:                               Arrays.             (line  278)
* FFTW:                                  Compiling from UNIX source.
                                                             (line   48)
* file:                                  Files.              (line    6)
* file <1>:                              Debugger.           (line   44)
* fill:                                  draw.               (line  154)
* fill <1>:                              fill.               (line    6)
* fill <2>:                              fill.               (line   17)
* Fill:                                  Frames and pictures.
                                                             (line  134)
* filldraw:                              fill.               (line   11)
* FillDraw:                              Frames and pictures.
                                                             (line  124)
* filloutside:                           fill.               (line   27)
* fillrule:                              Pens.               (line  164)
* filltype:                              Frames and pictures.
                                                             (line  123)
* find:                                  Data types.         (line  242)
* find <1>:                              Arrays.             (line  169)
* findall:                               Arrays.             (line  174)
* firstcut:                              Paths and guides.   (line  262)
* fit3:                                  three.              (line  361)
* fixedscaling:                          Frames and pictures.
                                                             (line   68)
* floor:                                 Mathematical functions.
                                                             (line   26)
* flowchart:                             flowchart.          (line    6)
* flush:                                 Files.              (line   65)
* flush <1>:                             Files.              (line  101)
* fmod:                                  Mathematical functions.
                                                             (line    6)
* font:                                  Pens.               (line  259)
* font <1>:                              Pens.               (line  259)
* font <2>:                              Pens.               (line  288)
* font encoding:                         Pens.               (line  288)
* fontcommand:                           Pens.               (line  272)
* fontsize:                              Pens.               (line  245)
* fontsize <1>:                          fontsize.           (line    6)
* for:                                   Programming.        (line   31)
* for, range-based:                      Iterators.          (line    6)
* format:                                Data types.         (line  290)
* format <1>:                            Options.            (line  200)
* forum:                                 Help.               (line    6)
* frame:                                 Frames and pictures.
                                                             (line    7)
* freshnel0:                             three.              (line   66)
* from:                                  Import.             (line   16)
* FrontView:                             three.              (line  473)
* function declarations:                 Functions.          (line   82)
* Function shading:                      fill.               (line  100)
* function shading:                      fill.               (line  100)
* functions:                             Functions.          (line    6)
* functions <1>:                         Mathematical functions.
                                                             (line    6)
* functionshade:                         fill.               (line  100)
* gamma:                                 Mathematical functions.
                                                             (line    6)
* Gaussrand:                             Mathematical functions.
                                                             (line   39)
* geometry:                              geometry.           (line    6)
* get:                                   Iterators.          (line   17)
* get <1>:                               collections.iter(T).
                                                             (line   16)
* get, Set.get:                          The Set API.        (line   70)
* getc:                                  Files.              (line   32)
* getint:                                Files.              (line  126)
* getpair:                               Files.              (line  126)
* getreal:                               Files.              (line  126)
* getstring:                             Files.              (line  126)
* gettriple:                             Files.              (line  126)
* git:                                   Git.                (line    6)
* globalwrite:                           Files.              (line   40)
* globalwrite <1>:                       Files.              (line  158)
* glOptions:                             three.              (line  282)
* glOptions <1>:                         Options.            (line  186)
* GNU Scientific Library:                Mathematical functions.
                                                             (line   59)
* gouraudshade:                          fill.               (line   63)
* Gradient:                              palette.            (line   25)
* gradient shading:                      fill.               (line   32)
* graph:                                 graph.              (line    6)
* graph3:                                graph3.             (line    6)
* graphic:                               label.              (line   85)
* graphic <1>:                           Options.            (line  205)
* graphical user interface:              GUI.                (line    6)
* graphwithderiv:                        graph.              (line  668)
* gray:                                  Pens.               (line   25)
* grayscale:                             Pens.               (line   25)
* Grayscale:                             palette.            (line    9)
* grid:                                  Pens.               (line  338)
* grid <1>:                              graph.              (line  764)
* grid3:                                 grid3.              (line    6)
* gs:                                    Configuring.        (line   15)
* GSL:                                   Compiling from UNIX source.
                                                             (line   48)
* gsl:                                   Mathematical functions.
                                                             (line   59)
* gsOptions:                             Options.            (line  186)
* GUI:                                   GUI.                (line    6)
* GUI installation:                      GUI installation.   (line    6)
* GUI usage:                             GUI usage.          (line    6)
* guide:                                 Paths and guides.   (line  315)
* guide3:                                three.              (line    6)
* hash:                                  Hash functions.     (line    6)
* hash function:                         Hash functions.     (line    6)
* HashMap_K_V:                           Initializing a Map. (line   10)
* hatch:                                 Pens.               (line  355)
* Headlamp:                              three.              (line   76)
* height:                                LaTeX usage.        (line   47)
* help:                                  Interactive mode.   (line   42)
* help <1>:                              Help.               (line    6)
* help <2>:                              Debugger.           (line   30)
* Hermite:                               graph.              (line   36)
* Hermite(splinetype splinetype:         graph.              (line   36)
* hex:                                   Data types.         (line  306)
* hex <1>:                               Pens.               (line   64)
* hexadecimal:                           Data types.         (line  306)
* hexadecimal <1>:                       Pens.               (line   62)
* hidden surface removal:                three.              (line  692)
* histogram:                             Mathematical functions.
                                                             (line   39)
* history:                               Files.              (line  151)
* history <1>:                           Interactive mode.   (line   54)
* historylines:                          Interactive mode.   (line   58)
* HookHead:                              draw.               (line   50)
* HookHead3:                             three.              (line  652)
* Horizontal:                            flowchart.          (line   77)
* HTML5:                                 three.              (line  247)
* htmlviewer:                            Configuring.        (line   15)
* htmlviewer <1>:                        Configuring.        (line   38)
* htmlviewerOptions:                     Options.            (line  186)
* hyperrefOptions:                       Options.            (line  186)
* hypot:                                 Mathematical functions.
                                                             (line    6)
* I:                                     Mathematical functions.
                                                             (line   59)
* i_scaled:                              Mathematical functions.
                                                             (line   59)
* ibl:                                   three.              (line  117)
* iconify:                               three.              (line  282)
* identity:                              Transforms.         (line   24)
* identity <1>:                          Mathematical functions.
                                                             (line    6)
* identity <2>:                          Arrays.             (line  332)
* identity4:                             three.              (line  524)
* if:                                    Programming.        (line   31)
* IgnoreAspect:                          Frames and pictures.
                                                             (line   50)
* image:                                 palette.            (line   33)
* image <1>:                             palette.            (line   61)
* image-based lighting:                  three.              (line  117)
* ImageMagick:                           Configuring.        (line   69)
* ImageMagick <1>:                       animation.          (line    6)
* ImageMagick <2>:                       Options.            (line  200)
* images:                                palette.            (line    6)
* implicit casts:                        Casts.              (line    6)
* implicit linear solver:                MetaPost.           (line   10)
* implicit scaling:                      Implicit scaling.   (line    6)
* implicitsurface:                       smoothcontour3.     (line   16)
* import:                                Import.             (line   45)
* importv3d:                             three.              (line  339)
* inches:                                Figure size.        (line   18)
* incircle:                              Data types.         (line  120)
* include:                               Import.             (line  139)
* including images:                      label.              (line   85)
* increasing:                            math.               (line   55)
* inf:                                   Data types.         (line   35)
* inheritance:                           Structures.         (line  146)
* initialized:                           Arrays.             (line   50)
* initializers:                          Variable initializers.
                                                             (line    6)
* inline:                                LaTeX usage.        (line   47)
* InOutTicks:                            graph3.             (line   39)
* input:                                 Files.              (line   10)
* input <1>:                             Files.              (line   12)
* input <2>:                             Interactive mode.   (line   45)
* input <3>:                             Interactive mode.   (line   49)
* input encoding:                        Pens.               (line  288)
* insert:                                Data types.         (line  253)
* insert <1>:                            Arrays.             (line   50)
* inside:                                Paths and guides.   (line  294)
* inside <1>:                            Paths and guides.   (line  299)
* inside <2>:                            Paths and guides.   (line  305)
* insphere:                              three.              (line  615)
* inst:                                  Debugger.           (line   35)
* installation:                          Installation.       (line    6)
* int:                                   Data types.         (line   30)
* integer division:                      Arithmetic & logical.
                                                             (line   20)
* integral:                              Mathematical functions.
                                                             (line   93)
* integrate:                             Mathematical functions.
                                                             (line   93)
* interactive mode:                      Drawing in interactive mode.
                                                             (line    6)
* interactive mode <1>:                  Interactive mode.   (line    6)
* interior:                              Paths and guides.   (line  290)
* interp:                                Arithmetic & logical.
                                                             (line   64)
* interpolate:                           interpolate.        (line    6)
* intersect:                             Paths and guides.   (line  195)
* intersect <1>:                         three.              (line  586)
* intersect <2>:                         math.               (line   13)
* intersection, Set:                     The Set API.        (line  127)
* intersectionpoint:                     Paths and guides.   (line  238)
* intersectionpoint <1>:                 three.              (line  586)
* intersectionpoint <2>:                 math.               (line   17)
* intersectionpoints:                    Paths and guides.   (line  242)
* intersectionpoints <1>:                three.              (line  586)
* intersectionpoints <2>:                three.              (line  599)
* intersections:                         Paths and guides.   (line  206)
* intersections <1>:                     Paths and guides.   (line  213)
* intersections <2>:                     three.              (line  586)
* intersections <3>:                     three.              (line  592)
* InTicks:                               graph3.             (line   39)
* intMax:                                Data types.         (line   30)
* intMin:                                Data types.         (line   30)
* inverse:                               Transforms.         (line   16)
* inverse <1>:                           Arrays.             (line  338)
* invert:                                three.              (line  514)
* invisible:                             Pens.               (line   43)
* isnan:                                 Data types.         (line   35)
* isNullValue:                           Initializing a Map. (line   26)
* isNullValue <1>:                       The Map API.        (line    9)
* Iter_array_T:                          Combining (zipping) iterators.
                                                             (line   60)
* Iter_T:                                Iterators.          (line   31)
* Iter_T <1>:                            collections.iter(T).
                                                             (line    6)
* Iter_T <2>:                            collections.iter(T).
                                                             (line   48)
* Iterable:                              collections.iter(T).
                                                             (line   41)
* Iterable_array_T:                      Combining (zipping) iterators.
                                                             (line   58)
* Iterable_T:                            collections.iter(T).
                                                             (line    6)
* iterators:                             Iterators.          (line    6)
* J:                                     Mathematical functions.
                                                             (line    6)
* J <1>:                                 Mathematical functions.
                                                             (line   59)
* Japanese:                              Pens.               (line  297)
* K:                                     Mathematical functions.
                                                             (line   59)
* k_scaled:                              Mathematical functions.
                                                             (line   59)
* Kate:                                  Editing modes.      (line   48)
* KDE editor:                            Editing modes.      (line   48)
* keepAspect:                            Frames and pictures.
                                                             (line   45)
* keepAspect <1>:                        Frames and pictures.
                                                             (line   50)
* keepAspect <2>:                        LaTeX usage.        (line   47)
* keyboard bindings::                    three.              (line  225)
* keys:                                  Arrays.             (line   50)
* keys, Map.keys:                        The Map API.        (line   63)
* keyword:                               Named arguments.    (line   37)
* keyword-only:                          Named arguments.    (line   37)
* keywords:                              Named arguments.    (line    6)
* Korean:                                Pens.               (line  297)
* label:                                 Labels.             (line    6)
* Label:                                 draw.               (line  137)
* label <1>:                             label.              (line    6)
* Label <1>:                             label.              (line   21)
* Label <2>:                             graph.              (line  328)
* label <2>:                             three.              (line  550)
* labelmargin:                           label.              (line    6)
* labelpath:                             labelpath.          (line    6)
* labelpath3:                            labelpath3.         (line    6)
* labelx:                                graph.              (line  328)
* labely:                                graph.              (line  328)
* Landscape:                             Frames and pictures.
                                                             (line   92)
* language context:                      Pens.               (line  288)
* language server protocol:              Language server protocol.
                                                             (line    6)
* lastcut:                               Paths and guides.   (line  266)
* lasy-mode:                             Editing modes.      (line    6)
* latex:                                 Options.            (line  200)
* LaTeX NFSS fonts:                      Pens.               (line  259)
* LaTeX usage:                           LaTeX usage.        (line    6)
* latexmk:                               LaTeX usage.        (line   30)
* latitude:                              Data types.         (line  164)
* latticeshade:                          fill.               (line   32)
* layer:                                 Drawing commands.   (line   16)
* leastsquares:                          graph.              (line  945)
* leastsquares <1>:                      stats.              (line    6)
* Left:                                  graph.              (line  267)
* LeftRight:                             graph.              (line  273)
* LeftSide:                              label.              (line   67)
* LeftTicks:                             graph.              (line  158)
* LeftTicks <1>:                         graph.              (line  231)
* LeftView:                              three.              (line  473)
* legend:                                Drawing commands.   (line   34)
* legend <1>:                            draw.               (line  101)
* legend <2>:                            graph.              (line  422)
* Legendre:                              Mathematical functions.
                                                             (line   59)
* length:                                Data types.         (line   65)
* length <1>:                            Data types.         (line  144)
* length <2>:                            Data types.         (line  239)
* length <3>:                            Paths and guides.   (line   76)
* length <4>:                            Paths and guides.   (line  374)
* length <5>:                            Arrays.             (line   50)
* length <6>:                            three.              (line  586)
* letter:                                Configuring.        (line   63)
* lexorder:                              math.               (line   63)
* lexorder <1>:                          math.               (line   66)
* libcurl:                               Import.             (line  102)
* libm routines:                         Mathematical functions.
                                                             (line    6)
* libsigsegv:                            Functions.          (line  103)
* libsigsegv <1>:                        Help.               (line   27)
* licenses:                              Options.            (line  271)
* light:                                 three.              (line   76)
* limits:                                graph.              (line  637)
* line:                                  Arrays.             (line  375)
* line <1>:                              Arrays.             (line  378)
* line <2>:                              Arrays.             (line  383)
* line mode:                             Arrays.             (line  375)
* linear:                                graph.              (line   36)
* Linear:                                graph.              (line  708)
* linecap:                               Pens.               (line  139)
* linejoin:                              Pens.               (line  149)
* lineskip:                              Pens.               (line  245)
* linetype:                              Pens.               (line  123)
* linewidth:                             Pens.               (line  127)
* lmfit:                                 lmfit.              (line    6)
* locale:                                Data types.         (line  316)
* log:                                   Mathematical functions.
                                                             (line    6)
* Log:                                   graph.              (line  708)
* log-log graph:                         graph.              (line  742)
* log10:                                 Mathematical functions.
                                                             (line    6)
* log1p:                                 Mathematical functions.
                                                             (line    6)
* log2 graph:                            graph.              (line  798)
* logarithmic graph:                     graph.              (line  742)
* logical operators:                     Arithmetic & logical.
                                                             (line    6)
* longdashdotted:                        Pens.               (line  102)
* longdashed:                            Pens.               (line  102)
* longitude:                             Data types.         (line  169)
* loop:                                  Programming.        (line   31)
* LSP:                                   Language server protocol.
                                                             (line    6)
* lualatex:                              Options.            (line  200)
* luatex:                                Options.            (line  200)
* macOS X binary distributions:          macOS X binary distributions.
                                                             (line    6)
* macOS X configuration:                 Compiling from UNIX source.
                                                             (line   34)
* magick:                                Configuring.        (line   69)
* magick <1>:                            Files.              (line  163)
* magick <2>:                            animation.          (line    6)
* magick <3>:                            Options.            (line  200)
* makePair:                              Pairs.              (line    6)
* makepen:                               Pens.               (line  391)
* makeQueue:                             Queues.             (line   12)
* map:                                   Arrays.             (line  146)
* map <1>:                               Arrays.             (line  151)
* Map:                                   Maps.               (line    6)
* Map_K_V:                               Maps.               (line    6)
* Map_K_V <1>:                           Maps.               (line   41)
* Map_K_V, .add:                         The Map API.        (line   68)
* Map_K_V, .contains:                    The Map API.        (line   22)
* Map_K_V, .delete:                      The Map API.        (line   39)
* Map_K_V, .empty:                       The Map API.        (line   19)
* Map_K_V, .keys:                        The Map API.        (line   63)
* Map_K_V, .operator []:                 The Map API.        (line   25)
* Map_K_V, .operator [=]:                The Map API.        (line   31)
* Map_K_V, .operator iter:               The Map API.        (line   44)
* Map_K_V, .pairs:                       The Map API.        (line   55)
* Map_K_V, .size:                        The Map API.        (line   16)
* Map_K_V, cast to Iterable:             The Map API.        (line   73)
* Map_K_V, iterating over:               The Map API.        (line   44)
* Map_K_V, iterating over <1>:           The Map API.        (line   55)
* Margin:                                draw.               (line   69)
* Margin3:                               three.              (line  668)
* Margin3 <1>:                           three.              (line  668)
* Margins:                               draw.               (line   70)
* margins:                               three.              (line  354)
* Margins3:                              three.              (line  668)
* mark:                                  graph.              (line  478)
* markangle:                             markers.            (line   35)
* marker:                                graph.              (line  478)
* markers:                               markers.            (line    6)
* marknodes:                             graph.              (line  478)
* markuniform:                           graph.              (line  478)
* mask:                                  Data types.         (line   35)
* material:                              three.              (line   66)
* math:                                  math.               (line    6)
* mathematical functions:                Mathematical functions.
                                                             (line    6)
* max:                                   Paths and guides.   (line  279)
* max <1>:                               Frames and pictures.
                                                             (line    7)
* max <2>:                               Arrays.             (line  241)
* max <3>:                               Arrays.             (line  251)
* max <4>:                               three.              (line  586)
* max, SortedSet.max:                    The SortedSet API.  (line   32)
* maxbound:                              Data types.         (line  134)
* maxbound <1>:                          Data types.         (line  205)
* maxtile:                               three.              (line  282)
* maxtimes:                              Paths and guides.   (line  233)
* maxviewport:                           three.              (line  282)
* metallic:                              three.              (line   66)
* MetaPost:                              MetaPost.           (line    6)
* MetaPost ... :                         Bezier curves.      (line   71)
* MetaPost cutafter:                     Paths and guides.   (line  267)
* MetaPost cutbefore:                    Paths and guides.   (line  263)
* MetaPost pickup:                       Pens.               (line    6)
* MetaPost whatever:                     MetaPost.           (line   10)
* Microsoft Windows:                     Microsoft Windows.  (line    6)
* MidArcArrow:                           draw.               (line   30)
* MidArcArrow3:                          three.              (line  652)
* MidArrow:                              draw.               (line   26)
* MidArrow3:                             three.              (line  652)
* MidPoint:                              label.              (line   62)
* midpoint:                              Paths and guides.   (line  180)
* min:                                   Paths and guides.   (line  275)
* min <1>:                               Frames and pictures.
                                                             (line    7)
* min <2>:                               Arrays.             (line  236)
* min <3>:                               Arrays.             (line  246)
* min <4>:                               three.              (line  586)
* min, SortedSet.min:                    The SortedSet API.  (line   27)
* minbound:                              Data types.         (line  131)
* minbound <1>:                          Data types.         (line  202)
* minipage:                              label.              (line  125)
* mintimes:                              Paths and guides.   (line  228)
* miterjoin:                             Pens.               (line  149)
* miterlimit:                            Pens.               (line  159)
* mktemp:                                Files.              (line   48)
* mm:                                    Figure size.        (line   18)
* mobile browser:                        three.              (line  247)
* mode:                                  Files.              (line   80)
* mode <1>:                              Files.              (line   98)
* monotonic:                             graph.              (line   36)
* mouse:                                 GUI.                (line    6)
* mouse bindings:                        three.              (line  206)
* mouse wheel:                           GUI usage.          (line    6)
* Move:                                  Pens.               (line  428)
* MoveQuiet:                             Pens.               (line  434)
* multisample:                           three.              (line  198)
* N:                                     Labels.             (line   18)
* name:                                  Files.              (line   98)
* named arguments:                       Named arguments.    (line    6)
* nan:                                   Data types.         (line   35)
* natural:                               graph.              (line   36)
* new:                                   Structures.         (line    6)
* new <1>:                               Arrays.             (line  111)
* new <2>:                               Arrays.             (line  114)
* newEmpty, Set.newEmpty:                The Set API.        (line   55)
* newframe:                              Frames and pictures.
                                                             (line    7)
* newl:                                  Files.              (line   65)
* newpage:                               Drawing commands.   (line   27)
* newton:                                Mathematical functions.
                                                             (line   77)
* newton <1>:                            Mathematical functions.
                                                             (line   84)
* next:                                  Debugger.           (line   41)
* nobasealign:                           Pens.               (line  181)
* NoFill:                                Frames and pictures.
                                                             (line  142)
* noglobalread:                          Files.              (line   26)
* noglobalread <1>:                      Files.              (line   40)
* nolight:                               three.              (line   76)
* NoMargin:                              draw.               (line   67)
* NoMargin3:                             three.              (line  668)
* None:                                  draw.               (line   19)
* None <1>:                              draw.               (line   26)
* none:                                  Files.              (line   65)
* normal:                                three.              (line  572)
* nosafe:                                Options.            (line  221)
* NOT:                                   Arithmetic & logical.
                                                             (line   68)
* notaknot:                              graph.              (line   36)
* NoTicks:                               graph.              (line  158)
* NoTicks3:                              graph3.             (line   39)
* null:                                  Structures.         (line    6)
* nullpen:                               label.              (line   21)
* nullpen <1>:                           Frames and pictures.
                                                             (line  128)
* nullpen <2>:                           Frames and pictures.
                                                             (line  137)
* nullValue:                             Initializing a Map. (line   26)
* nullValue <1>:                         The Map API.        (line    9)
* O:                                     three.              (line  366)
* obj:                                   obj.                (line    6)
* object:                                label.              (line  111)
* oblique:                               three.              (line  409)
* obliqueX:                              three.              (line  416)
* obliqueY:                              three.              (line  422)
* obliqueZ:                              three.              (line  409)
* ode:                                   ode.                (line    6)
* offset:                                Pens.               (line  123)
* offset <1>:                            Options.            (line  226)
* OmitTick:                              graph.              (line  221)
* OmitTickInterval:                      graph.              (line  221)
* OmitTickIntervals:                     graph.              (line  221)
* opacity:                               Pens.               (line  307)
* opacity <1>:                           three.              (line   66)
* open:                                  Files.              (line   12)
* OpenGL:                                three.              (line  198)
* operator:                              User-defined operators.
                                                             (line    6)
* operator --:                           graph.              (line   30)
* operator ..:                           graph.              (line   33)
* operator []:                           Bracket operators.  (line    6)
* operator [], Map.operator []:          The Map API.        (line   25)
* operator [=]:                          Bracket operators.  (line    6)
* operator [=], Map.operator [=]:        The Map API.        (line   31)
* operator +(...string[] a).:            Data types.         (line  284)
* operator ==:                           Hashing user-defined structures.
                                                             (line   34)
* operator answer:                       Interactive mode.   (line   35)
* operator cast:                         Casts.              (line   38)
* operator ecast:                        Casts.              (line   65)
* operator init:                         Variable initializers.
                                                             (line    6)
* operator init <1>:                     Structures.         (line  104)
* operator iter:                         Iterators.          (line    6)
* operator iter <1>:                     collections.iter(T).
                                                             (line   20)
* operator iter, Map.operator iter:      The Map API.        (line   44)
* operator iter, Queue_T.operator iter:  Queues.             (line   34)
* operator iter, Set.operator iter:      The Set API.        (line   44)
* operators:                             Operators.          (line    6)
* options:                               Options.            (line    6)
* OR:                                    Arithmetic & logical.
                                                             (line   68)
* orient:                                Data types.         (line  108)
* orient <1>:                            three.              (line  603)
* orientation:                           Frames and pictures.
                                                             (line   92)
* orthographic:                          three.              (line  426)
* outformat:                             three.              (line  198)
* outprefix:                             Frames and pictures.
                                                             (line   78)
* output:                                Files.              (line   38)
* output <1>:                            Options.            (line  200)
* OutTicks:                              graph3.             (line   39)
* overloading functions:                 Functions.          (line   58)
* overwrite:                             Pens.               (line  413)
* P:                                     Mathematical functions.
                                                             (line   59)
* pack:                                  label.              (line  109)
* packing:                               Rest arguments.     (line   37)
* pad:                                   Frames and pictures.
                                                             (line  174)
* pair:                                  Figure size.        (line    6)
* pair <1>:                              Data types.         (line   46)
* Pair_K_V:                              Pairs.              (line    6)
* pairs:                                 Arrays.             (line  256)
* pairs, Map.pairs:                      The Map API.        (line   55)
* paperheight:                           Configuring.        (line   63)
* papertype:                             Configuring.        (line   63)
* paperwidth:                            Configuring.        (line   63)
* parallelogram:                         flowchart.          (line   47)
* parametric surface:                    graph3.             (line  103)
* parametrized curve:                    graph.              (line  637)
* partialsum:                            math.               (line   49)
* partialsum <1>:                        math.               (line   52)
* patch-dependent colors:                three.              (line  132)
* path:                                  Paths.              (line    6)
* path <1>:                              Paths and guides.   (line    7)
* path <2>:                              three.              (line   42)
* path <3>:                              flowchart.          (line   77)
* path markers:                          graph.              (line  478)
* path[]:                                Paths.              (line   23)
* path3:                                 three.              (line    6)
* path3 <1>:                             three.              (line   42)
* patterns:                              Pens.               (line  324)
* patterns <1>:                          patterns.           (line    6)
* PBR:                                   three.              (line   74)
* PDF:                                   Options.            (line  200)
* pdflatex:                              Options.            (line  200)
* pdfreloadOptions:                      Options.            (line  186)
* pdfviewer:                             Configuring.        (line   15)
* pdfviewerOptions:                      Options.            (line  186)
* peek, Queue_T.peek:                    Queues.             (line   27)
* pen:                                   Pens.               (line    6)
* PenMargin:                             draw.               (line   78)
* PenMargin2:                            three.              (line  668)
* PenMargin3:                            three.              (line  668)
* PenMargins:                            draw.               (line   79)
* PenMargins2:                           three.              (line  668)
* PenMargins3:                           three.              (line  668)
* periodic:                              graph.              (line   36)
* perl:                                  LaTeX usage.        (line   30)
* perpendicular:                         geometry.           (line    6)
* perspective:                           three.              (line  453)
* physically based rendering:            three.              (line   74)
* picture:                               Frames and pictures.
                                                             (line   25)
* picture alignment:                     Frames and pictures.
                                                             (line  219)
* picture.add:                           Deferred drawing.   (line   31)
* picture.addPoint:                      Deferred drawing.   (line   51)
* picture.calculateTransform:            Frames and pictures.
                                                             (line  106)
* picture.fit:                           Frames and pictures.
                                                             (line  101)
* picture.scale:                         Frames and pictures.
                                                             (line  111)
* piecewisestraight:                     Paths and guides.   (line   92)
* pixel:                                 three.              (line  675)
* Pl:                                    Mathematical functions.
                                                             (line   59)
* plain:                                 plain.              (line    6)
* planar:                                three.              (line  142)
* plane:                                 three.              (line  392)
* planeproject:                          three.              (line  569)
* point:                                 Paths and guides.   (line   95)
* point <1>:                             Paths and guides.   (line  380)
* point <2>:                             three.              (line  586)
* polar:                                 Data types.         (line  149)
* polargraph:                            graph.              (line   89)
* polygon:                               graph.              (line  478)
* pop:                                   Arrays.             (line   50)
* pop, Queue_T.pop:                      Queues.             (line   23)
* popMax, SortedSet.popMax:              The SortedSet API.  (line   42)
* popMin, SortedSet.popMin:              The SortedSet API.  (line   37)
* Portrait:                              Frames and pictures.
                                                             (line   92)
* position:                              three.              (line   76)
* position <1>:                          three.              (line  198)
* postcontrol:                           Paths and guides.   (line  146)
* postcontrol <1>:                       three.              (line  586)
* postfix operators:                     Self & prefix operators.
                                                             (line   19)
* PostScript:                            Frames and pictures.
                                                             (line  285)
* PostScript fonts:                      Pens.               (line  275)
* PostScript subpath:                    Paths.              (line   23)
* pow10:                                 Mathematical functions.
                                                             (line    6)
* prc:                                   three.              (line  300)
* precision:                             Files.              (line  101)
* precontrol:                            Paths and guides.   (line  139)
* precontrol <1>:                        three.              (line  586)
* prefix operators:                      Self & prefix operators.
                                                             (line    6)
* private:                               Structures.         (line    6)
* programming:                           Programming.        (line    6)
* pstoedit:                              pstoedit.           (line    6)
* pstoedit <1>:                          PostScript to Asymptote.
                                                             (line    6)
* psviewer:                              Configuring.        (line   15)
* psviewerOptions:                       Options.            (line  186)
* pt:                                    Figure size.        (line   18)
* public:                                Structures.         (line    6)
* push:                                  Arrays.             (line   50)
* push, Queue_T.push:                    Queues.             (line   20)
* push, Set.push:                        The Set API.        (line   75)
* Python usage:                          Interactive mode.   (line   72)
* quadraticroots:                        Arrays.             (line  341)
* quadraticroots <1>:                    Arrays.             (line  346)
* quarticroots:                          math.               (line   22)
* Queue_T, .operator iter:               Queues.             (line   34)
* Queue_T, .peek:                        Queues.             (line   27)
* Queue_T, .pop:                         Queues.             (line   23)
* Queue_T, .push:                        Queues.             (line   20)
* Queue_T, .size:                        Queues.             (line   31)
* Queue_T, cast to Iterable:             Queues.             (line   44)
* Queue_T, iterating over:               Queues.             (line   34)
* quick reference:                       Description.        (line   92)
* quit:                                  Drawing in interactive mode.
                                                             (line   11)
* quit <1>:                              Interactive mode.   (line   54)
* quit <2>:                              Debugger.           (line   53)
* quote:                                 Import.             (line  128)
* quotient:                              Arithmetic & logical.
                                                             (line    6)
* radialshade:                           fill.               (line   52)
* RadialShade:                           Frames and pictures.
                                                             (line  160)
* RadialShadeDraw:                       Frames and pictures.
                                                             (line  164)
* radians:                               Mathematical functions.
                                                             (line   17)
* radius:                                Paths and guides.   (line  135)
* radius <1>:                            three.              (line  586)
* Rainbow:                               palette.            (line   12)
* rand:                                  Mathematical functions.
                                                             (line   39)
* randMax:                               Mathematical functions.
                                                             (line   39)
* randString:                            Mathematical functions.
                                                             (line   39)
* range:                                 range.              (line    6)
* range <1>:                             collections.iter(T).
                                                             (line   45)
* rational:                              rational.           (line    6)
* rationalSimplex:                       rationalSimplex.    (line    6)
* read:                                  Arrays.             (line  378)
* read <1>:                              Arrays.             (line  418)
* reading:                               Files.              (line   12)
* reading string arrays:                 Arrays.             (line  388)
* readline:                              Files.              (line  143)
* real:                                  Data types.         (line   35)
* realDigits:                            Data types.         (line   35)
* realEpsilon:                           Data types.         (line   35)
* realMax:                               Data types.         (line   35)
* realMin:                               Data types.         (line   35)
* realmult:                              Data types.         (line  100)
* realschur:                             Arrays.             (line  282)
* rectangle:                             flowchart.          (line   34)
* recursion:                             Functions.          (line  103)
* reference:                             Description.        (line   92)
* reflect:                               Transforms.         (line   42)
* Relative:                              label.              (line   57)
* Relative <1>:                          label.              (line   67)
* relpoint:                              Paths and guides.   (line  176)
* reltime:                               Paths and guides.   (line  172)
* remainder:                             Mathematical functions.
                                                             (line    6)
* rename:                                Files.              (line  160)
* render:                                three.              (line   46)
* render <1>:                            three.              (line  198)
* render <2>:                            Options.            (line  200)
* replace:                               Data types.         (line  270)
* resetdefaultpen:                       Pens.               (line  440)
* rest arguments:                        Rest arguments.     (line    6)
* restore:                               Frames and pictures.
                                                             (line  279)
* restricted:                            Structures.         (line    6)
* return:                                Debugger.           (line   47)
* reverse:                               Data types.         (line  266)
* reverse <1>:                           Paths and guides.   (line  183)
* reverse <2>:                           Paths and guides.   (line  383)
* reverse <3>:                           Arrays.             (line  156)
* reverse <4>:                           three.              (line  586)
* rewind:                                Files.              (line  101)
* rfind:                                 Data types.         (line  247)
* rgb:                                   Pens.               (line   30)
* rgb <1>:                               Pens.               (line   34)
* rgb <2>:                               Pens.               (line   62)
* Riemann zeta function:                 Mathematical functions.
                                                             (line   59)
* Right:                                 graph.              (line  270)
* RightSide:                             label.              (line   67)
* RightTicks:                            graph.              (line  158)
* RightTicks <1>:                        graph.              (line  231)
* RightView:                             three.              (line  473)
* Rotate:                                label.              (line   43)
* rotate:                                Transforms.         (line   40)
* rotate <1>:                            three.              (line  540)
* Rotate(pair z):                        label.              (line   46)
* round:                                 Mathematical functions.
                                                             (line   26)
* roundcap:                              Pens.               (line  139)
* roundedpath:                           roundedpath.        (line    6)
* roundjoin:                             Pens.               (line  149)
* roundrectangle:                        flowchart.          (line   66)
* RPM:                                   UNIX binary distributions.
                                                             (line    6)
* runtime imports:                       Import.             (line  110)
* Russian:                               Pens.               (line  291)
* S:                                     Labels.             (line   18)
* safe:                                  Options.            (line  221)
* sameElementsInOrder:                   The Set API.        (line  111)
* save:                                  Frames and pictures.
                                                             (line  276)
* saveline:                              Files.              (line  143)
* Scale:                                 label.              (line   52)
* scale:                                 Pens.               (line  123)
* scale <1>:                             Transforms.         (line   34)
* scale <2>:                             Transforms.         (line   36)
* scale <3>:                             graph.              (line  708)
* Scale <1>:                             graph.              (line  725)
* scale <4>:                             three.              (line  539)
* scale3:                                three.              (line  537)
* scaled graph:                          graph.              (line  688)
* schur:                                 Arrays.             (line  282)
* schur <1>:                             Arrays.             (line  286)
* scientific graph:                      graph.              (line  385)
* scroll:                                Files.              (line  117)
* search:                                Arrays.             (line  177)
* search <1>:                            Arrays.             (line  183)
* search paths:                          Search paths.       (line    6)
* Seascape:                              Frames and pictures.
                                                             (line   98)
* secondary axis:                        graph.              (line  851)
* secondaryX:                            graph.              (line  851)
* secondaryY:                            graph.              (line  851)
* seconds:                               Data types.         (line  330)
* seek:                                  Files.              (line  101)
* seekeof:                               Files.              (line  101)
* segment:                               math.               (line   46)
* segmentation fault:                    Help.               (line   27)
* self operators:                        Self & prefix operators.
                                                             (line    6)
* sequence:                              Arrays.             (line  129)
* Set, -:                                The Set API.        (line  123)
* Set, !=:                               The Set API.        (line  107)
* Set, .add:                             The Set API.        (line   24)
* Set, .add <1>:                         The Set API.        (line   30)
* Set, .contains:                        The Set API.        (line   19)
* Set, .delete:                          The Set API.        (line   34)
* Set, .delete <1>:                      The Set API.        (line   39)
* Set, .empty:                           The Set API.        (line   16)
* Set, .extract:                         The Set API.        (line   85)
* Set, .get:                             The Set API.        (line   70)
* Set, .newEmpty:                        The Set API.        (line   55)
* Set, .operator iter:                   The Set API.        (line   44)
* Set, .push:                            The Set API.        (line   75)
* Set, .size:                            The Set API.        (line   13)
* Set, &:                                The Set API.        (line  127)
* Set, ^:                                The Set API.        (line  130)
* Set, +:                                The Set API.        (line  120)
* Set, <=:                               The Set API.        (line   98)
* Set, ==:                               The Set API.        (line  104)
* Set, >=:                               The Set API.        (line  101)
* Set, difference:                       The Set API.        (line  123)
* Set, intersection:                     The Set API.        (line  127)
* Set, iterating over:                   The Set API.        (line   44)
* Set, sameElementsInOrder:              The Set API.        (line  111)
* Set, symmetric difference:             The Set API.        (line  130)
* Set, union:                            The Set API.        (line  120)
* settings:                              Configuring.        (line   15)
* settings <1>:                          Options.            (line  171)
* sgn:                                   Mathematical functions.
                                                             (line   26)
* shading:                               fill.               (line   32)
* Shift:                                 label.              (line   40)
* shift:                                 Transforms.         (line   26)
* shift <1>:                             Transforms.         (line   28)
* shift <2>:                             Transforms.         (line   46)
* shift <3>:                             three.              (line  529)
* shiftless:                             Transforms.         (line   46)
* shininess:                             three.              (line   66)
* shipout:                               Frames and pictures.
                                                             (line   78)
* showtarget:                            three.              (line  426)
* Si:                                    Mathematical functions.
                                                             (line   59)
* signedint:                             Files.              (line   80)
* signedint <1>:                         Files.              (line   98)
* SimpleHead:                            draw.               (line   50)
* simplex:                               simplex.            (line    6)
* simplex2:                              simplex2.           (line    6)
* simpson:                               Mathematical functions.
                                                             (line   93)
* sin:                                   Mathematical functions.
                                                             (line    6)
* Sin:                                   Mathematical functions.
                                                             (line   20)
* single precision:                      Files.              (line   80)
* singleint:                             Files.              (line   80)
* singleint <1>:                         Files.              (line   98)
* singlereal:                            Files.              (line   80)
* singlereal <1>:                        Files.              (line   98)
* sinh:                                  Mathematical functions.
                                                             (line    6)
* SixViews:                              three.              (line  488)
* SixViewsFR:                            three.              (line  488)
* SixViewsUS:                            three.              (line  488)
* size:                                  Figure size.        (line    6)
* size <1>:                              Paths and guides.   (line   81)
* size <2>:                              Paths and guides.   (line  371)
* size <3>:                              Frames and pictures.
                                                             (line   34)
* size <4>:                              Frames and pictures.
                                                             (line   61)
* size <5>:                              three.              (line  586)
* size <6>:                              Options.            (line  200)
* size, Map.size:                        The Map API.        (line   16)
* size, Queue_T.size:                    Queues.             (line   31)
* size, Set.size:                        The Set API.        (line   13)
* size10:                                size10.             (line    6)
* size11:                                size11.             (line    6)
* size3:                                 three.              (line  351)
* Slant:                                 label.              (line   49)
* slant:                                 Transforms.         (line   38)
* sleep:                                 Data types.         (line  376)
* slice:                                 Paths and guides.   (line  251)
* slice <1>:                             Paths and guides.   (line  262)
* slices:                                Slices.             (line    6)
* slide:                                 slide.              (line    6)
* slope:                                 math.               (line   40)
* slope <1>:                             math.               (line   43)
* slopefield:                            slopefield.         (line    6)
* smoothcontour3:                        smoothcontour3.     (line    6)
* sncndn:                                Mathematical functions.
                                                             (line   59)
* solid:                                 Pens.               (line  102)
* solids:                                solids.             (line    6)
* solve:                                 Arrays.             (line  310)
* solve <1>:                             Arrays.             (line  326)
* sort:                                  Arrays.             (line  197)
* sort <1>:                              Arrays.             (line  201)
* sort <2>:                              Arrays.             (line  216)
* SortedSet:                             The Set API.        (line  139)
* SortedSet, .after:                     The SortedSet API.  (line   47)
* SortedSet, .atOrAfter:                 The SortedSet API.  (line   52)
* SortedSet, .atOrBefore:                The SortedSet API.  (line   62)
* SortedSet, .before:                    The SortedSet API.  (line   57)
* SortedSet, .max:                       The SortedSet API.  (line   32)
* SortedSet, .min:                       The SortedSet API.  (line   27)
* SortedSet, .popMax:                    The SortedSet API.  (line   42)
* SortedSet, .popMin:                    The SortedSet API.  (line   37)
* SortedSet, iterating over:             The SortedSet API.  (line   67)
* SortedSet_T:                           The Set API.        (line  140)
* specialty modules:                     Specialty modules.  (line    6)
* specular:                              three.              (line   76)
* specularfactor:                        three.              (line   76)
* specularpen:                           three.              (line   66)
* Spline:                                graph.              (line   33)
* Spline <1>:                            graph3.             (line  103)
* split:                                 Data types.         (line  279)
* sqrt:                                  Mathematical functions.
                                                             (line    6)
* squarecap:                             Pens.               (line  139)
* srand:                                 Mathematical functions.
                                                             (line   39)
* stack overflow:                        Functions.          (line  103)
* stack overflow <1>:                    Functions.          (line  103)
* stack overflow <2>:                    Help.               (line   27)
* static:                                Static.             (line    6)
* stats:                                 stats.              (line    6)
* stdin:                                 Files.              (line   52)
* stdout:                                Files.              (line   52)
* step:                                  Debugger.           (line   38)
* stickframe:                            markers.            (line   16)
* stop:                                  Debugger.           (line   10)
* straight:                              Paths and guides.   (line   88)
* Straight:                              graph.              (line   30)
* straight <1>:                          three.              (line  586)
* strftime:                              Data types.         (line  321)
* strftime <1>:                          Data types.         (line  346)
* string:                                Data types.         (line  208)
* string <1>:                            Data types.         (line  312)
* stroke:                                fill.               (line   36)
* stroke <1>:                            clip.               (line    6)
* strokepath:                            Paths and guides.   (line  310)
* strptime:                              Data types.         (line  330)
* struct:                                Structures.         (line    6)
* structures:                            Structures.         (line    6)
* subpath:                               Paths and guides.   (line  186)
* subpath <1>:                           three.              (line  586)
* subpictures:                           Frames and pictures.
                                                             (line  101)
* substr:                                Data types.         (line  262)
* sum:                                   Arrays.             (line  231)
* superpath:                             Paths.              (line   23)
* Suppress:                              Pens.               (line  420)
* SuppressQuiet:                         Pens.               (line  424)
* surface:                               three.              (line   46)
* surface <1>:                           three.              (line  117)
* surface <2>:                           three.              (line  142)
* surface <3>:                           three.              (line  156)
* surface <4>:                           graph3.             (line  103)
* surface <5>:                           graph3.             (line  132)
* SVG:                                   Options.            (line  205)
* symmetric difference:                  The Set API.        (line  130)
* system:                                Data types.         (line  354)
* system <1>:                            Options.            (line  221)
* syzygy:                                syzygy.             (line    6)
* tab:                                   Files.              (line   65)
* tab completion:                        Drawing in interactive mode.
                                                             (line   11)
* tan:                                   Mathematical functions.
                                                             (line    6)
* Tan:                                   Mathematical functions.
                                                             (line   20)
* tanh:                                  Mathematical functions.
                                                             (line    6)
* target:                                three.              (line  426)
* tell:                                  Files.              (line  101)
* template:                              Templated imports.  (line    6)
* tension:                               Bezier curves.      (line   56)
* tension <1>:                           three.              (line    6)
* tensionSpecifier:                      Paths and guides.   (line  403)
* tensor product shading:                fill.               (line   78)
* tensorshade:                           fill.               (line   78)
* tessellation:                          three.              (line  168)
* tex:                                   Frames and pictures.
                                                             (line  293)
* tex <1>:                               Options.            (line  200)
* TeX fonts:                             Pens.               (line  266)
* TeX string:                            Data types.         (line  208)
* texcolors:                             texcolors.          (line    6)
* texcommand:                            Configuring.        (line   69)
* TeXHead:                               draw.               (line   50)
* TeXHead3:                              three.              (line  652)
* texpath:                               Configuring.        (line   69)
* texpath <1>:                           label.              (line  122)
* texpreamble:                           Frames and pictures.
                                                             (line  302)
* texreset:                              Frames and pictures.
                                                             (line  306)
* textbook graph:                        graph.              (line  358)
* tgz:                                   UNIX binary distributions.
                                                             (line    6)
* thick:                                 three.              (line  180)
* thin:                                  three.              (line  180)
* third-party licenses:                  Options.            (line  271)
* this:                                  Structures.         (line    6)
* three:                                 three.              (line    6)
* ThreeViews:                            three.              (line  488)
* ThreeViewsFR:                          three.              (line  488)
* ThreeViewsUS:                          three.              (line  488)
* tick:                                  graph.              (line  328)
* ticks:                                 graph.              (line  158)
* Ticks:                                 graph.              (line  158)
* Ticks <1>:                             graph.              (line  231)
* tildeframe:                            markers.            (line   24)
* tile:                                  Pens.               (line  338)
* tilings:                               Pens.               (line  324)
* time:                                  Data types.         (line  321)
* time <1>:                              Data types.         (line  346)
* time <2>:                              math.               (line   26)
* time <3>:                              math.               (line   30)
* times:                                 Paths and guides.   (line  220)
* times <1>:                             Paths and guides.   (line  224)
* Top:                                   graph.              (line  133)
* TopView:                               three.              (line  473)
* trace:                                 Debugger.           (line   50)
* trailingzero:                          graph.              (line  173)
* transform:                             Transforms.         (line    6)
* transform <1>:                         three.              (line  561)
* transform3:                            three.              (line  524)
* transparency:                          Pens.               (line  307)
* transparent:                           three.              (line   97)
* transpose:                             Arrays.             (line  223)
* transpose <1>:                         Arrays.             (line  226)
* tree:                                  tree.               (line    6)
* trembling:                             trembling.          (line    6)
* triangle:                              geometry.           (line    6)
* triangles:                             three.              (line  168)
* triangulate:                           contour.            (line  189)
* tridiagonal:                           Arrays.             (line  297)
* trigonometric integrals:               Mathematical functions.
                                                             (line   59)
* triple:                                Data types.         (line  137)
* TrueMargin:                            draw.               (line   90)
* TrueMargin3:                           three.              (line  668)
* tube:                                  three.              (line  180)
* tube <1>:                              tube.               (line    6)
* tutorial:                              Tutorial.           (line    6)
* type1cm:                               Pens.               (line  245)
* typedef:                               Data types.         (line  385)
* typedef <1>:                           Functions.          (line   48)
* U3D:                                   embed.              (line   22)
* undefined:                             Paths and guides.   (line  283)
* unfill:                                fill.               (line  110)
* UnFill:                                Frames and pictures.
                                                             (line  153)
* UnFill <1>:                            Frames and pictures.
                                                             (line  156)
* uniform:                               Arrays.             (line  165)
* uninstall:                             Uninstall.          (line    6)
* union:                                 The Set API.        (line  120)
* unique:                                math.               (line   59)
* unit:                                  Data types.         (line   83)
* unit <1>:                              Data types.         (line  174)
* unitbox:                               Paths.              (line   44)
* unitbox <1>:                           three.              (line  398)
* unitcircle:                            Paths.              (line   17)
* unitcircle <1>:                        Paths.              (line   17)
* unitcircle <2>:                        three.              (line  366)
* unitrand:                              Mathematical functions.
                                                             (line   39)
* unitsize:                              Figure size.        (line   39)
* unitsize <1>:                          Frames and pictures.
                                                             (line   56)
* UNIX binary distributions:             UNIX binary distributions.
                                                             (line    6)
* unpacking:                             Rest arguments.     (line   46)
* unravel:                               Import.             (line   29)
* up:                                    three.              (line  426)
* update:                                Files.              (line   38)
* UpsideDown:                            Frames and pictures.
                                                             (line   92)
* UpsideDown <1>:                        Frames and pictures.
                                                             (line   99)
* URL:                                   Import.             (line  102)
* usepackage:                            Frames and pictures.
                                                             (line  309)
* user coordinates:                      Figure size.        (line   39)
* user modules:                          User modules.       (line    6)
* user-defined operators:                User-defined operators.
                                                             (line    6)
* using:                                 Data types.         (line  385)
* using <1>:                             Functions.          (line   48)
* usleep:                                Data types.         (line  379)
* v3d:                                   three.              (line  321)
* v3d <1>:                               v3d.                (line    6)
* valid:                                 Iterators.          (line   25)
* valid <1>:                             collections.iter(T).
                                                             (line   17)
* value:                                 math.               (line   34)
* value <1>:                             math.               (line   37)
* var:                                   Variable initializers.
                                                             (line   55)
* variable initializers:                 Variable initializers.
                                                             (line    6)
* vectorfield:                           graph.              (line 1020)
* vectorfield <1>:                       graph.              (line 1059)
* vectorfield3:                          graph3.             (line  171)
* vectorization:                         Arrays.             (line  354)
* verbatim:                              Frames and pictures.
                                                             (line  285)
* vertex-dependent colors:               three.              (line  132)
* Vertical:                              flowchart.          (line   77)
* Viewport:                              three.              (line   76)
* viewportheight:                        LaTeX usage.        (line   47)
* viewportmargin:                        three.              (line  354)
* viewportsize:                          three.              (line  354)
* viewportwidth:                         LaTeX usage.        (line   47)
* views:                                 three.              (line  300)
* vim:                                   Editing modes.      (line   32)
* virtual functions:                     Structures.         (line  146)
* void:                                  Data types.         (line   10)
* W:                                     Labels.             (line   18)
* warn:                                  Configuring.        (line   79)
* WebGL:                                 three.              (line  247)
* whatever:                              Paths and guides.   (line  246)
* Wheel:                                 palette.            (line   22)
* wheel mouse:                           GUI.                (line    6)
* while:                                 Programming.        (line   52)
* White:                                 three.              (line   76)
* white-space string delimiter mode:     Arrays.             (line  388)
* width:                                 LaTeX usage.        (line   47)
* windingnumber:                         Paths and guides.   (line  283)
* word:                                  Arrays.             (line  378)
* word <1>:                              Arrays.             (line  388)
* write:                                 Files.              (line   57)
* write <1>:                             Arrays.             (line  427)
* X:                                     three.              (line  366)
* x11colors:                             x11colors.          (line    6)
* xasy:                                  GUI.                (line    6)
* xaxis3:                                graph3.             (line    7)
* xdr:                                   Files.              (line   80)
* xelatex:                               Options.            (line  200)
* XEquals:                               graph.              (line  263)
* xequals:                               graph.              (line  276)
* xlimits:                               graph.              (line  637)
* XOR:                                   Arithmetic & logical.
                                                             (line   68)
* xpart:                                 Data types.         (line   94)
* xpart <1>:                             Data types.         (line  185)
* xscale:                                Transforms.         (line   30)
* xscale3:                               three.              (line  531)
* xtick:                                 graph.              (line  328)
* XY:                                    three.              (line  546)
* XY <1>:                                three.              (line  561)
* XYEquals:                              graph3.             (line   25)
* XYZero:                                graph3.             (line   25)
* XZEquals:                              graph3.             (line   25)
* XZero:                                 graph.              (line  258)
* XZZero:                                graph3.             (line   25)
* Y:                                     Mathematical functions.
                                                             (line    6)
* Y <1>:                                 Mathematical functions.
                                                             (line   59)
* Y <2>:                                 three.              (line  366)
* yaxis3:                                graph3.             (line    7)
* YEquals:                               graph.              (line  126)
* yequals:                               graph.              (line  276)
* ylimits:                               graph.              (line  637)
* ypart:                                 Data types.         (line   97)
* ypart <1>:                             Data types.         (line  188)
* yscale:                                Transforms.         (line   32)
* yscale3:                               three.              (line  533)
* ytick:                                 graph.              (line  328)
* YX:                                    three.              (line  561)
* YZ:                                    three.              (line  561)
* YZEquals:                              graph3.             (line   25)
* YZero:                                 graph.              (line  121)
* YZZero:                                graph3.             (line   25)
* Z:                                     three.              (line  366)
* zaxis3:                                graph3.             (line    7)
* zero_Ai:                               Mathematical functions.
                                                             (line   59)
* zero_Ai_deriv:                         Mathematical functions.
                                                             (line   59)
* zero_Bi:                               Mathematical functions.
                                                             (line   59)
* zero_Bi_deriv:                         Mathematical functions.
                                                             (line   59)
* zero_J:                                Mathematical functions.
                                                             (line   59)
* zeroTransform:                         Transforms.         (line   44)
* zerowinding:                           Pens.               (line  164)
* zeta:                                  Mathematical functions.
                                                             (line   59)
* zip:                                   Combining (zipping) iterators.
                                                             (line    6)
* zip <1>:                               Combining (zipping) iterators.
                                                             (line   64)
* zpart:                                 Data types.         (line  191)
* zscale3:                               three.              (line  535)
* ZX:                                    three.              (line  561)
* ZX <1>:                                three.              (line  561)
* ZY:                                    three.              (line  561)


Tag Table:
Node: Top571
Node: Description10551
Node: Installation14916
Node: UNIX binary distributions16061
Node: macOS X binary distributions17204
Node: Microsoft Windows17816
Node: Configuring19053
Node: Search paths23568
Node: Compiling from UNIX source24660
Node: Editing modes27402
Node: Git29965
Node: Building the documentation30465
Node: Uninstall30851
Node: Tutorial31212
Node: Drawing in batch mode32077
Node: Drawing in interactive mode32984
Node: Figure size34055
Node: Labels35740
Node: Paths36626
Ref: unitcircle37266
Node: Drawing commands39219
Node: draw41087
Ref: arrows42338
Node: fill48759
Ref: gradient shading49841
Node: clip54716
Node: label55493
Ref: Label56407
Ref: baseline60321
Ref: envelope61058
Node: Bezier curves62610
Node: Programming66587
Ref: array iteration68727
Node: Data types68894
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Ref: extension96104
Node: Pens103297
Ref: fillrule111404
Ref: basealign112362
Ref: transparency117721
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Ref: overwrite122388
Node: Transforms123647
Node: Frames and pictures125742
Ref: size127356
Ref: unitsize128450
Ref: shipout129579
Ref: filltype132068
Ref: add135697
Ref: add about136670
Ref: tex139788
Ref: deferred drawing140687
Node: Deferred drawing140687
Node: Files143978
Ref: cd145013
Ref: scroll150432
Node: Variable initializers153535
Node: Structures156409
Node: Operators162478
Node: Arithmetic & logical162892
Node: Self & prefix operators165430
Node: User-defined operators166290
Node: Bracket operators167329
Node: Cast operators169207
Node: Implicit scaling169421
Node: Functions169984
Ref: stack overflow173319
Node: Default arguments173608
Node: Named arguments174362
Node: Rest arguments177012
Node: Mathematical functions180482
Node: Arrays187063
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Node: Static222843
Node: Autounravel225824
Node: Where fields are autounraveled227362
Node: Where autounravel is legal228278
Ref: Where autounravel is legal-Footnote-1229103
Node: Hash functions229378
Node: Hashing user-defined structures230951
Node: Collections (containers)233514
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Node: collections.iter(T)236797
Node: Combining (zipping) iterators239311
Node: Attaching a counter to an iterator243202
Node: Maps244324
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Node: The Map API248547
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Node: The Set API258003
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Node: LaTeX usage274189
Node: User modules280735
Node: graph282157
Ref: ticks289753
Ref: pathmarkers303936
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Ref: markuniform304776
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Node: three323675
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Node: markers383799
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Node: patterns388597
Node: roundedpath388845
Node: slopefield389137
Node: smoothcontour3390690
Node: solids392455
Node: stats393489
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Node: size10412991
Node: size11413364
Node: slide413651
Node: syzygy413998
Node: tree414279
Node: trembling414470
Node: Developer modules414736
Node: plain415182
Node: bezulate415843
Node: simplex2416262
Node: v3d416567
Node: Options417013
Ref: configuration file426283
Ref: settings426283
Ref: texengines427631
Ref: magick427631
Node: Interactive mode431298
Ref: history433527
Node: GUI434898
Node: GUI installation435468
Node: GUI usage436613
Node: Command-Line Interface437680
Node: Language server protocol439125
Node: PostScript to Asymptote440606
Node: Help441400
Node: Debugger443124
Node: Credits444972
Node: General Index446238

End Tag Table


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coding: utf-8
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