\input texinfo @c -*-texinfo-*- @setfilename asymptote.info @settitle Asymptote: the Vector Graphics Language @include version.texi @finalout @codequoteundirected on @copying This file documents @code{Asymptote}, version @value{VERSION}. @url{https://asymptote.sourceforge.io} Copyright @copyright{} 2004-21 Andy Hammerlindl, John Bowman, and Tom Prince. @quotation Permission is granted to copy, distribute and/or modify this document under the terms of the @acronym{GNU} Lesser General Public License (see the file LICENSE in the top-level source directory). @end quotation @end copying @dircategory Languages @direntry * asymptote: (asymptote/asymptote). Vector graphics language. @end direntry @titlepage @title Asymptote: the Vector Graphics Language @subtitle For version @value{VERSION} @sp 1 @center @image{./logo} @page @vskip 0pt plus 1filll @insertcopying @end titlepage @c So the toc is printed at the start. @contents @ifnottex @node Top, Description, (dir), (dir) @top Asymptote @insertcopying @end ifnottex @menu * Description:: What is @code{Asymptote}? * Installation:: Downloading and installing * Tutorial:: Getting started * Drawing commands:: Four primitive graphics commands * Bezier curves:: Path connectors and direction specifiers * Programming:: The @code{Asymptote} vector graphics language * LaTeX usage:: Embedding @code{Asymptote} commands within @code{LaTeX} * Base modules:: Base modules shipped with @code{Asymptote} * Options:: Command-line options * Interactive mode:: Typing @code{Asymptote} commands interactively * GUI:: Graphical user interface * Command-Line Interface:: Remote command-line interface * Language server protocol:: Help when writing code * PostScript to Asymptote:: @code{Asymptote} backend to @code{pstoedit} * Help:: Where to get help and submit bug reports * Debugger:: Squish those bugs! * Credits:: Contributions and acknowledgments * Index:: General index @detailmenu --- The Detailed Node Listing --- Installation * UNIX binary distributions:: Prebuilt @code{UNIX} binaries * MacOS X binary distributions:: Prebuilt @code{MacOS X} binaries * Microsoft Windows:: Prebuilt @code{Microsoft Windows} binary * Configuring:: Configuring @code{Asymptote} for your system * Search paths:: Where @code{Asymptote} looks for your files * Compiling from UNIX source:: Building @code{Asymptote} from scratch * Editing modes:: Convenient @code{emacs} and @code{vim} modes * Git:: Getting the latest development source * Uninstall:: Goodbye, @code{Asymptote}! Tutorial * Drawing in batch mode:: Run @code{Asymptote} on a text file * Drawing in interactive mode:: Running @code{Asymptote} interactively * Figure size:: Specifying the figure size * Labels:: Adding @code{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 * 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 high-order functions * Arrays:: Dynamic vectors * Casts:: Implicit and explicit casts * Import:: Importing external @code{Asymptote} modules * Static:: Where to allocate your variable? Operators * Arithmetic & logical:: Basic mathematical operators * Self & prefix operators:: Increment and decrement * User-defined operators:: Overloading operators 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 Base modules * plain:: Default @code{Asymptote} base file * simplex:: Linear programming: simplex method * math:: Extend @code{Asymptote}'s math capabilities * interpolate:: Interpolation routines * geometry:: Geometry routines * trembling:: Wavy lines * stats:: Statistics routines and histograms * patterns:: Custom fill and draw patterns * markers:: Custom path marker routines * map:: Map keys to values * tree:: Dynamic binary search tree * binarytree:: Binary tree drawing module * drawtree:: Tree drawing module * syzygy:: Syzygy and braid drawing module * feynman:: Feynman diagrams * roundedpath:: Round the sharp corners of paths * animation:: Embedded @acronym{PDF} and @acronym{MPEG} movies * embed:: Embedding movies, sounds, and 3D objects * slide:: Making presentations with @code{Asymptote} * MetaPost:: @code{MetaPost} compatibility routines * babel:: Interface to @code{LaTeX} @code{babel} package * labelpath:: Drawing curved labels * labelpath3:: Drawing curved labels in 3D * annotate:: Annotate your @acronym{PDF} files * CAD:: 2D CAD pen and measurement functions (DIN 15) * graph:: 2D linear & logarithmic graphs * palette:: Color density images and palettes * three:: 3D vector graphics * obj:: 3D obj files * graph3:: 3D linear & logarithmic graphs * grid3:: 3D grids * solids:: 3D solid geometry * tube:: 3D rotation minimizing tubes * flowchart:: Flowchart drawing routines * contour:: Contour lines * contour3:: Contour surfaces * smoothcontour3:: Smooth implicit surfaces * slopefield:: Slope fields * ode:: Ordinary differential equations Graphical User Interface * GUI installation:: Installing @code{xasy} * GUI usage:: Using @code{xasy} to edit objects @end detailmenu @end menu @node Description, Installation, Top, Top @chapter Description @cindex description @cindex @code{Asymptote Web Application} @code{Asymptote} is a powerful descriptive vector graphics language that provides a mathematical coordinate-based framework for technical drawing. Labels and equations are typeset with @code{LaTeX}, for overall document consistency, yielding the same high-quality level of typesetting that @code{LaTeX} provides for scientific text. By default it produces @code{PostScript} output, but it can also generate @code{PDF}, @code{SVG}, @code{WebGL}, @code{V3D}, @code{PRC}, and any format that the @code{ImageMagick} package can produce. You can even try it out in your Web browser without installing it, using the @code{Asymptote Web Application} @url{http://asymptote.ualberta.ca} It is also possible to send remote commands to this server via the curl utility (@pxref{Command-Line Interface}). A major advantage of @code{Asymptote} over other graphics packages is that it is a high-level programming language, as opposed to just a graphics program: it can therefore exploit the best features of the script (command-driven) and graphical-user-interface (@acronym{GUI}) methods for producing figures. The rudimentary @acronym{GUI} @code{xasy} included with the package allows one to move script-generated objects around. To make @code{Asymptote} accessible to the average user, this @acronym{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 @code{Asymptote} graphics commands (@pxref{Drawing commands}). @code{Asymptote} is mathematically oriented (e.g.@ one can use complex multiplication to rotate a vector) and uses @code{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 @code{MetaPost}. The @code{Asymptote} vector graphics language provides: @itemize @bullet @item a standard for typesetting mathematical figures, just as @TeX{}/@code{LaTeX} is the de-facto standard for typesetting equations. @item @code{LaTeX} typesetting of labels, for overall document consistency; @item the ability to generate and embed 3D vector @acronym{WebGL} graphics within @acronym{HTML} files; @item the ability to generate and embed 3D vector @acronym{PRC} graphics within @acronym{PDF} files; @item a natural coordinate-based framework for technical drawing, inspired by @code{MetaPost}, with a much cleaner, powerful C++-like programming syntax; @item compilation of figures into virtual machine code for speed, without sacrificing portability; @item the power of a script-based language coupled to the convenience of a @acronym{GUI}; @item customization using its own C++-like graphics programming language; @item sensible defaults for graphical features, with the ability to override; @item a high-level mathematically oriented interface to the @code{PostScript} language for vector graphics, including affine transforms and complex variables; @item functions that can create new (anonymous) functions; @item 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; @end itemize Many of the features of @code{Asymptote} are written in the @code{Asymptote} language itself. While the stock version of @code{Asymptote} is designed for mathematics typesetting needs, one can write @code{Asymptote} modules that tailor it to specific applications; for example, a scientific graphing module is available (@pxref{graph}). Examples of @code{Asymptote} code and output, including animations, are available at @quotation @url{https://asymptote.sourceforge.io/gallery/} @end quotation @noindent Clicking on an example file name in this manual, like @code{@uref{https://asymptote.sourceforge.io/gallery/Pythagoras.svg,,Pythagoras}}, will display the @acronym{PDF} output, whereas clicking on its @code{@uref{https://asymptote.sourceforge.io/gallery/Pythagoras.asy,,.asy}} extension will show the corresponding @code{Asymptote} code in a separate window. Links to many external resources, including an excellent user-written @code{Asymptote} tutorial can be found at @quotation @url{https://asymptote.sourceforge.io/links.html} @end quotation @cindex reference @cindex quick reference A quick reference card for @code{Asymptote} is available at @quotation @url{https://asymptote.sourceforge.io/asyRefCard.pdf} @end quotation @node Installation, Tutorial, Description, Top @chapter Installation @cindex installation @menu * UNIX binary distributions:: Prebuilt @code{UNIX} binaries * MacOS X binary distributions:: Prebuilt @code{MacOS X} binaries * Microsoft Windows:: Prebuilt @code{Microsoft Windows} binary * Configuring:: Configuring @code{Asymptote} for your system * Search paths:: Where @code{Asymptote} looks for your files * Compiling from UNIX source:: Building @code{Asymptote} from scratch * Editing modes:: Convenient @code{emacs} and @code{vim} modes * Git:: Getting the latest development source * Uninstall:: Goodbye, @code{Asymptote}! @end menu After following the instructions for your specific distribution, please see also @ref{Configuring}. @noindent We recommend subscribing to new release announcements at @quotation @url{https://sourceforge.net/projects/asymptote} @end quotation @noindent Users may also wish to monitor the @code{Asymptote} forum: @quotation @url{https://sourceforge.net/p/asymptote/discussion/409349} @end quotation @noindent @node UNIX binary distributions, MacOS X binary distributions, Installation, Installation @section UNIX binary distributions @cindex UNIX binary distributions @cindex @acronym{RPM} @cindex @code{tgz} We release both @code{tgz} and @acronym{RPM} binary distributions of @code{Asymptote}. The root user can install the @code{Linux x86_64} @code{tgz} distribution of version @code{x.xx} of @code{Asymptote} with the commands: @verbatim tar -C / -zxf asymptote-x.xx.x86_64.tgz texhash @end verbatim @noindent The @code{texhash} command, which installs LaTeX style files, is optional. The executable file will be @code{/usr/local/bin/asy}) and example code will be installed by default in @code{@value{Docdir}/examples}. @noindent @cindex Fedora Fedora users can easily install a recent version of @code{Asymptote} with the command @verbatim dnf --enablerepo=rawhide install asymptote @end verbatim @cindex Debian @noindent To install the latest version of @code{Asymptote} on a Debian-based distribution (e.g.@ Ubuntu, Mepis, Linspire) follow the instructions for compiling from @code{UNIX} source (@pxref{Compiling from UNIX source}). Alternatively, Debian users can install one of Hubert Chan's prebuilt @code{Asymptote} binaries from @quotation @url{http://ftp.debian.org/debian/pool/main/a/asymptote} @end quotation @node MacOS X binary distributions, Microsoft Windows, UNIX binary distributions, Installation @section MacOS X binary distributions @cindex @code{MacOS X} binary distributions @code{MacOS X} users can either compile the @code{UNIX} source code (@pxref{Compiling from UNIX source}) or install the @code{Asymptote} binary available at @url{https://www.macports.org/} or at @url{https://brew.sh/} @noindent Note that many @code{MacOS X} (and FreeBSD) systems lack the @acronym{GNU} @code{readline} library. For full interactive functionality, @acronym{GNU} @code{readline} version 4.3 or later must be installed. @node Microsoft Windows, Configuring, MacOS X binary distributions, Installation @section Microsoft Windows @cindex Microsoft Windows Users of the @code{Microsoft Windows} operating system can install the self-extracting @code{Asymptote} executable @code{asymptote-x.xx-setup.exe}, where @code{x.xx} denotes the latest version. A working @TeX{} implementation (we recommend @url{https://www.tug.org/texlive} or @url{http://www.miktex.org}) will be required to typeset labels. You will also need to install @code{GPL Ghostscript} version 9.52 or later from @url{http://downloads.ghostscript.com/public}. To view @code{PostScript} output, you can install the program @code{Sumatra PDF} available from @url{https://www.sumatrapdfreader.org/}. The @code{ImageMagick} package from @url{https://www.imagemagick.org/script/binary-releases.php} @noindent is required to support output formats other than @acronym{HTML}, @acronym{PDF}, @acronym{SVG}, and @acronym{PNG} (@pxref{convert}). The @code{Python 3} interpreter from @url{https://www.python.org} is only required if you wish to try out the graphical user interface (@pxref{GUI}). @noindent Example code will be installed by default in the @code{examples} subdirectory of the installation directory (by default, @code{C:\Program Files\Asymptote}). @node Configuring, Search paths, Microsoft Windows, Installation @section Configuring @cindex configuring @cindex @code{-V} In interactive mode, or when given the @code{-V} option (the default when running @code{Asymptote} on a single file under @code{MSDOS}), @code{Asymptote} will automatically invoke the @code{PostScript} viewer @code{gv} (under @code{UNIX}) or @code{gsview} (under @code{MSDOS} to display graphical output. The @code{PostScript} viewer should be capable of automatically redrawing whenever the output file is updated. The default @code{UNIX} @code{PostScript} viewer @code{gv} supports this (via a @code{SIGHUP} signal). Version @code{gv-3.6.3} or later (from @url{https://ftp.gnu.org/gnu/gv/}) is required for interactive mode to work properly. Users of @code{ggv} will need to enable @code{Watch file} under @code{Edit/PostScript Viewer Preferences}. Users of @code{gsview} will need to enable @code{Options/Auto Redisplay} (however, under @code{MSDOS} it is still necessary to click on the @code{gsview} window; under @code{UNIX} one must manually redisplay by pressing the @code{r} key). @cindex @code{psviewer} @cindex @code{pdfviewer} @cindex @code{htmlviewer} @cindex @code{gs} @cindex @code{display} @cindex @code{animate} @cindex @code{settings} @cindex configuration file Configuration variables are most easily set as @code{Asymptote} variables in an optional configuration file @code{config.asy} @pxref{configuration file}). For example, the setting @code{pdfviewer} specifies the location of the @acronym{PDF} viewer. Here are the default values of several important configuration variables under @code{UNIX}: @noindent @verbatim import settings; pdfviewer="acroread"; htmlviewer="google-chrome"; psviewer="gv"; display="display"; animate="animate"; gs="gs"; libgs=""; @end verbatim @noindent @cindex @code{cmd} Under @code{MSDOS}, the viewer settings @code{htmlviewer}, @code{pdfviewer}, @code{psviewer}, @code{display}, and @code{animate} default to the string @code{cmd}, requesting the application normally associated with each file type. The (installation-dependent) default values of @code{gs} and @code{libgs} are determined automatically from the @code{Microsoft Windows} registry. The @code{gs} setting specifies the location of the @code{PostScript} processor @code{Ghostscript}, available from @url{https://www.ghostscript.com/}. @noindent @cindex @code{htmlviewer} @cindex @code{absolute} The configuration variable @code{htmlviewer} specifies the browser to use to display 3D @code{WebGL} output. The default setting is @code{google-chrome} under @code{UNIX} and @code{cmd} under @code{Microsoft Windows}. Note that @code{Internet Explorer} does not support @code{WebGL}; @code{Microsoft Windows} users should set their default html browser to @code{chrome} or @code{microsoft-edge}. By default, 2D and 3D @code{HTML} images expand to the enclosing canvas; this can be disabled by setting the configuration variable @code{absolute} to @code{true}. On @code{UNIX} systems, to support automatic document reloading of @code{PDF} files in @code{Adobe Reader}, we recommend copying the file @code{reload.js} from the @code{Asymptote} system directory (by default, @code{@value{Datadir}/asymptote} under @code{UNIX} to @code{~/.adobe/Acrobat/x.x/JavaScripts/}, where @code{x.x} represents the appropriate @code{Adobe Reader} version number. The automatic document reload feature must then be explicitly enabled by putting @verbatim import settings; pdfreload=true; pdfreloadOptions="-tempFile"; @end verbatim @noindent in the @code{Asymptote} configuration file. This reload feature is not useful under @code{MSDOS} since the document cannot be updated anyway on that operating system until it is first closed by @code{Adobe Reader}. The configuration variable @code{dir} can be used to adjust the search path (@pxref{Search paths}). @cindex @code{papertype} @cindex @code{paperwidth} @cindex @code{paperheight} @cindex @code{letter} @cindex @code{a4} By default, @code{Asymptote} attempts to center the figure on the page, assuming that the paper type is @code{letter}. The default paper type may be changed to @code{a4} with the configuration variable @code{papertype}. Alignment to other paper sizes can be obtained by setting the configuration variables @code{paperwidth} and @code{paperheight}. @cindex @code{config} @cindex @code{texpath} @cindex @code{texcommand} @cindex @code{dvips} @cindex @code{dvisvgm} @cindex @code{convert} @cindex @code{ImageMagick} @cindex @code{asygl} These additional configuration variables normally do not require adjustment: @verbatim config texpath texcommand dvips dvisvgm convert asygl @end verbatim @noindent Warnings (such as "unbounded" and "offaxis") may be enabled or disabled with the functions @verbatim warn(string s); nowarn(string s); @end verbatim @noindent or by directly modifying the string array @code{settings.suppress}, which lists all disabled warnings. @cindex command-line options Configuration variables may also be set or overwritten with a command-line option: @verbatim asy -psviewer=gsview -V venn @end verbatim @cindex environment variables 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 @code{ASYMPTOTE_}: for example, to set the environment variable @verbatim ASYMPTOTE_PSVIEWER="C:\Program Files\Ghostgum\gsview\gsview32.exe"; @end verbatim @noindent under @code{Microsoft Windows XP}: @enumerate @item Click on the @code{Start} button; @item Right-click on @code{My Computer}; @item Choose @code{View system information}; @item Click the @code{Advanced} tab; @item Click the @code{Environment Variables} button. @end enumerate @node Search paths, Compiling from UNIX source, Configuring, Installation @section Search paths @cindex search paths In looking for @code{Asymptote} files, @code{asy} will search the following paths, in the order listed: @enumerate @item The current directory; @item @cindex @code{dir} A list of one or more directories specified by the configuration variable @code{dir} or environment variable @code{ASYMPTOTE_DIR} (separated by @code{:} under UNIX and @code{;} under @code{MSDOS}); @item @cindex @code{.asy} The directory specified by the environment variable @code{ASYMPTOTE_HOME}; if this variable is not set, the directory @code{.asy} in the user's home directory (@code{%USERPROFILE%\.asy} under @code{MSDOS}) is used; @item The @code{Asymptote} system directory (by default, @code{@value{Datadir}/asymptote} under @code{UNIX} and @code{C:\Program Files\Asymptote} under @code{MSDOS}). @item The @code{Asymptote} examples directory (by default, @code{@value{Docdir}/examples} under @code{UNIX} and @code{C:\Program Files\Asymptote\examples} under @code{MSDOS}). @end enumerate @node Compiling from UNIX source, Editing modes, Search paths, Installation @section Compiling from UNIX source @cindex Compiling from UNIX source To compile and install a @code{UNIX} executable from the source release @code{asymptote-x.xx.src.tgz} in the subdirectory @code{x.xx} under @url{https://sourceforge.net/projects/asymptote/files/} execute the commands: @verbatim gunzip asymptote-x.xx.src.tgz tar -xf asymptote-x.xx.src.tar cd asymptote-x.xx @end verbatim By default the system version of the Boehm garbage collector will be used; if it is old we recommend first putting @url{https://github.com/ivmai/bdwgc/releases/download/v8.0.4/gc-8.0.4.tar.gz} @url{https://www.ivmaisoft.com/_bin/atomic_ops/libatomic_ops-7.6.10.tar.gz} in the @code{Asymptote} source directory. On @code{UNIX} platforms (other than @code{MacOS X}), we recommend using version @code{3.2.1} of the @code{freeglut} library. To compile @code{freeglut}, download @quotation @url{https://prdownloads.sourceforge.net/freeglut/freeglut-3.2.1.tar.gz} @end quotation @noindent and type (as the root user): @verbatim gunzip freeglut-3.2.1.tar.gz tar -xf freeglut-3.2.1.tar cd freeglut-3.2.1 cmake -DCMAKE_INSTALL_PREFIX=/usr -DCMAKE_C_FLAGS=-fcommon . make make install @end verbatim @noindent Then compile @code{Asymptote} with the commands @verbatim ./configure make all make install @end verbatim @noindent Be sure to use @acronym{GNU} @code{make} (on non-@acronym{GNU} systems this command may be called @code{gmake}). To build the documentation, you may need to install the @code{texinfo-tex} package. If you get errors from a broken @code{texinfo} or @code{pdftex} installation, simply put @quotation @url{https://asymptote.sourceforge.io/asymptote.pdf} @end quotation @noindent in the directory @code{doc} and repeat the command @code{make all}. @noindent For a (default) system-wide installation, the last command should be done as the root user. To install without root privileges, change the @code{./configure} command to @verbatim ./configure --prefix=$HOME/asymptote @end verbatim @cindex @code{MacOS X} configuration @cindex @code{clang} One can disable use of the Boehm garbage collector by configuring with @code{./configure --disable-gc}. For a list of other configuration options, say @code{./configure --help}. For example, under @code{MacOS X}, one can tell configure to use the @code{clang} compilers and look for header files and libraries in nonstandard locations: @verbatim ./configure CC=clang CXX=clang++ CPPFLAGS=-I/opt/local/include LDFLAGS=-L/opt/local/lib @end verbatim If you are compiling @code{Asymptote} with @code{gcc}, you will need a relatively recent version (e.g.@ 3.4.4 or later). For full interactive functionality, you will need version 4.3 or later of the @acronym{GNU} @code{readline} library. The file @code{gcc3.3.2curses.patch} in the @code{patches} directory can be used to patch the broken curses.h header file (or a local copy thereof in the current directory) on some @code{AIX} and @code{IRIX} systems. @cindex @code{FFTW} @cindex @code{GSL} The @code{FFTW} library is only required if you want @code{Asymptote} to be able to take Fourier transforms of data (say, to compute an audio power spectrum). The @code{GSL} library is only required if you require the special functions that it supports. If you don't want to install @code{Asymptote} system wide, just make sure the compiled binary @code{asy} and @acronym{GUI} script @code{xasy} are in your path and set the configuration variable @code{dir} to point to the directory @code{base} (in the top level directory of the @code{Asymptote} source code). @node Editing modes, Git, Compiling from UNIX source, Installation @section Editing modes @cindex Editing modes @cindex @code{emacs} @cindex @code{asy-mode} @cindex @code{lasy-mode} Users of @code{emacs} can edit @code{Asymptote} code with the mode @code{asy-mode}, after enabling it by putting the following lines in their @code{.emacs} initialization file, replacing @code{ASYDIR} with the location of the @code{Asymptote} system directory (by default, @code{@value{Datadir}/asymptote} or @code{C:\Program Files\Asymptote} under @code{MSDOS}): @verbatim (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)) @end verbatim @noindent Particularly useful key bindings in this mode are @code{C-c C-c}, which compiles and displays the current buffer, and the key binding @code{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 @code{two-mode-mode}: @quotation @url{https://www.dedasys.com/freesoftware/files/two-mode-mode.el} @end quotation @noindent Once installed, you can use the hybrid mode @code{lasy-mode} to edit a LaTeX file containing embedded @code{Asymptote} code (@pxref{LaTeX usage}). This mode can be enabled within @code{latex-mode} with the key sequence @code{M-x lasy-mode }. On @code{UNIX} systems, additional keywords will be generated from all @code{asy} files in the space-separated list of directories specified by the environment variable @code{ASYMPTOTE_SITEDIR}. Further documentation of @code{asy-mode} is available within @code{emacs} by pressing the sequence keys @code{C-h f asy-mode }. @cindex @code{vim} @cindex @code{asy.vim} Fans of @code{vim} can customize @code{vim} for @code{Asymptote} with @noindent @code{cp @value{Datadir}/asymptote/asy.vim ~/.vim/syntax/asy.vim} @noindent and add the following to their @code{~/.vimrc} file: @verbatim augroup filetypedetect au BufNewFile,BufRead *.asy setf asy augroup END filetype plugin on @end verbatim If any of these directories or files don't exist, just create them. To set @code{vim} up to run the current asymptote script using @code{:make} just add to @code{~/.vim/ftplugin/asy.vim}: @verbatim setlocal makeprg=asy\ % setlocal errorformat=%f:\ %l.%c:\ %m @end verbatim @cindex @code{KDE editor} @cindex @code{Kate} @cindex @code{asymptote.xml} Syntax highlighting support for the @acronym{KDE} editor @code{Kate} can be enabled by running @code{asy-kate.sh} in the @code{@value{Datadir}/asymptote} directory and putting the generated @code{asymptote.xml} file in @code{~/.kde/share/apps/katepart/syntax/}. @node Git, Uninstall, Editing modes, Installation @section Git @cindex git The following commands are needed to install the latest development version of @code{Asymptote} using @code{git}: @verbatim git clone https://github.com/vectorgraphics/asymptote cd asymptote ./autogen.sh ./configure make all make install @end verbatim @noindent To compile without optimization, use the command @code{make CFLAGS=-g}. On @code{Ubuntu} systems, you may need to first install the required dependencies: @verbatim apt-get build-dep asymptote @end verbatim @noindent @node Uninstall, , Git, Installation @section Uninstall @cindex uninstall To uninstall a @code{Linux x86_64} binary distribution, use the commands @verbatim tar -zxvf asymptote-x.xx.x86_64.tgz | xargs --replace=% rm /% texhash @end verbatim @noindent To uninstall all @code{Asymptote} files installed from a source distribution, use the command @verbatim make uninstall @end verbatim @node Tutorial, Drawing commands, Installation, Top @chapter Tutorial @cindex tutorial @menu * Drawing in batch mode:: Run @code{Asymptote} on a text file * Drawing in interactive mode:: Running @code{Asymptote} interactively * Figure size:: Specifying the figure size * Labels:: Adding @code{LaTeX} labels * Paths:: Drawing lines and curves @end menu A concise introduction to @code{Asymptote} is given here. For a more thorough introduction, see the excellent @code{Asymptote} tutorial written by Charles Staats: @url{https://asymptote.sourceforge.io/asymptote_tutorial.pdf} Another @code{Asymptote} tutorial is available as a wiki, with images rendered by an online Asymptote engine: @url{https://www.artofproblemsolving.com/wiki/?title=Asymptote_(Vector_Graphics_Language)} @node Drawing in batch mode, Drawing in interactive mode, Tutorial, Tutorial @section Drawing in batch mode @cindex batch mode To draw a line from coordinate (0,0) to coordinate (100,100), create a text file @code{test.asy} containing @verbatiminclude diagonal.asy @noindent Then execute the command @verbatim asy -V test @end verbatim @noindent Alternatively, @code{MSDOS} users can drag and drop @code{test.asy} onto the Desktop @code{asy} icon (or make @code{Asymptote} the default application for the extension @code{asy}). @noindent @cindex @code{-V} This method, known as @emph{batch mode}, outputs a @code{PostScript} file @code{test.eps}. If you prefer @acronym{PDF} output, use the command line @verbatim asy -V -f pdf test @end verbatim In either case, the @code{-V} option opens up a viewer window so you can immediately view the result: @sp 1 @center @image{./diagonal} @cindex @code{bp} @noindent Here, the @code{--} connector joins the two points @code{(0,0)} and @code{(100,100)} with a line segment. @node Drawing in interactive mode, Figure size, Drawing in batch mode, Tutorial @section Drawing in interactive mode @cindex interactive mode Another method is @emph{interactive mode}, where @code{Asymptote} reads individual commands as they are entered by the user. To try this out, enter @code{Asymptote}'s interactive mode by clicking on the @code{Asymptote} icon or typing the command @code{asy}. Then type @verbatim draw((0,0)--(100,100)); @end verbatim @noindent followed by @code{Enter}, to obtain the above image. @cindex tab completion @cindex arrow keys @cindex erase @cindex quit @noindent At this point you can type further @code{draw} commands, which will be added to the displayed figure, @code{erase} to clear the canvas, @verbatim input test; @end verbatim @noindent to execute all of the commands contained in the file @code{test.asy}, or @code{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 @ref{Interactive mode}. @node Figure size, Labels, Drawing in interactive mode, Tutorial @section Figure size @cindex @code{size} @cindex @code{pair} In @code{Asymptote}, coordinates like @code{(0,0)} and @code{(100,100)}, called @emph{pairs}, are expressed in @code{PostScript} "big points" (1 @code{bp} = 1/72 @code{inch}) and the default line width is @code{0.5bp}. However, it is often inconvenient to work directly in @code{PostScript} coordinates. The next example produces identical output to the previous example, by scaling the line @code{(0,0)--(1,1)} to fit a rectangle of width @code{100.5 bp} and height @code{100.5 bp} (the extra @code{0.5bp} accounts for the line width): @verbatim size(100.5,100.5); draw((0,0)--(1,1)); @end verbatim @sp 1 @center @image{./diagonal} @cindex @code{inches} @cindex @code{cm} @cindex @code{mm} @cindex @code{pt} One can also specify the size in @code{pt} (1 @code{pt} = 1/72.27 @code{inch}), @code{cm}, @code{mm}, or @code{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 (@pxref{size}): @verbatiminclude bigdiagonal.asy @sp 1 @center @image{./bigdiagonal} @cindex @code{cycle} To connect several points and create a cyclic path, use the @code{cycle} keyword: @verbatiminclude square.asy @sp 1 @center @image{./square} @noindent For convenience, the path @code{(0,0)--(1,0)--(1,1)--(0,1)--cycle} may be replaced with the predefined variable @code{unitsquare}, or equivalently, @code{box((0,0),(1,1))}. @cindex user coordinates @cindex @code{unitsize} To make the user coordinates represent multiples of exactly @code{1cm}: @verbatim unitsize(1cm); draw(unitsquare); @end verbatim @noindent @node Labels, Paths, Figure size, Tutorial @section Labels @cindex @code{label} Adding labels is easy in @code{Asymptote}; one specifies the label as a double-quoted @code{LaTeX} string, a coordinate, and an optional alignment direction: @verbatiminclude labelsquare.asy @sp 1 @center @image{./labelsquare} @cindex compass directions @cindex @code{N} @cindex @code{E} @cindex @code{W} @cindex @code{S} @code{Asymptote} uses the standard compass directions @code{E=(1,0)}, @code{N=(0,1)}, @code{NE=unit(N+E)}, and @code{ENE=unit(E+NE)}, etc., which along with the directions @code{up}, @code{down}, @code{right}, and @code{left} are defined as pairs in the @code{Asymptote} base module @code{plain} (a user who has a local variable named @code{E} may access the compass direction @code{E} by prefixing it with the name of the module where it is defined: @code{plain.E}). @node Paths, , Labels, Tutorial @section Paths @cindex @code{path} This example draws a path that approximates a quarter circle, terminated with an arrowhead: @verbatiminclude quartercircle.asy @sp 1 @center @image{./quartercircle} @noindent Here the directions @code{up} and @code{left} in braces specify the outgoing and incoming directions at the points @code{(1,0)} and @code{(0,1)}, respectively. In general, a path is specified as a list of points (or other paths) interconnected with @cindex @code{cycle} @cindex @code{--} @cindex @code{..} @code{--}, which denotes a straight line segment, or @code{..}, which denotes a cubic spline (@pxref{Bezier curves}). @cindex @code{unitcircle} @anchor{unitcircle} @cindex @code{unitcircle} Specifying a final @code{..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: @verbatim path unitcircle=E..N..W..S..cycle; @end verbatim @cindex @code{PostScript} subpath @cindex @code{^^} @cindex @code{path[]} @cindex superpath @noindent An @code{Asymptote} path, being connected, is equivalent to a @code{PostScript subpath}. The @code{^^} 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 @code{Asymptote} paths into a @code{path[]} array (equivalent to a @code{PostScript} path): @verbatiminclude superpath.asy @sp 1 @center @image{./superpath} @cindex evenodd @noindent The @code{PostScript} even-odd fill rule here specifies that only the region bounded between the two unit circles is filled (@pxref{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: @verbatim filldraw(unitcircle^^reverse(g),yellow,black); @end verbatim @cindex @code{unitbox} The @code{^^} operator is used by the @code{box(triple, triple)} function in the module @code{three} to construct the edges of a cube @code{unitbox} without retracing steps (@pxref{three}): @verbatiminclude cube.asy @sp 1 @center @image{./cube} See section @ref{graph} (or the online @code{Asymptote} @uref{https://asymptote.sourceforge.io/gallery,,gallery} and external links posted at @url{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 @url{http://www.piprime.fr/asymptote}. @node Drawing commands, Bezier curves, Tutorial, Top @chapter Drawing commands @cindex drawing commands All of @code{Asymptote}'s graphical capabilities are based on four primitive commands. The three @code{PostScript} drawing commands @code{draw}, @code{fill}, and @code{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 @code{label} can be used to add text labels and external @acronym{EPS} images, which will appear on top of the @code{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 @code{shipout} function (@pxref{shipout}). @cindex @code{layer} If you wish to draw @code{PostScript} objects on top of labels (or verbatim @code{tex} commands; @pxref{tex}), the @code{layer} command may be used to start a new @code{PostScript/LaTeX} layer: @verbatim void layer(picture pic=currentpicture); @end verbatim The @code{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 @code{tex} commands are always drawn after the @code{PostScript} objects in that layer. @cindex @code{newpage} A page break can be generated with the command @verbatim void newpage(picture pic=currentpicture); @end verbatim While some of these drawing commands take many options, they all have sensible default values (for example, the picture argument defaults to currentpicture). @cindex legend @cindex @code{draw} @cindex @code{arrow} @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 @end menu @node draw, fill, Drawing commands, Drawing commands @section draw @cindex @code{draw} @verbatim 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); @end verbatim Draw the path @code{g} on the picture @code{pic} using pen @code{p} for drawing, with optional drawing attributes (Label @code{L}, explicit label alignment @code{align}, arrows and bars @code{arrow} and @code{bar}, margins @code{margin}, legend, and markers @code{marker}). Only one parameter, the path, is required. For convenience, the arguments @code{arrow} and @code{bar} may be specified in either order. The argument @code{legend} is a Label to use in constructing an optional legend entry. @cindex @code{None} @cindex @code{BeginBar} @cindex @code{EndBar} @cindex @code{Bar} @cindex @code{Bars} @cindex @code{barsize} Bars are useful for indicating dimensions. The possible values of @code{bar} are @code{None}, @code{BeginBar}, @code{EndBar} (or equivalently @code{Bar}), and @code{Bars} (which draws a bar at both ends of the path). Each of these bar specifiers (except for @code{None}) will accept an optional real argument that denotes the length of the bar in @code{PostScript} coordinates. The default bar length is @code{barsize(pen)}. @cindex arrows @anchor{arrows} @cindex @code{None} @cindex @code{Blank} @cindex @code{BeginArrow} @cindex @code{MidArrow} @cindex @code{EndArrow} @cindex @code{Arrow} @cindex @code{Arrows} @cindex @code{FillDraw} @cindex @code{Fill} @cindex @code{Draw} @cindex @code{NoFill} @cindex @code{UnFill} @cindex @code{BeginArcArrow} @cindex @code{MidArcArrow} @cindex @code{EndArcArrow} @cindex @code{ArcArrow} @cindex @code{ArcArrows} @cindex @code{DefaultHead} @cindex @code{SimpleHead} @cindex @code{HookHead} @cindex @code{TeXHead} The possible values of @code{arrow} are @code{None}, @code{Blank} (which draws no arrows or path), @code{BeginArrow}, @code{MidArrow}, @code{EndArrow} (or equivalently @code{Arrow}), and @code{Arrows} (which draws an arrow at both ends of the path). All of the arrow specifiers except for @code{None} and @code{Blank} may be given the optional arguments arrowhead @code{arrowhead} (one of the predefined arrowhead styles @code{DefaultHead}, @code{SimpleHead}, @code{HookHead}, @code{TeXHead}), real @code{size} (arrowhead size in @code{PostScript} coordinates), real @code{angle} (arrowhead angle in degrees), filltype @code{filltype} (one of @code{FillDraw}, @code{Fill}, @code{NoFill}, @code{UnFill}, @code{Draw}) and (except for @code{MidArrow} and @code{Arrows}) a real @code{position} (in the sense of @code{point(path p, real t)}) along the path where the tip of the arrow should be placed. The default arrowhead size when drawn with a pen @code{p} is @code{arrowsize(p)}. There are also arrow versions with slightly modified default values of @code{size} and @code{angle} suitable for curved arrows: @code{BeginArcArrow}, @code{EndArcArrow} (or equivalently @code{ArcArrow}), @code{MidArcArrow}, and @code{ArcArrows}. @cindex @code{NoMargin} @cindex @code{BeginMargin} @cindex @code{EndMargin} @cindex @code{Margin} @cindex @code{Margins} @cindex @code{BeginPenMargin} @cindex @code{EndPenMargin} @cindex @code{PenMargin} @cindex @code{PenMargins} @cindex @code{BeginDotMargin} @cindex @code{EndDotMargin} @cindex @code{DotMargin} @cindex @code{DotMargins} @cindex @code{Margin} @cindex @code{TrueMargin} Margins can be used to shrink the visible portion of a path by @code{labelmargin(p)} to avoid overlap with other drawn objects. Typical values of @code{margin} are @code{NoMargin}, @code{BeginMargin}, @code{EndMargin} (or equivalently @code{Margin}), and @code{Margins} (which leaves a margin at both ends of the path). One may use @code{Margin(real begin, real end=begin)} to specify the size of the beginning and ending margin, respectively, in multiples of the units @code{labelmargin(p)} used for aligning labels. Alternatively, @code{BeginPenMargin}, @code{EndPenMargin} (or equivalently @code{PenMargin}), @code{PenMargins}, @code{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. For example, use @code{DotMargin}, an abbreviation for @code{PenMargin(-0.5*dotfactor,0.5*dotfactor)}, to draw from the usual beginning point just up to the boundary of an end dot of width @code{dotfactor*linewidth(p)}. The qualifiers @code{BeginDotMargin}, @code{EndDotMargin}, and @code{DotMargins} work similarly. The qualifier @code{TrueMargin(real begin, real end=begin)} allows one to specify a margin directly in @code{PostScript} units, independent of the pen line width. The use of arrows, bars, and margins is illustrated by the examples @code{@uref{https://asymptote.sourceforge.io/gallery/Pythagoras.svg,,Pythagoras}@uref{https://asymptote.sourceforge.io/gallery/Pythagoras.asy,,.asy}} and @code{@uref{https://asymptote.sourceforge.io/gallery/3Dgraphs/sqrtx01.html,,sqrtx01}@uref{https://asymptote.sourceforge.io/gallery/sqrtx01.asy,,.asy}}. The legend for a picture @code{pic} can be fit and aligned to a frame with the routine: @cindex @code{legend} @verbatim 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); @end verbatim @noindent Here @code{xmargin} and @code{ymargin} specify the surrounding @math{x} and @math{y} margins, @code{perline} specifies the number of entries per line (default 1; 0 means choose this number automatically), @code{linelength} specifies the length of the path lines, @code{hskip} and @code{vskip} specify the line skip (as a multiple of the legend entry size), @code{maxwidth} and @code{maxheight} specify optional upper limits on the width and height of the resulting legend (0 means unlimited), @code{hstretch} and @code{vstretch} allow the legend to stretch horizontally or vertically, and @code{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 @code{dest} using @code{add} or @code{attach} (@pxref{add about}). @cindex @code{dot} To draw a dot, simply draw a path containing a single point. The @code{dot} command defined in the module @code{plain} draws a dot having a diameter equal to an explicit pen line width or the default line width magnified by @code{dotfactor} (6 by default), using the specified filltype (@pxref{filltype}) or @code{dotfilltype} (@code{Fill} by default): @verbatim 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); @end verbatim @cindex @code{Label} If the variable @code{Label} is given as the @code{Label} argument to the third routine, the @code{format} argument will be used to format a string based on the dot location (here @code{defaultformat} is @code{"$%.4g$"}). The fourth routine draws a dot at every point of a pair array @code{z}. One can also draw a dot at every node of a path: @verbatim void dot(picture pic=currentpicture, Label[] L=new Label[], explicit path g, align align=RightSide, string format=defaultformat, pen p=currentpen, filltype filltype=dotfilltype); @end verbatim See @ref{pathmarkers} and @ref{markers} for more general methods for marking path nodes. To draw a fixed-sized object (in @code{PostScript} coordinates) about the user coordinate @code{origin}, use the routine @cindex @code{draw} @verbatim 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); @end verbatim @cindex @code{fill} @node fill, clip, draw, Drawing commands @section fill @cindex @code{fill} @verbatim void fill(picture pic=currentpicture, path g, pen p=currentpen); @end verbatim Fill the interior region bounded by the cyclic path @code{g} on the picture @code{pic}, using the pen @code{p}. @cindex @code{filldraw} There is also a convenient @code{filldraw} command, which fills the path and then draws in the boundary. One can specify separate pens for each operation: @verbatim void filldraw(picture pic=currentpicture, path g, pen fillpen=currentpen, pen drawpen=currentpen); @end verbatim @cindex @code{fill} This fixed-size version of @code{fill} allows one to fill an object described in @code{PostScript} coordinates about the user coordinate @code{origin}: @verbatim void fill(pair origin, picture pic=currentpicture, path g, pen p=currentpen); @end verbatim @noindent This is just a convenient abbreviation for the commands: @verbatim picture opic; fill(opic,g,p); add(pic,opic,origin); @end verbatim The routine @cindex @code{filloutside} @verbatim void filloutside(picture pic=currentpicture, path g, pen p=currentpen); @end verbatim @noindent fills the region exterior to the path @code{g}, out to the current boundary of picture @code{pic}. @anchor{gradient shading} @cindex gradient shading @cindex shading @cindex @code{latticeshade} Lattice gradient shading varying smoothly over a two-dimensional array of pens @code{p}, using fill rule @code{fillrule}, can be produced with @verbatim void latticeshade(picture pic=currentpicture, path g, bool stroke=false, pen fillrule=currentpen, pen[][] p) @end verbatim @cindex @code{stroke} If @code{stroke=true}, the region filled is the same as the region that would be drawn by @code{draw(pic,g,zerowinding)}; in this case the path @code{g} need not be cyclic. The pens in @code{p} must belong to the same color space. One can use the functions @code{rgb(pen)} or @code{cmyk(pen)} to promote pens to a higher color space, as illustrated in the example file @code{@uref{https://asymptote.sourceforge.io/gallery/latticeshading.svg,,latticeshading}@uref{https://asymptote.sourceforge.io/gallery/latticeshading.asy,,.asy}}. @cindex @code{axialshade} Axial gradient shading varying smoothly from @code{pena} to @code{penb} in the direction of the line segment @code{a--b} can be achieved with @verbatim void axialshade(picture pic=currentpicture, path g, bool stroke=false, pen pena, pair a, bool extenda=true, pen penb, pair b, bool extendb=true); @end verbatim @noindent The boolean parameters @code{extenda} and @code{extendb} indicate whether the shading should extend beyond the axis endpoints @code{a} and @code{b}. An example of axial shading is provided in the example file @code{@uref{https://asymptote.sourceforge.io/gallery/axialshade.svg,,axialshade}@uref{https://asymptote.sourceforge.io/gallery/axialshade.asy,,.asy}}. @cindex @code{radialshade} Radial gradient shading varying smoothly from @code{pena} on the circle with center @code{a} and radius @code{ra} to @code{penb} on the circle with center @code{b} and radius @code{rb} is similar: @verbatim 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); @end verbatim @noindent The boolean parameters @code{extenda} and @code{extendb} indicate whether the shading should extend beyond the radii @code{a} and @code{b}. Illustrations of radial shading are provided in the example files @code{@uref{https://asymptote.sourceforge.io/gallery/shade.svg,,shade}@uref{https://asymptote.sourceforge.io/gallery/shade.asy,,.asy}}, @code{@uref{https://asymptote.sourceforge.io/gallery/PDFs/ring.pdf,,ring}@uref{https://asymptote.sourceforge.io/gallery/PDFs/ring.asy,,.asy}}, and @code{@uref{https://asymptote.sourceforge.io/gallery/PDFs/shadestroke.pdf,,shadestroke}@uref{https://asymptote.sourceforge.io/gallery/PDFs/shadestroke.asy,,.asy}}. @cindex @code{gouraudshade} Gouraud shading using fill rule @code{fillrule} and the vertex colors in the pen array @code{p} on a triangular lattice defined by the vertices @code{z} and edge flags @code{edges} is implemented with @verbatim 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); @end verbatim @noindent In the second form, the elements of @code{z} are taken to be successive nodes of path @code{g}. The pens in @code{p} must belong to the same color space. Illustrations of Gouraud shading are provided in the example file @code{@uref{https://asymptote.sourceforge.io/gallery/PDFs/Gouraud.pdf,,Gouraud}@uref{https://asymptote.sourceforge.io/gallery/PDFs/Gouraud.asy,,.asy}}. The edge flags used in Gouraud shading are documented here: @quotation @url{https://www.adobe.com/content/dam/acom/en/devnet/postscript/pdfs/TN5600.SmoothShading.pdf} @end quotation @cindex Coons shading @cindex tensor product shading @cindex @code{tensorshade} Tensor product shading using clipping path @code{g}, fill rule @code{fillrule} on patches bounded by the @math{n} cyclic paths of length 4 in path array @code{b}, using the vertex colors specified in the @math{n \times 4} pen array @code{p} and internal control points in the @math{n \times 4} array @code{z}, is implemented with @verbatim void tensorshade(picture pic=currentpicture, path[] g, bool stroke=false, pen fillrule=currentpen, pen[][] p, path[] b=g, pair[][] z=new pair[][]); @end verbatim @noindent If the array @code{z} is empty, Coons shading, in which the color control points are calculated automatically, is used. The pens in @code{p} must belong to the same color space. A simpler interface for the case of a single patch (@math{n=1}) is also available: @verbatim void tensorshade(picture pic=currentpicture, path g, bool stroke=false, pen fillrule=currentpen, pen[] p, path b=g, pair[] z=new pair[]); @end verbatim One can also smoothly shade the regions between consecutive paths of a sequence using a given array of pens: @verbatim void draw(picture pic=currentpicture, pen fillrule=currentpen, path[] g, pen[] p); @end verbatim @noindent Illustrations of tensor product and Coons shading are provided in the example files @code{@uref{https://asymptote.sourceforge.io/gallery/PDFs/tensor.pdf,,tensor}@uref{https://asymptote.sourceforge.io/gallery/PDFs/tensor.asy,,.asy}}, @code{@uref{https://asymptote.sourceforge.io/gallery/PDFs/Coons.pdf,,Coons}@uref{https://asymptote.sourceforge.io/gallery/PDFs/Coons.asy,,.asy}}, @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/BezierPatch.html,,BezierPatch}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/BezierPatch.asy,,.asy}}, and @code{@uref{https://asymptote.sourceforge.io/gallery/PDFs/rainbow.pdf,,rainbow}@uref{https://asymptote.sourceforge.io/gallery/PDFs/rainbow.asy,,.asy}}. @cindex Function shading @cindex function shading @cindex @code{functionshade} More general shading possibilities are available using @TeX{} engines that produce PDF output (@pxref{texengines}): the routine @verbatim void functionshade(picture pic=currentpicture, path[] g, bool stroke=false, pen fillrule=currentpen, string shader); @end verbatim @noindent shades on picture @code{pic} the interior of path @code{g} according to fill rule @code{fillrule} using the @code{PostScript} calculator routine specified by the string @code{shader}; this routine takes 2 arguments, each in [0,1], and returns @code{colors(fillrule).length} color components. Function shading is illustrated in the example @code{@uref{https://asymptote.sourceforge.io/gallery/PDFs/functionshading.pdf,,functionshading}@uref{https://asymptote.sourceforge.io/gallery/PDFs/functionshading.asy,,.asy}}. @cindex unfill The following routine uses @code{evenodd} clipping together with the @code{^^} operator to unfill a region: @verbatim void unfill(picture pic=currentpicture, path g); @end verbatim @node clip, label, fill, Drawing commands @section clip @cindex @code{clip} @cindex @code{stroke} @verbatim void clip(picture pic=currentpicture, path g, stroke=false, pen fillrule=currentpen); @end verbatim Clip the current contents of picture @code{pic} to the region bounded by the path @code{g}, using fill rule @code{fillrule} (@pxref{fillrule}). If @code{stroke=true}, the clipped portion is the same as the region that would be drawn with @code{draw(pic,g,zerowinding)}; in this case the path @code{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 @code{pic}. For an illustration of picture clipping, see the first example in @ref{LaTeX usage}. @node label, , clip, Drawing commands @section label @cindex @code{label} @verbatim void label(picture pic=currentpicture, Label L, pair position, align align=NoAlign, pen p=currentpen, filltype filltype=NoFill) @end verbatim Draw Label @code{L} on picture @code{pic} using pen @code{p}. If @code{align} is @code{NoAlign}, the label will be centered at user coordinate @code{position}; otherwise it will be aligned in the direction of @code{align} and displaced from @code{position} by the @code{PostScript} offset @code{align*labelmargin(p)}. @cindex @code{Align} The constant @code{Align} can be used to align the bottom-left corner of the label at @code{position}. @cindex @code{nullpen} @cindex @code{Label} @anchor{Label} The Label @code{L} can either be a string or the structure obtained by calling one of the functions @verbatim 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); @end verbatim The text of a Label can be scaled, slanted, rotated, or shifted by multiplying it on the left by an affine transform (@pxref{Transforms}). For example, @code{rotate(45)*xscale(2)*L} first scales @code{L} in the @math{x} direction and then rotates it counterclockwise by 45 degrees. The final position of a Label can also be shifted by a @code{PostScript} coordinate translation: @code{shift(10,0)*L}. An explicit pen specified within the Label overrides other pen arguments. The @code{embed} argument determines how the Label should transform with the embedding picture: @table @code @item Shift @cindex @code{Shift} only shift with embedding picture; @item Rotate @cindex @code{Rotate} only shift and rotate with embedding picture (default); @item Rotate(pair z) @cindex @code{Rotate(pair z)} rotate with (picture-transformed) vector @code{z}. @item Slant @cindex @code{Slant} only shift, rotate, slant, and reflect with embedding picture; @item Scale @cindex @code{Scale} shift, rotate, slant, reflect, and scale with embedding picture. @end table To add a label to a path, use @verbatim void label(picture pic=currentpicture, Label L, path g, align align=NoAlign, pen p=currentpen, filltype filltype=NoFill); @end verbatim @cindex @code{Relative} By default the label will be positioned at the midpoint of the path. An alternative label position (in the sense of @code{point(path p, real t)}) may be specified as a real value for @code{position} in constructing the Label. The position @code{Relative(real)} specifies a location relative to the total arclength of the path. These convenient abbreviations are predefined: @cindex @code{BeginPoint} @cindex @code{MidPoint} @cindex @code{EndPoint} @verbatim position BeginPoint=Relative(0); position MidPoint=Relative(0.5); position EndPoint=Relative(1); @end verbatim @cindex @code{Relative} @cindex @code{LeftSide} @cindex @code{Center} @cindex @code{RightSide} Path labels are aligned in the direction @code{align}, which may be specified as an absolute compass direction (pair) or a direction @code{Relative(pair)} measured relative to a north axis in the local direction of the path. For convenience @code{LeftSide}, @code{Center}, and @code{RightSide} are defined as @code{Relative(W)}, @code{Relative((0,0))}, and @code{Relative(E)}, respectively. Multiplying @code{LeftSide} and @code{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 @code{arrowlength} pointing to @code{b} from direction @code{dir} can be produced with the routine @cindex @code{arrow} @verbatim 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); @end verbatim If no alignment is specified (either in the Label or as an explicit argument), the optional Label will be aligned in the direction @code{dir}, using margin @code{margin}. @cindex including images @cindex @code{graphic} @cindex @acronym{EPS} The function @code{string graphic(string name, string options="")} returns a string that can be used to include an encapsulated @code{PostScript} (@acronym{EPS}) file. Here, @code{name} is the name of the file to include and @code{options} is a string containing a comma-separated list of optional bounding box (@code{bb=llx lly urx ury}), width (@code{width=value}), height (@code{height=value}), rotation (@code{angle=value}), scaling (@code{scale=factor}), clipping (@code{clip=bool}), and draft mode (@code{draft=bool}) parameters. The @code{layer()} function can be used to force future objects to be drawn on top of the included image: @verbatim label(graphic("file.eps","width=1cm"),(0,0),NE); layer(); @end verbatim @cindex @code{baseline} The @code{string baseline(string s, string template="\strut")} function can be used to enlarge the bounding box of labels to match a given template, so that their baselines will be typeset on a horizontal line. See @code{@uref{https://asymptote.sourceforge.io/gallery/Pythagoras.svg,,Pythagoras}@uref{https://asymptote.sourceforge.io/gallery/Pythagoras.asy,,.asy}} for an example. One can prevent labels from overwriting one another with the @code{overwrite} pen attribute (@pxref{overwrite}). The structure @code{object} defined in @code{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 @cindex @code{pack} @verbatim frame pack(pair align=2S ... object inset[]); @end verbatim @noindent To draw or fill a box (or ellipse or other path) around a Label and return the bounding object, use one of the routines @verbatim 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); @end verbatim @noindent Here @code{envelope} is a boundary-drawing routine such as @code{box}, @code{roundbox}, or @code{ellipse} defined in @code{plain_boxes.asy} (@pxref{envelope}). @cindex @code{texpath} The function @code{path[] texpath(Label L)} returns the path array that @TeX{} would fill to draw the Label @code{L}. @cindex @code{minipage} The @code{string minipage(string s, width=100pt)} function can be used to format string @code{s} into a paragraph of width @code{width}. This example uses @code{minipage}, @code{clip}, and @code{graphic} to produce a CD label: @sp 1 @center @image{./CDlabel} @verbatiminclude CDlabel.asy @node Bezier curves, Programming, Drawing commands, Top @chapter Bezier curves @cindex Bezier curves @cindex direction specifier Each interior node of a cubic spline may be given a direction prefix or suffix @code{@{dir@}}: the direction of the pair @code{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 @math{z_0}, with postcontrol point @math{c_0}, and the node @math{z_1}, with precontrol point @math{c_1}, is computed as the Bezier curve @sp 1 @center @image{./bezier,,,(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 (@math{m_5}) constructed from two endpoints @math{z_0} and @math{z_1} and two control points @math{c_0} and @math{c_1}, is the point corresponding to @math{t=1/2} on the Bezier curve formed by the quadruple (@math{z_0}, @math{c_0}, @math{c_1}, @math{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: @sp 1 @center @image{./bezier2} Here @math{m_0}, @math{m_1} and @math{m_2} are the first-order midpoints, @math{m_3} and @math{m_4} are the second-order midpoints, and @math{m_5} is the third-order midpoint. The curve is then constructed by recursively applying the algorithm to (@math{z_0}, @math{m_0}, @math{m_3}, @math{m_5}) and (@math{m_5}, @math{m_4}, @math{m_2}, @math{z_1}). In fact, an analogous property holds for points located at any fraction @math{t} in @math{[0,1]} of each segment, not just for midpoints (@math{t=1/2}). The Bezier curve constructed in this manner has the following properties: @itemize @bullet @item It is entirely contained in the convex hull of the given four points. @item It starts heading from the first endpoint to the first control point and finishes heading from the second control point to the second endpoint. @end itemize @cindex @code{controls} The user can specify explicit control points between two nodes like this: @verbatim draw((0,0)..controls (0,100) and (100,100)..(100,0)); @end verbatim However, it is usually more convenient to just use the @code{..} operator, which tells @code{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. @cindex @code{tension} @cindex @code{and} @cindex @code{atleast} One can change the spline tension from its default value of 1 to any real value greater than or equal to 0.75 (cf. John D. Hobby, Discrete and Computational Geometry 1, 1986): @verbatim 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)); @end verbatim In these examples there is a space between @code{2} and @code{..}. This is needed as @code{2.} is interpreted as a numerical constant. @cindex @code{curl} The curl parameter specifies the curvature at the endpoints of a path (0 means straight; the default value of 1 means approximately circular): @verbatim draw((100,0){curl 0}..(100,100)..{curl 0}(0,100)); @end verbatim @cindex @code{MetaPost ...@ } @cindex @code{::} The @code{MetaPost ...} path connector, which requests, when possible, an inflection-free curve confined to a triangle defined by the endpoints and directions, is implemented in @code{Asymptote} as the convenient abbreviation @code{::} for @code{..tension atleast 1 ..} (the ellipsis @code{...} is used in @code{Asymptote} to indicate a variable number of arguments; @pxref{Rest arguments}). For example, compare @verbatiminclude dots.asy @sp 1 @center @image{./dots} @noindent with @verbatiminclude colons.asy @sp 1 @center @image{./colons} @cindex @code{---} @cindex @code{&} The @code{---} connector is an abbreviation for @code{..tension atleast infinity..} and the @code{&} 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). @node Programming, LaTeX usage, Bezier curves, Top @chapter Programming @cindex 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 * 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 high-order functions * Arrays:: Dynamic vectors * Casts:: Implicit and explicit casts * Import:: Importing external @code{Asymptote} modules * Static:: Where to allocate your variable? @end menu Here is a short introductory example to the @code{Asymptote} programming language that highlights the similarity of its control structures with those of C, C++, and Java: @cindex declaration @cindex assignment @cindex conditional @cindex loop @cindex @code{if} @cindex @code{else} @cindex @code{for} @verbatim // 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); } @end verbatim @cindex @code{while} @cindex @code{do} @cindex @code{break} @cindex @code{continue} @code{Asymptote} supports @code{while}, @code{do}, @code{break}, and @code{continue} statements just as in C/C++. It also supports the Java-style shorthand for iterating over all elements of an array: @cindex array iteration @anchor{array iteration} @verbatim // Iterate over an array int[] array={1,1,2,3,5}; for(int k : array) { write(k); } @end verbatim @noindent In addition, it supports many features beyond the ones found in those languages. @node Data types, Paths and guides, Programming, Programming @section Data types @cindex data types @code{Asymptote} supports the following data types (in addition to user-defined types): @table @code @item void @cindex @code{void} The void type is used only by functions that take or return no arguments. @item bool @cindex @code{bool} a boolean type that can only take on the values @code{true} or @code{false}. For example: @verbatim bool b=true; @end verbatim @noindent defines a boolean variable @code{b} and initializes it to the value @code{true}. If no initializer is given: @verbatim bool b; @end verbatim @noindent the value @code{false} is assumed. @item bool3 @cindex @code{bool3} an extended boolean type that can take on the values @code{true}, @code{default}, or @code{false}. A bool3 type can be cast to or from a bool. The default initializer for bool3 is @code{default}. @item int @cindex @code{int} @cindex @code{intMin} @cindex @code{intMax} an integer type; if no initializer is given, the implicit value @code{0} is assumed. The minimum allowed value of an integer is @code{intMin} and the maximum value is @code{intMax}. @item real @cindex @code{real} @cindex @code{realMin} @cindex @code{realMax} @cindex @code{realEpsilon} @cindex @code{realDigits} @cindex @code{mask} @cindex @code{inf} @cindex @code{nan} @cindex @code{isnan} a real number; this should be set to the highest-precision native floating-point type on the architecture. The implicit initializer for reals is @code{0.0}. Real numbers have precision @code{realEpsilon}, with @code{realDigits} significant digits. The smallest positive real number is @code{realMin} and the largest positive real number is @code{realMax}. The variables @code{inf} and @code{nan}, along with the function @code{bool isnan(real x)} are useful when floating-point exceptions are masked with the @code{-mask} command-line option (the default in interactive mode). @item pair @cindex @code{pair} complex number, that is, an ordered pair of real components @code{(x,y)}. The real and imaginary parts of a pair @code{z} can read as @code{z.x} and @code{z.y}. We say that @code{x} and @code{y} are virtual members of the data element pair; they cannot be directly modified, however. The implicit initializer for pairs is @code{(0.0,0.0)}. There are a number of ways to take the complex conjugate of a pair: @example pair z=(3,4); z=(z.x,-z.y); z=z.x-I*z.y; z=conj(z); @end example Here @code{I} is the pair @code{(0,1)}. A number of built-in functions are defined for pairs: @table @code @item pair conj(pair z) @cindex @code{conj} returns the conjugate of @code{z}; @item real length(pair z) @cindex @code{length} @cindex @code{abs} @cindex @code{abs2} returns the complex modulus @math{|@code{z}|} of its argument @code{z}. For example, @example pair z=(3,4); length(z); @end example returns the result 5. A synonym for @code{length(pair)} is @code{abs(pair)}. The function @code{abs2(pair z)} returns @math{|@code{z}|^2}; @item real angle(pair z, bool warn=true) @cindex @code{angle} returns the angle of @code{z} in radians in the interval [-@code{pi},@code{pi}] or @code{0} if @code{warn} is @code{false} and @code{z=(0,0)} (rather than producing an error); @item real degrees(pair z, bool warn=true) @cindex @code{degrees} returns the angle of @code{z} in degrees in the interval [0,360) or @code{0} if @code{warn} is @code{false} and @code{z=(0,0)} (rather than producing an error); @item pair unit(pair z) @cindex @code{unit} returns a unit vector in the direction of the pair @code{z}; @item pair expi(real angle) @cindex @code{expi} returns a unit vector in the direction @code{angle} measured in radians; @item pair dir(real degrees) @cindex @code{dir} returns a unit vector in the direction @code{degrees} measured in degrees; @item real xpart(pair z) @cindex @code{xpart} returns @code{z.x}; @item real ypart(pair z) @cindex @code{ypart} returns @code{z.y}; @item pair realmult(pair z, pair w) @cindex @code{realmult} returns the element-by-element product @code{(z.x*w.x,z.y*w.y)}; @item real dot(explicit pair z, explicit pair w) @cindex @code{dot} returns the dot product @code{z.x*w.x+z.y*w.y}; @item real cross(explicit pair z, explicit pair w) @cindex @code{cross} returns the 2D scalar product @code{z.x*w.y-z.y*w.x}; @cindex @code{orient} @item real orient(pair a, pair b, pair c); returns a positive (negative) value if @code{a--b--c--cycle} is oriented counterclockwise (clockwise) or zero if all three points are colinear. Equivalently, a positive (negative) value is returned if @code{c} lies to the left (right) of the line through @code{a} and @code{b} or zero if @code{c} lies on this line. The value returned can be expressed in terms of the 2D scalar cross product as @code{cross(a-c,b-c)}, which is the determinant @verbatim |a.x a.y 1| |b.x b.y 1| |c.x c.y 1| @end verbatim @cindex @code{incircle} @item real incircle(pair a, pair b, pair c, pair d); returns a positive (negative) value if @code{d} lies inside (outside) the circle passing through the counterclockwise-oriented points @code{a,b,c} or zero if @code{d} lies on the this circle. The value returned is the determinant @verbatim |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| @end verbatim @item pair minbound(pair z, pair w) @cindex @code{minbound} returns @code{(min(z.x,w.x),min(z.y,w.y))}; @item pair maxbound(pair z, pair w) @cindex @code{maxbound} returns @code{(max(z.x,w.x),max(z.y,w.y))}. @end table @item triple @cindex @code{triple} an ordered triple of real components @code{(x,y,z)} used for three-dimensional drawings. The respective components of a triple @code{v} can read as @code{v.x}, @code{v.y}, and @code{v.z}. The implicit initializer for triples is @code{(0.0,0.0,0.0)}. Here are the built-in functions for triples: @table @code @item real length(triple v) @cindex @code{length} @cindex @code{abs} @cindex @code{abs2} returns the length @math{|@code{v}|} of its argument @code{v}. A synonym for @code{length(triple)} is @code{abs(triple)}. The function @code{abs2(triple v)} returns @math{|@code{v}|^2}; @item real polar(triple v, bool warn=true) @cindex @code{polar} returns the colatitude of @code{v} measured from the @math{z} axis in radians or @code{0} if @code{warn} is @code{false} and @code{v=O} (rather than producing an error); @item real azimuth(triple v, bool warn=true) @cindex @code{azimuth} returns the longitude of @code{v} measured from the @math{x} axis in radians or @code{0} if @code{warn} is @code{false} and @code{v.x=v.y=0} (rather than producing an error); @item real colatitude(triple v, bool warn=true) @cindex @code{colatitude} returns the colatitude of @code{v} measured from the @math{z} axis in degrees or @code{0} if @code{warn} is @code{false} and @code{v=O} (rather than producing an error); @item real latitude(triple v, bool warn=true) @cindex @code{latitude} returns the latitude of @code{v} measured from the @math{xy} plane in degrees or @code{0} if @code{warn} is @code{false} and @code{v=O} (rather than producing an error); @item real longitude(triple v, bool warn=true) @cindex @code{longitude} returns the longitude of @code{v} measured from the @math{x} axis in degrees or @code{0} if @code{warn} is @code{false} and @code{v.x=v.y=0} (rather than producing an error); @item triple unit(triple v) @cindex @code{unit} returns a unit triple in the direction of the triple @code{v}; @item triple expi(real polar, real azimuth) @cindex @code{expi} returns a unit triple in the direction @code{(polar,azimuth)} measured in radians; @item triple dir(real colatitude, real longitude) @cindex @code{dir} returns a unit triple in the direction @code{(colatitude,longitude)} measured in degrees; @item real xpart(triple v) @cindex @code{xpart} returns @code{v.x}; @item real ypart(triple v) @cindex @code{ypart} returns @code{v.y}; @item real zpart(triple v) @cindex @code{zpart} returns @code{v.z}; @item real dot(triple u, triple v) @cindex @code{dot} returns the dot product @code{u.x*v.x+u.y*v.y+u.z*v.z}; @item triple cross(triple u, triple v) @cindex @code{cross} returns the cross product @code{(u.y*v.z-u.z*v.y,u.z*v.x-u.x*v.z,u.x*v.y-v.x*u.y)}; @item triple minbound(triple u, triple v) @cindex @code{minbound} returns @code{(min(u.x,v.x),min(u.y,v.y),min(u.z,v.z))}; @item triple maxbound(triple u, triple v) @cindex @code{maxbound} returns @code{(max(u.x,v.x),max(u.y,v.y),max(u.z,v.z)}). @end table @item string @cindex @code{string} @cindex @TeX{} string a character string, implemented using the STL @code{string} class. Strings delimited by double quotes (@code{"}) are subject to the following mappings to allow the use of double quotes in @TeX{} (e.g.@ for using the @code{babel} package, @pxref{babel}): @itemize @bullet @item \" maps to " @item \\ maps to \\ @end itemize @cindex @code{C} string Strings delimited by single quotes (@code{'}) have the same mappings as character strings in ANSI @code{C}: @itemize @bullet @item \' maps to ' @item \" maps to " @item \? maps to ? @item \\ maps to backslash @item \a maps to alert @item \b maps to backspace @item \f maps to form feed @item \n maps to newline @item \r maps to carriage return @item \t maps to tab @item \v maps to vertical tab @item \0-\377 map to corresponding octal byte @item \x0-\xFF map to corresponding hexadecimal byte @end itemize The implicit initializer for strings is the empty string @code{""}. Strings may be concatenated with the @code{+} operator. In the following string functions, position @code{0} denotes the start of the string: @table @code @cindex @code{length} @item int length(string s) returns the length of the string @code{s}; @cindex @code{find} @item int find(string s, string t, int pos=0) returns the position of the first occurrence of string @code{t} in string @code{s} at or after position @code{pos}, or -1 if @code{t} is not a substring of @code{s}; @cindex @code{rfind} @item int rfind(string s, string t, int pos=-1) returns the position of the last occurrence of string @code{t} in string @code{s} at or before position @code{pos} (if @code{pos}=-1, at the end of the string @code{s}), or -1 if @code{t} is not a substring of @code{s}; @cindex @code{insert} @item string insert(string s, int pos, string t) returns the string formed by inserting string @code{t} at position @code{pos} in @code{s}; @cindex @code{erase} @item string erase(string s, int pos, int n) returns the string formed by erasing the string of length @code{n} (if @code{n}=-1, to the end of the string @code{s}) at position @code{pos} in @code{s}; @cindex @code{substr} @item string substr(string s, int pos, int n=-1) returns the substring of @code{s} starting at position @code{pos} and of length @code{n} (if @code{n}=-1, until the end of the string @code{s}); @cindex @code{reverse} @item string reverse(string s) returns the string formed by reversing string @code{s}; @item string replace(string s, string before, string after) @cindex @code{replace} returns a string with all occurrences of the string @code{before} in the string @code{s} changed to the string @code{after}; @item string replace(string s, string[][] table) returns a string constructed by translating in string @code{s} all occurrences of the string @code{before} in an array @code{table} of string pairs @{@code{before},@code{after}@} to the corresponding string @code{after}; @cindex @code{split} @item string[] split(string s, string delimiter="") returns an array of strings obtained by splitting @code{s} into substrings delimited by @code{delimiter} (an empty delimiter signifies a space, but with duplicate delimiters discarded); @cindex @code{array} @cindex @code{operator +(...string[] a)}. @item string[] array(string s) returns an array of strings obtained by splitting @code{s} into individual characters. The inverse operation is provided by @code{operator +(...string[] a)}. @anchor{format} @item string format(string s, int n, string locale="") @cindex @code{format} returns a string containing @code{n} formatted according to the C-style format string @code{s} using locale @code{locale} (or the current locale if an empty string is specified), following the behaviour of the C function @code{fprintf}), except that only one data field is allowed. @item string format(string s=defaultformat, bool forcemath=false, string s=defaultseparator, real x, string locale="") returns a string containing @code{x} formatted according to the C-style format string @code{s} using locale @code{locale} (or the current locale if an empty string is specified), following the behaviour of the C function @code{fprintf}), except that only one data field is allowed, trailing zeros are removed by default (unless @code{#} is specified), and if @code{s} specifies math mode or @code{forcemath=true}, @TeX{} is used to typeset scientific notation using the @code{defaultseparator="\!\times\!";}; @cindex @code{hex} @cindex @code{hexadecimal} @item int hex(string s); casts a hexadecimal string @code{s} to an integer; @cindex @code{ascii} @cindex @code{ascii} @item int ascii(string s); returns the ASCII code for the first character of string @code{s}; @cindex @code{string} @item string string(real x, int digits=realDigits) casts @code{x} to a string using precision @code{digits} and the C locale; @cindex @code{locale} @item string locale(string s="") sets the locale to the given string, if nonempty, and returns the current locale; @item string time(string format="%a %b %d %T %Z %Y") @cindex @code{time} @cindex date @cindex @code{strftime} returns the current time formatted by the ANSI C routine @code{strftime} according to the string @code{format} using the current locale. Thus @verbatim time(); time("%a %b %d %H:%M:%S %Z %Y"); @end verbatim @noindent are equivalent ways of returning the current time in the default format used by the @code{UNIX} @code{date} command; @cindex @code{seconds} @cindex @code{strptime} @item 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 @code{strptime} according to the string @code{format} using the current locale, or the current time if @code{t} is the empty string. Note that the @code{"%Z"} extension to the POSIX @code{strptime} specification is ignored by the current GNU C Library. If an error occurs, the value -1 is returned. Here are some examples: @verbatim 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); @end verbatim The last example returns today's ordinal date, measured from the beginning of the year. @cindex @code{time} @cindex @code{strftime} @item string time(int seconds, string format="%a %b %d %T %Z %Y") returns the time corresponding to @code{seconds} seconds after the Epoch (Thu Jan 01 00:00:00 UTC 1970) formatted by the ANSI C routine @code{strftime} according to the string @code{format} using the current locale. For example, to return the date corresponding to 24 hours ago: @verbatim time(seconds()-24*60*60); @end verbatim @cindex @code{system} @item int system(string s) @item int system(string[] s) if the setting @code{safe} is false, call the arbitrary system command @code{s}; @cindex @code{asy} @item void asy(string format, bool overwrite=false ... string[] s) conditionally process each file name in array @code{s} in a new environment, using format @code{format}, overwriting the output file only if @code{overwrite} is true; @cindex @code{abort} @item void abort(string s="") aborts execution (with a non-zero return code in batch mode); if string @code{s} is nonempty, a diagnostic message constructed from the source file, line number, and @code{s} is printed; @cindex @code{assert} @item void assert(bool b, string s="") aborts execution with an error message constructed from @code{s} if @code{b=false}; @cindex @code{exit} @item void exit() exits (with a zero error return code in batch mode); @cindex @code{sleep} @item void sleep(int seconds) pauses for the given number of seconds; @cindex @code{usleep} @item void usleep(int microseconds) pauses for the given number of microseconds; @cindex @code{beep} @item void beep() produces a beep on the console; @end table @cindex @code{typedef} @end table As in C/C++, complicated types may be abbreviated with @code{typedef} (see the example in @ref{Functions}). @node Paths and guides, Pens, Data types, Programming @section Paths and guides @table @code @item path @cindex @code{path} a cubic spline resolved into a fixed path. The implicit initializer for paths is @code{nullpath}. @cindex @code{circle} @anchor{circle} For example, the routine @code{circle(pair c, real r)}, which returns a Bezier curve approximating a circle of radius @code{r} centered on @code{c}, is based on @code{unitcircle} (@pxref{unitcircle}): @verbatim path circle(pair c, real r) { return shift(c)*scale(r)*unitcircle; } @end verbatim If high accuracy is needed, a true circle may be produced with the routine @code{Circle} defined in the module @code{graph}: @cindex @code{Circle} @verbatim import graph; path Circle(pair c, real r, int n=nCircle); @end verbatim A circular arc consistent with @code{circle} centered on @code{c} with radius @code{r} from @code{angle1} to @code{angle2} degrees, drawing counterclockwise if @code{angle2 >= angle1}, can be constructed with @cindex @code{arc} @verbatim path arc(pair c, real r, real angle1, real angle2); @end verbatim One may also specify the direction explicitly: @verbatim path arc(pair c, real r, real angle1, real angle2, bool direction); @end verbatim Here the direction can be specified as CCW (counter-clockwise) or CW (clockwise). For convenience, an arc centered at @code{c} from pair @code{z1} to @code{z2} (assuming @code{|z2-c|=|z1-c|}) in the may also be constructed with @verbatim path arc(pair c, explicit pair z1, explicit pair z2, bool direction=CCW) @end verbatim If high accuracy is needed, true arcs may be produced with routines in the module @code{graph} that produce Bezier curves with @code{n} control points: @cindex @code{Arc} @verbatim 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); @end verbatim An ellipse can be drawn with the routine @cindex @code{ellipse} @verbatim path ellipse(pair c, real a, real b) { return shift(c)*scale(a,b)*unitcircle; } @end verbatim A brace can be constructed between pairs @code{a} and @code{b} with @cindex @code{brace} @verbatim path brace(pair a, pair b, real amplitude=bracedefaultratio*length(b-a)); @end verbatim This example illustrates the use of all five guide connectors discussed in @ref{Tutorial} and @ref{Bezier curves}: @verbatiminclude join.asy @sp 1 @center @image{./join} Here are some useful functions for paths: @table @code @cindex @code{length} @item int length(path p); This is the number of (linear or cubic) segments in path @code{p}. If @code{p} is cyclic, this is the same as the number of nodes in @code{p}. @cindex @code{size} @item int size(path p); This is the number of nodes in the path @code{p}. If @code{p} is cyclic, this is the same as @code{length(p)}. @cindex @code{cyclic} @item bool cyclic(path p); returns @code{true} iff path @code{p} is cyclic. @cindex @code{straight} @item bool straight(path p, int i); returns @code{true} iff the segment of path @code{p} between node @code{i} and node @code{i+1} is straight. @cindex @code{piecewisestraight} @item bool piecewisestraight(path p) returns @code{true} iff the path @code{p} is piecewise straight. @cindex @code{point} @item pair point(path p, int t); If @code{p} is cyclic, return the coordinates of node @code{t} mod @code{length(p)}. Otherwise, return the coordinates of node @code{t}, unless @code{t} < 0 (in which case @code{point(0)} is returned) or @code{t} > @code{length(p)} (in which case @code{point(length(p))} is returned). @item pair point(path p, real t); This returns the coordinates of the point between node @code{floor(t)} and @code{floor(t)+1} corresponding to the cubic spline parameter @code{t-floor(t)} (@pxref{Bezier curves}). If @code{t} lies outside the range [0,@code{length(p)}], it is first reduced modulo @code{length(p)} in the case where @code{p} is cyclic or else converted to the corresponding endpoint of @code{p}. @cindex @code{dir} @item pair dir(path p, int t, int sign=0, bool normalize=true); If @code{sign < 0}, return the direction (as a pair) of the incoming tangent to path @code{p} at node @code{t}; if @code{sign > 0}, return the direction of the outgoing tangent. If @code{sign=0}, the mean of these two directions is returned. @item pair dir(path p, real t, bool normalize=true); returns the direction of the tangent to path @code{p} at the point between node @code{floor(t)} and @code{floor(t)+1} corresponding to the cubic spline parameter @code{t-floor(t)} (@pxref{Bezier curves}). @item pair dir(path p) returns dir(p,length(p)). @item pair dir(path p, path q) returns unit(dir(p)+dir(q)). @cindex @code{accel} @item pair accel(path p, int t, int sign=0); If @code{sign < 0}, return the acceleration of the incoming path @code{p} at node @code{t}; if @code{sign > 0}, return the acceleration of the outgoing path. If @code{sign=0}, the mean of these two accelerations is returned. @cindex @code{accel} @item pair accel(path p, real t); returns the acceleration of the path @code{p} at the point @code{t}. @cindex @code{radius} @item real radius(path p, real t); returns the radius of curvature of the path @code{p} at the point @code{t}. @cindex @code{precontrol} @item pair precontrol(path p, int t); returns the precontrol point of @code{p} at node @code{t}. @item pair precontrol(path p, real t); returns the effective precontrol point of @code{p} at parameter @code{t}. @cindex @code{postcontrol} @item pair postcontrol(path p, int t); returns the postcontrol point of @code{p} at node @code{t}. @item pair postcontrol(path p, real t); returns the effective postcontrol point of @code{p} at parameter @code{t}. @cindex @code{arclength} @item real arclength(path p); returns the length (in user coordinates) of the piecewise linear or cubic curve that path @code{p} represents. @cindex @code{arctime} @item 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 @code{point(path p, real t)}, at which the cumulative arclength (measured from the beginning of the path) equals @code{L}. @cindex @code{arcpoint} @item pair arcpoint(path p, real L); returns @code{point(p,arctime(p,L))}. @cindex @code{dirtime} @item 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 @code{point(path, real)}, at which the tangent to the path has the direction of pair @code{z}, or -1 if this never happens. @cindex @code{reltime} @item real reltime(path p, real l); returns the time on path @code{p} at the relative fraction @code{l} of its arclength. @cindex @code{relpoint} @item pair relpoint(path p, real l); returns the point on path @code{p} at the relative fraction @code{l} of its arclength. @cindex @code{midpoint} @item pair midpoint(path p); returns the point on path @code{p} at half of its arclength. @cindex @code{reverse} @item path reverse(path p); returns a path running backwards along @code{p}. @cindex @code{subpath} @item path subpath(path p, int a, int b); returns the subpath of @code{p} running from node @code{a} to node @code{b}. If @code{a} > @code{b}, the direction of the subpath is reversed. @item path subpath(path p, real a, real b); returns the subpath of @code{p} running from path time @code{a} to path time @code{b}, in the sense of @code{point(path, real)}. If @code{a} > @code{b}, the direction of the subpath is reversed. @cindex @code{intersect} @item real[] intersect(path p, path q, real fuzz=-1); If @code{p} and @code{q} have at least one intersection point, return a real array of length 2 containing the times representing the respective path times along @code{p} and @code{q}, in the sense of @code{point(path, real)}, for one such intersection point (as chosen by the algorithm described on page 137 of @code{The MetaFontbook}). The computations are performed to the absolute error specified by @code{fuzz}, or if @code{fuzz < 0}, to machine precision. If the paths do not intersect, return a real array of length 0. @cindex @code{intersections} @item real[][] intersections(path p, path q, real fuzz=-1); Return all (unless there are infinitely many) intersection times of paths @code{p} and @code{q} as a sorted array of real arrays of length 2 (@pxref{sort}). The computations are performed to the absolute error specified by @code{fuzz}, or if @code{fuzz < 0}, to machine precision. @cindex @code{intersections} @item real[] intersections(path p, explicit pair a, explicit pair b, real fuzz=-1); Return all (unless there are infinitely many) intersection times of path @code{p} with the (infinite) line through points @code{a} and @code{b} as a sorted array. The intersections returned are guaranteed to be correct to within the absolute error specified by @code{fuzz}, or if @code{fuzz < 0}, to machine precision. @cindex @code{times} @item real[] times(path p, real x) returns all intersection times of path @code{p} with the vertical line through @code{(x,0)}. @cindex @code{times} @item real[] times(path p, explicit pair z) returns all intersection times of path @code{p} with the horizontal line through @code{(0,z.y)}. @cindex @code{mintimes} @item real[] mintimes(path p) returns an array of length 2 containing times at which path @code{p} reaches its minimal horizontal and vertical extents, respectively. @cindex @code{maxtimes} @item real[] maxtimes(path p) returns an array of length 2 containing times at which path @code{p} reaches its maximal horizontal and vertical extents, respectively. @cindex @code{intersectionpoint} @item pair intersectionpoint(path p, path q, real fuzz=-1); returns the intersection point @code{point(p,intersect(p,q,fuzz)[0])}. @cindex @code{intersectionpoints} @item pair[] intersectionpoints(path p, path q, real fuzz=-1); returns an array containing all intersection points of the paths @code{p} and @code{q}. @anchor{extension} @cindex @code{whatever} @cindex @code{extension} @item pair extension(pair P, pair Q, pair p, pair q); returns the intersection point of the extensions of the line segments @code{P--Q} and @code{p--q}, or if the lines are parallel, @code{(infinity,infinity)}. @cindex @code{cut} @cindex @code{slice} @item slice cut(path p, path knife, int n); returns the portions of path @code{p} before and after the @code{n}th intersection of @code{p} with path @code{knife} as a structure @code{slice} (if no intersection exist is found, the entire path is considered to be `before' the intersection): @verbatim struct slice { path before,after; } @end verbatim The argument @code{n} is treated as modulo the number of intersections. @cindex @code{firstcut} @cindex @code{slice} @item slice firstcut(path p, path knife); equivalent to @code{cut(p,knife,0);} @cindex @code{MetaPost cutbefore} Note that @code{firstcut.after} plays the role of the @code{MetaPost cutbefore} command. @cindex @code{lastcut} @item slice lastcut(path p, path knife); equivalent to @code{cut(p,knife,-1);} @cindex @code{MetaPost cutafter} Note that @code{lastcut.before} plays the role of the @code{MetaPost cutafter} command. @cindex @code{buildcycle} @item path buildcycle(... path[] p); This returns the path surrounding a region bounded by a list of two or more consecutively intersecting paths, following the behaviour of the @code{MetaPost buildcycle} command. @cindex @code{min} @item pair min(path p); returns the pair (left,bottom) for the path bounding box of path @code{p}. @cindex @code{max} @item pair max(path p); returns the pair (right,top) for the path bounding box of path @code{p}. @cindex @code{windingnumber} @cindex @code{undefined} @item int windingnumber(path p, pair z); returns the winding number of the cyclic path @code{p} relative to the point @code{z}. The winding number is positive if the path encircles @code{z} in the counterclockwise direction. If @code{z} lies on @code{p} the constant @code{undefined} (defined to be the largest odd integer) is returned. @cindex @code{interior} @item bool interior(int windingnumber, pen fillrule) returns true if @code{windingnumber} corresponds to an interior point according to @code{fillrule}. @cindex @code{inside} @item bool inside(path p, pair z, pen fillrule=currentpen); returns @code{true} iff the point @code{z} lies inside or on the edge of the region bounded by the cyclic path @code{p} according to the fill rule @code{fillrule} (@pxref{fillrule}). @cindex @code{inside} @item int inside(path p, path q, pen fillrule=currentpen); returns @code{1} if the cyclic path @code{p} strictly contains @code{q} according to the fill rule @code{fillrule} (@pxref{fillrule}), @code{-1} if the cyclic path @code{q} strictly contains @code{p}, and @code{0} otherwise. @cindex @code{inside} @item pair inside(path p, pen fillrule=currentpen); returns an arbitrary point strictly inside a cyclic path @code{p} according to the fill rule @code{fillrule} (@pxref{fillrule}). @cindex @code{strokepath} @item path[] strokepath(path g, pen p=currentpen); returns the path array that @code{PostScript} would fill in drawing path @code{g} with pen @code{p}. @end table @item guide @cindex @code{guide} an unresolved cubic spline (list of cubic-spline nodes and control points). The implicit initializer for a guide is @code{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: @verbatiminclude mexicanhat.asy @sp 1 @center @image{./mexicanhat} We point out an efficiency distinction in the use of guides and paths: @verbatim guide g; for(int i=0; i < 10; ++i) g=g--(i,i); path p=g; @end verbatim @noindent runs in linear time, whereas @verbatim path p; for(int i=0; i < 10; ++i) p=p--(i,i); @end verbatim @noindent 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): @table @code @cindex @code{size} @item int size(guide g); Analogous to @code{size(path p)}. @cindex @code{length} @item int length(guide g); Analogous to @code{length(path p)}. @cindex @code{cyclic} @item bool cyclic(path p); Analogous to @code{cyclic(path p)}. @cindex @code{point} @item pair point(guide g, int t); Analogous to @code{point(path p, int t)}. @cindex @code{reverse} @item guide reverse(guide g); Analogous to @code{reverse(path p)}. If @code{g} is cyclic and also contains a secondary cycle, it is first solved to a path, then reversed. If @code{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. @cindex @code{dirSpecifier} @item 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 @code{(0,0)} if none specified) for the segment of guide @code{g} between nodes @code{i} and @code{i+1}. @cindex @code{controlSpecifier} @item pair[] controlSpecifier(guide g, int i); If the segment of guide @code{g} between nodes @code{i} and @code{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. @cindex @code{tensionSpecifier} @item tensionSpecifier tensionSpecifier(guide g, int i); This returns the tension specifier for the segment of guide @code{g} between nodes @code{i} and @code{i+1}. The individual components of the @code{tensionSpecifier} type can be accessed as the virtual members @code{in}, @code{out}, and @code{atLeast}. @cindex @code{curlSpecifier} @item 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 @code{g}. @end table As a technical detail we note that a direction specifier given to @code{nullpath} modifies the node on the other side: the guides @verbatim a..{up}nullpath..b; c..nullpath{up}..d; e..{up}nullpath{down}..f; @end verbatim are respectively equivalent to @verbatim a..nullpath..{up}b; c{up}..nullpath..d; e{down}..nullpath..{up}f; @end verbatim @end table @node Pens, Transforms, Paths and guides, Programming @section Pens @cindex @code{pen} @cindex @code{currentpen} @cindex @code{MetaPost pickup} In @code{Asymptote}, pens provide a context for the four basic drawing commands (@pxref{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 @code{currentpen}. This provides the same functionality as the @code{MetaPost} command @code{pickup}. The implicit initializer for pens is @code{defaultpen}. @cindex @code{+} @cindex @code{*} Pens may be added together with the nonassociative binary operator @code{+}. 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 @code{dashed+red+green} or @code{red+green+dashed} or @code{red+dashed+green}. The binary operator @code{*} 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. @itemize @bullet @item Colors are specified using one of the following colorspaces: @cindex color @table @code @item pen gray(real g); @cindex @code{gray} @cindex grayscale This produces a grayscale color, where the intensity @code{g} lies in the interval [0,1], with 0.0 denoting black and 1.0 denoting white. @item pen rgb(real r, real g, real b); @cindex @code{rgb} This produces an @acronym{RGB} color, where each of the red, green, and blue intensities @code{r}, @code{g}, @code{b}, lies in the interval [0,1]. @item pen RGB(int r, int g, int b); @cindex @code{rgb} This produces an @acronym{RGB} color, where each of the red, green, and blue intensities @code{r}, @code{g}, @code{b}, lies in the interval [0,255]. @item pen cmyk(real c, real m, real y, real k); @cindex @code{cmyk} This produces a @acronym{CMYK} color, where each of the cyan, magenta, yellow, and black intensities @code{c}, @code{m}, @code{y}, @code{k}, lies in the interval [0,1]. @item pen invisible; @cindex @code{invisible} This special pen writes in invisible ink, but adjusts the bounding box as if something had been drawn (like the @code{\phantom} command in @TeX{}). The function @code{bool invisible(pen)} can be used to test whether a pen is invisible. @end table @cindex @code{defaultpen} The default color is @code{black}; this may be changed with the routine @code{defaultpen(pen)}. The function @code{colorspace(pen p)} returns the colorspace of pen @code{p} as a string (@code{"gray"}, @code{"rgb"}, @code{"cmyk"}, or @code{""}). @cindex @code{colors} The function @code{real[] colors(pen)} returns the color components of a pen. The functions @code{pen gray(pen)}, @code{pen rgb(pen)}, and @code{pen cmyk(pen)} return new pens obtained by converting their arguments to the respective color spaces. @cindex @code{colorless} The function @code{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 @cindex @code{rgb} @cindex @code{hexadecimal} @verbatim pen rgb(string s); @end verbatim @noindent A pen can be converted to a hexadecimal string with @cindex @code{hex} @item string hex(pen p); Various shades and mixtures of the grayscale primary colors @code{black} and @code{white}, @acronym{RGB} primary colors @code{red}, @code{green}, and @code{blue}, and @acronym{RGB} secondary colors @code{cyan}, @code{magenta}, and @code{yellow} are defined as named colors, along with the @acronym{CMYK} primary colors @code{Cyan}, @code{Magenta}, @code{Yellow}, and @code{Black}, in the module @code{plain}: @sp 1 @center @image{./colors} The standard 140 @acronym{RGB} @code{X11} colors can be imported with the command @verbatim import x11colors; @end verbatim and the standard 68 @acronym{CMYK} @TeX{} colors can be imported with the command @verbatim import texcolors; @end verbatim Note that there is some overlap between these two standards and the definitions of some colors (e.g.@ @code{Green}) actually disagree. @code{Asymptote} also comes with a @code{asycolors.sty} @code{LaTeX} package that defines to @code{LaTeX} @acronym{CMYK} versions of @code{Asymptote}'s predefined colors, so that they can be used directly within @code{LaTeX} strings. Normally, such colors are passed to @code{LaTeX} via a pen argument; however, to change the color of only a portion of a string, say for a slide presentation, (@pxref{slide}) it may be desirable to specify the color directly to @code{LaTeX}. This file can be passed to @code{LaTeX} with the @code{Asymptote} command @verbatim usepackage("asycolors"); @end verbatim The structure @code{hsv} defined in @code{plain_pens.asy} may be used to convert between @acronym{HSV} and @acronym{RGB} spaces, where the hue @code{h} is an angle in @math{[0,360)} and the saturation @code{s} and value @code{v} lie in @code{[0,1]}: @verbatim 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 @end verbatim @item Line types are specified with the function @code{pen linetype(real[] a, real offset=0, bool scale=true, bool adjust=true)}, @cindex @code{solid} @cindex @code{dashed} @cindex @code{dotted} @cindex @code{longdashed} @cindex @code{dashdotted} @cindex @code{longdashdotted} where @code{a} is an array of real array numbers. The optional parameter @code{offset} specifies where in the pattern to begin. The first number specifies how far (if @code{scale} is @code{true}, in units of the pen line width; otherwise in @code{PostScript} units) to draw with the pen on, the second number specifies how far to draw with the pen off, and so on. If @code{adjust} is @code{true}, these spacings are automatically adjusted by @code{Asymptote} to fit the arclength of the path. Here are the predefined line types: @verbatim 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(); @end verbatim @sp 1 @center @image{./linetype} @cindex @code{defaultpen} The default line type is @code{solid}; this may be changed with @code{defaultpen(pen)}. @cindex @code{linetype} @cindex @code{offset} @cindex @code{scale} @cindex @code{adjust} The line type of a pen can be determined with the functions @code{real[] linetype(pen p=currentpen)}, @code{real offset(pen p)}, @code{bool scale(pen p)}, and @code{bool adjust(pen p)}. @cindex @code{linewidth} @cindex @code{defaultpen} @item The pen line width is specified in @code{PostScript} units with @code{pen linewidth(real)}. The default line width is 0.5 bp; this value may be changed with @code{defaultpen(pen)}. The line width of a pen is returned by @code{real linewidth(pen p=currentpen)}. For convenience, in the module @code{plain_pens} we define @verbatim 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;} @end verbatim so that one may set the line width like this: @verbatim defaultpen(2); pen p=red+0.5; @end verbatim @cindex @code{linecap} @cindex @code{squarecap} @cindex @code{roundcap} @cindex @code{extendcap} @cindex @code{defaultpen} @item A pen with a specific @code{PostScript} line cap is returned on calling @code{linecap} with an integer argument: @verbatim pen squarecap=linecap(0); pen roundcap=linecap(1); pen extendcap=linecap(2); @end verbatim @noindent The default line cap, @code{roundcap}, may be changed with @code{defaultpen(pen)}. The line cap of a pen is returned by @code{int linecap(pen p=currentpen)}. @cindex @code{linejoin} @cindex @code{miterjoin} @cindex @code{roundjoin} @cindex @code{beveljoin} @item A pen with a specific @code{PostScript} join style is returned on calling @code{linejoin} with an integer argument: @verbatim pen miterjoin=linejoin(0); pen roundjoin=linejoin(1); pen beveljoin=linejoin(2); @end verbatim @noindent The default join style, @code{roundjoin}, may be changed with @code{defaultpen(pen)}.The join style of a pen is returned by @code{int linejoin(pen p=currentpen)}. @cindex @code{miterlimit} @item A pen with a specific @code{PostScript} miter limit is returned by calling @code{miterlimit(real)}. The default miterlimit, @code{10.0}, may be changed with @code{defaultpen(pen)}. The miter limit of a pen is returned by @code{real miterlimit(pen p=currentpen)}. @cindex @code{fillrule} @cindex @code{zerowinding} @cindex @code{evenodd} @anchor{fillrule} @item A pen with a specific @code{PostScript} fill rule is returned on calling @code{fillrule} with an integer argument: @verbatim pen zerowinding=fillrule(0); pen evenodd=fillrule(1); @end verbatim @noindent The fill rule, which identifies the algorithm used to determine the insideness of a path or array of paths, only affects the @code{clip}, @code{fill}, and @code{inside} functions. For the @code{zerowinding} fill rule, a point @code{z} is outside the region bounded by a path if the number of upward intersections of the path with the horizontal line @code{z--z+infinity} minus the number of downward intersections is zero. For the @code{evenodd} fill rule, @code{z} is considered to be outside the region if the total number of such intersections is even. The default fill rule, @code{zerowinding}, may be changed with @code{defaultpen(pen)}. The fill rule of a pen is returned by @code{int fillrule(pen p=currentpen)}. @cindex @code{nobasealign} @cindex @code{basealign} @anchor{basealign} @item A pen with a specific text alignment setting is returned on calling @code{basealign} with an integer argument: @verbatim pen nobasealign=basealign(0); pen basealign=basealign(1); @end verbatim @noindent The default setting, @code{nobasealign},which may be changed with @code{defaultpen(pen)}, causes the label alignment routines to use the full label bounding box for alignment. In contrast, @code{basealign} requests that the @TeX{} baseline be respected. The base align setting of a pen is returned by @code{int basealign(pen p=currentpen)}. @cindex @code{fontsize} @cindex @code{lineskip} @cindex @code{defaultpen} @cindex @code{type1cm} @item The font size is specified in @TeX{} points (1 pt = 1/72.27 inches) with the function @code{pen fontsize(real size, real lineskip=1.2*size)}. The default font size, 12pt, may be changed with @code{defaultpen(pen)}. Nonstandard font sizes may require inserting @verbatim import fontsize; @end verbatim at the beginning of the file (this requires the @code{type1cm} package available from @quotation @url{http://mirror.ctan.org/macros/latex/contrib/type1cm/} @end quotation and included in recent @code{LaTeX} distributions). The font size and line skip of a pen can be examined with the routines @code{real fontsize(pen p=currentpen)} and @code{real lineskip(pen p=currentpen)}, respectively. @cindex font @cindex @LaTeX{} NFSS fonts @cindex @code{font} @item A pen using a specific @LaTeX{} NFSS font is returned by calling the function @code{pen font(string encoding, string family, string series, string shape)}. The default setting, @code{font("OT1","cmr","m","n")}, corresponds to 12pt Computer Modern Roman; this may be changed with @code{defaultpen(pen)}. The font setting of a pen is returned by @code{string font(pen p=currentpen)}. @cindex @TeX{} fonts Alternatively, one may select a fixed-size @TeX{} font (on which @code{fontsize} has no effect) like @code{"cmr12"} (12pt Computer Modern Roman) or @code{"pcrr"} (Courier) using the function @code{pen font(string name)}. An optional size argument can also be given to scale the font to the requested size: @code{pen font(string name, real size)}. @cindex @code{fontcommand} A nonstandard font command can be generated with @code{pen fontcommand(string)}. @cindex @code{PostScript} fonts A convenient interface to the following standard @code{PostScript} fonts is also provided: @verbatim 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"); @end verbatim @cindex font @cindex font encoding @cindex input encoding @cindex language context @item 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). @noindent @cindex Cyrillic @cindex Russian Here is an example for @code{cp1251} and Russian language in Cyrillic script (font encoding @code{T2A}): @verbatim texpreamble("\usepackage[math]{anttor}"); texpreamble("\usepackage[T2A]{fontenc}"); texpreamble("\usepackage[cp1251]{inputenc}"); texpreamble("\usepackage[russian]{babel}"); @end verbatim @noindent @cindex Chinese @cindex Japanese @cindex Korean @cindex CJK Support for Chinese, Japanese, and Korean fonts is provided by the CJK package: @quotation @url{https://ctan.org/pkg/cjk} @end quotation @noindent The following commands enable the CJK song family (within a label, you can also temporarily switch to another family, say kai, by prepending @code{"\CJKfamily@{kai@}"} to the label string): @verbatim texpreamble("\usepackage{CJK} \AtBeginDocument{\begin{CJK*}{GBK}{song}} \AtEndDocument{\clearpage\end{CJK*}}"); @end verbatim @anchor{transparency} @cindex transparency @cindex @code{opacity} @item The transparency of a pen can be changed with the command: @verbatim pen opacity(real opacity=1, string blend="Compatible"); @end verbatim The opacity can be varied from @code{0} (fully transparent) to the default value of @code{1} (opaque), and @code{blend} specifies one of the following foreground--background blending operations: @verbatim "Compatible","Normal","Multiply","Screen","Overlay","SoftLight", "HardLight","ColorDodge","ColorBurn","Darken","Lighten","Difference", "Exclusion","Hue","Saturation","Color","Luminosity", @end verbatim as described in @url{https://www.adobe.com/content/dam/acom/en/devnet/pdf/pdfs/PDF32000_2008.pdf}. Since @code{PostScript} does not support transparency, this feature is only effective with the @code{-f pdf} output format option; other formats can be produced from the resulting @acronym{PDF} file with the @code{ImageMagick} @code{convert} program. Labels are always drawn with an @code{opacity} of 1. A simple example of transparent filling is provided in the example file @code{@uref{https://asymptote.sourceforge.io/gallery/transparency.svg,,transparency}@uref{https://asymptote.sourceforge.io/gallery/transparency.asy,,.asy}}. @cindex patterns @cindex tilings @item @code{PostScript} commands within a @code{picture} may be used to create a tiling pattern, identified by the string @code{name}, for @code{fill} and @code{draw} operations by adding it to the global @code{PostScript} frame @code{currentpatterns}, with optional left-bottom margin @code{lb} and right-top margin @code{rt}. @verbatim import patterns; void add(string name, picture pic, pair lb=0, pair rt=0); @end verbatim To @code{fill} or @code{draw} using pattern @code{name}, use the pen @code{pattern("name")}. For example, rectangular tilings can be constructed using the routines @code{picture tile(real Hx=5mm, real Hy=0, pen p=currentpen, filltype filltype=NoFill)}, @code{picture checker(real Hx=5mm, real Hy=0, pen p=currentpen)}, and @code{picture brick(real Hx=5mm, real Hy=0, pen p=currentpen)} defined in module @code{patterns}: @cindex grid @cindex tile @cindex checker @cindex brick @verbatiminclude tile.asy @sp 1 @center @image{./tile} @cindex hatch @cindex crosshatch Hatch patterns can be generated with the routines @code{picture hatch(real H=5mm, pair dir=NE, pen p=currentpen)}, @code{picture crosshatch(real H=5mm, pen p=currentpen)}: @verbatiminclude hatch.asy @sp 1 @center @image{./hatch} You may need to turn off aliasing in your @code{PostScript} viewer for patterns to appear correctly. Custom patterns can easily be constructed, following the examples in module @code{patterns}. The tiled pattern can even incorporate shading (@pxref{gradient shading}), as illustrated in this example (not included in the manual because not all printers support @code{PostScript} 3): @verbatiminclude shadedtiling.asy @anchor{makepen} @cindex @code{makepen} @item One can specify a custom pen nib as an arbitrary polygonal path with @code{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 @code{path nib(pen)}. Unlike in @code{MetaPost}, the path need not be convex: @verbatiminclude makepen.asy @sp 1 @center @image{./makepen} The value @code{nullpath} represents a circular pen nib (the default); an elliptical pen can be achieved simply by multiplying the pen by a transform: @code{yscale(2)*currentpen}. @anchor{overwrite} @cindex @code{overwrite} @item One can prevent labels from overwriting one another by using the pen attribute @code{overwrite}, which takes a single argument: @table @code @cindex @code{Allow} @cindex @code{defaultpen} @item Allow Allow labels to overwrite one another. This is the default behaviour (unless overridden with @code{defaultpen(pen)}. @cindex @code{Suppress} @item Suppress Suppress, with a warning, each label that would overwrite another label. @cindex @code{SuppressQuiet} @item SuppressQuiet Suppress, without warning, each label that would overwrite another label. @cindex @code{Move} @item 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 @code{PostScript} coordinates) it could result in a larger figure than requested. @cindex @code{MoveQuiet} @item MoveQuiet Move a label that would overwrite another out of the way, without warning. As this adjustment is during the final output phase (in @code{PostScript} coordinates) it could result in a larger figure than requested. @end table @end itemize @cindex @code{defaultpen} @cindex @code{resetdefaultpen} The routine @code{defaultpen()} returns the current default pen attributes. Calling the routine @code{resetdefaultpen()} resets all pen default attributes to their initial values. @node Transforms, Frames and pictures, Pens, Programming @section Transforms @cindex @code{transform} @code{Asymptote} makes extensive use of affine transforms. A pair @code{(x,y)} is transformed by the transform @code{t=(t.x,t.y,t.xx,t.xy,t.yx,t.yy)} to @code{(x',y')}, where @verbatim x' = t.x + t.xx * x + t.xy * y y' = t.y + t.yx * x + t.yy * y @end verbatim @noindent This is equivalent to the @code{PostScript} transformation @code{[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 @code{*}) on the left (@pxref{circle} for an example). @cindex @code{inverse} Transforms can be composed with one another and inverted with the function @code{transform inverse(transform t)}; they can also be raised to any integer power with the @code{^} operator. The built-in transforms are: @table @code @item transform identity; @cindex @code{identity} the identity transform; @item transform shift(pair z); @cindex @code{shift} translates by the pair @code{z}; @item transform shift(real x, real y); @cindex @code{shift} translates by the pair @code{(x,y)}; @item transform xscale(real x); @cindex @code{xscale} scales by @code{x} in the @math{x} direction; @item transform yscale(real y); @cindex @code{yscale} scales by @code{y} in the @math{y} direction; @item transform scale(real s); @cindex @code{scale} scale by @code{s} in both @math{x} and @math{y} directions; @item transform scale(real x, real y); @cindex @code{scale} scale by @code{x} in the @math{x} direction and by @code{y} in the @math{y} direction; @item transform slant(real s); @cindex @code{slant} maps @code{(x,y)} --> @code{(x+s*y,y)}; @item transform rotate(real angle, pair z=(0,0)); rotates by @code{angle} in degrees about @code{z}; @item transform reflect(pair a, pair b); @cindex @code{reflect} reflects about the line @code{a--b}. @item transform zeroTransform; @cindex @code{zeroTransform} the zero transform; @end table @cindex @code{shift} @cindex @code{shiftless} The implicit initializer for transforms is @code{identity()}. The routines @code{shift(transform t)} and @code{shiftless(transform t)} return the transforms @code{(t.x,t.y,0,0,0,0)} and @code{(0,0,t.xx,t.xy,t.yx,t.yy)} respectively. The function @code{bool isometry(transform t)} can be used to test if @code{t} is an isometry (preserves distance). @node Frames and pictures, Files, Transforms, Programming @section Frames and pictures @table @code @item frame @cindex @code{frame} @cindex @code{newframe} @cindex @code{empty} @cindex @code{erase} @cindex @code{min} @cindex @code{max} Frames are canvases for drawing in @code{PostScript} coordinates. While working with frames directly is occasionally necessary for constructing deferred drawing routines, pictures are usually more convenient to work with. The implicit initializer for frames is @code{newframe}. The function @code{bool empty(frame f)} returns @code{true} only if the frame @code{f} is empty. A frame may be erased with the @code{erase(frame)} routine. The functions @code{pair min(frame)} and @code{pair max(frame)} return the (left,bottom) and (right,top) coordinates of the frame bounding box, respectively. The contents of frame @code{src} may be appended to frame @code{dest} with the command @verbatim void add(frame dest, frame src); @end verbatim or prepended with @verbatim void prepend(frame dest, frame src); @end verbatim A frame obtained by aligning frame @code{f} in the direction @code{align}, in a manner analogous to the @code{align} argument of @code{label} (@pxref{label}), is returned by @verbatim frame align(frame f, pair align); @end verbatim @cindex @code{box} @cindex @code{ellipse} @anchor{envelope} @cindex @code{envelope} To draw or fill a box or ellipse around a label or frame and return the boundary as a path, use one of the predefined @code{envelope} routines @verbatim path box(frame f, Label L="", real xmargin=0, real ymargin=xmargin, pen p=currentpen, filltype filltype=NoFill, bool above=true); path roundbox(frame f, Label L="", real xmargin=0, real ymargin=xmargin, pen p=currentpen, filltype filltype=NoFill, bool above=true); path ellipse(frame f, Label L="", real xmargin=0, real ymargin=xmargin, pen p=currentpen, filltype filltype=NoFill, bool above=true); @end verbatim @item picture @cindex @code{picture} Pictures are high-level structures (@pxref{Structures}) defined in the module @code{plain} that provide canvases for drawing in user coordinates. The default picture is called @code{currentpicture}. A new picture can be created like this: @verbatim picture pic; @end verbatim @noindent Anonymous pictures can be made by the expression @code{new picture}. The @code{size} routine specifies the dimensions of the desired picture: @anchor{size} @cindex @code{size} @verbatim void size(picture pic=currentpicture, real x, real y=x, bool keepAspect=Aspect); @end verbatim If the @code{x} and @code{y} sizes are both 0, user coordinates will be interpreted as @code{PostScript} coordinates. In this case, the transform mapping @code{pic} to the final output frame is @code{identity()}. If exactly one of @code{x} or @code{y} is 0, no size restriction is imposed in that direction; it will be scaled the same as the other direction. @cindex @code{keepAspect} @cindex @code{Aspect} If @code{keepAspect} is set to @code{Aspect} or @code{true}, the picture will be scaled with its aspect ratio preserved such that the final width is no more than @code{x} and the final height is no more than @code{y}. @cindex @code{keepAspect} @cindex @code{IgnoreAspect} If @code{keepAspect} is set to @code{IgnoreAspect} or @code{false}, the picture will be scaled in both directions so that the final width is @code{x} and the height is @code{y}. To make the user coordinates of picture @code{pic} represent multiples of @code{x} units in the @math{x} direction and @code{y} units in the @math{y} direction, use @anchor{unitsize} @cindex @code{unitsize} @verbatim void unitsize(picture pic=currentpicture, real x, real y=x); @end verbatim When nonzero, these @code{x} and @code{y} values override the corresponding size parameters of picture @code{pic}. The routine @cindex @code{size} @verbatim void size(picture pic=currentpicture, real xsize, real ysize, pair min, pair max); @end verbatim forces the final picture scaling to map the user coordinates @code{box(min,max)} to a region of width @code{xsize} and height @code{ysize} (when these parameters are nonzero). Alternatively, calling the routine @cindex @code{fixedscaling} @verbatim transform fixedscaling(picture pic=currentpicture, pair min, pair max, pen p=nullpen, bool warn=false); @end verbatim will cause picture @code{pic} to use a fixed scaling to map user coordinates in @code{box(min,max)} to the (already specified) picture size, taking account of the width of pen @code{p}. A warning will be issued if the final picture exceeds the specified size. A picture @code{pic} can be fit to a frame and output to a file @code{prefix}.@code{format} using image format @code{format} by calling the @code{shipout} function: @anchor{shipout} @cindex @code{shipout} @cindex @code{outprefix} @verbatim 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) @end verbatim @noindent The default output format, @code{PostScript}, may be changed with the @code{-f} or @code{-tex} command-line options. The @code{options}, @code{script}, and @code{projection} parameters are only relevant for 3D pictures. If @code{defaultfilename} is an empty string, the prefix @code{outprefix()} will be used. A @code{shipout()} command is added implicitly at file exit if no previous @code{shipout} commands have been executed. @cindex @code{orientation} @cindex @code{Portrait} @cindex @code{Landscape} @cindex @code{UpsideDown} The default page orientation is @code{Portrait}; this may be modified by changing the variable @code{orientation}. To output in landscape mode, simply set the variable @code{orientation=Landscape} or issue the command @verbatim shipout(Landscape); @end verbatim @cindex @code{Seascape} To rotate the page by @math{-90} degrees, use the orientation @code{Seascape}. @cindex @code{UpsideDown} The orientation @code{UpsideDown} rotates the page by 180 degrees. @cindex subpictures @cindex @code{fit} A picture @code{pic} can be explicitly fit to a frame by calling @verbatim frame pic.fit(real xsize=pic.xsize, real ysize=pic.ysize, bool keepAspect=pic.keepAspect); @end verbatim The default size and aspect ratio settings are those given to the @code{size} command (which default to @code{0}, @code{0}, and @code{true}, respectively). @cindex @code{calculateTransform} The transformation that would currently be used to fit a picture @code{pic} to a frame is returned by the member function @code{pic.calculateTransform()}. In certain cases (e.g.@ 2D graphs) where only an approximate size estimate for @code{pic} is available, the picture fitting routine @verbatim frame pic.scale(real xsize=this.xsize, real ysize=this.ysize, bool keepAspect=this.keepAspect); @end verbatim (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. @cindex @code{box} To draw a bounding box with margins around a picture, fit the picture to a frame using the function @verbatim frame bbox(picture pic=currentpicture, real xmargin=0, real ymargin=xmargin, pen p=currentpen, filltype filltype=NoFill); @end verbatim @anchor{filltype} Here @code{filltype} specifies one of the following fill types: @table @code @cindex @code{FillDraw} @item FillDraw Fill the interior and draw the boundary. @item FillDraw(real xmargin=0, real ymargin=xmargin, pen fillpen=nullpen, @code{pen drawpen=nullpen)} @cindex @code{nullpen} If @code{fillpen} is @code{nullpen}, fill with the drawing pen; otherwise fill with pen @code{fillpen}. If @code{drawpen} is @code{nullpen}, draw the boundary with @code{fillpen}; otherwise with @code{drawpen}. An optional margin of @code{xmargin} and @code{ymargin} can be specified. @cindex @code{Fill} @item Fill Fill the interior. @cindex @code{nullpen} @item Fill(real xmargin=0, real ymargin=xmargin, pen p=nullpen) If @code{p} is @code{nullpen}, fill with the drawing pen; otherwise fill with pen @code{p}. An optional margin of @code{xmargin} and @code{ymargin} can be specified. @cindex @code{NoFill} @item NoFill Do not fill. @item Draw Draw only the boundary. @cindex @code{Draw} @item Draw(real xmargin=0, real ymargin=xmargin, pen p=nullpen) If @code{p} is @code{nullpen}, draw the boundary with the drawing pen; otherwise draw with pen @code{p}. An optional margin of @code{xmargin} and @code{ymargin} can be specified. @cindex @code{UnFill} @item UnFill Clip the region. @cindex @code{UnFill} @item UnFill(real xmargin=0, real ymargin=xmargin) Clip the region and surrounding margins @code{xmargin} and @code{ymargin}. @cindex @code{RadialShade} @item RadialShade(pen penc, pen penr) Fill varying radially from @code{penc} at the center of the bounding box to @code{penr} at the edge. @cindex @code{RadialShadeDraw} @item RadialShadeDraw(real xmargin=0, real ymargin=xmargin, pen penc, @code{pen penr, pen drawpen=nullpen)} Fill with RadialShade and draw the boundary. @end table @cindex bounding box @cindex background color For example, to draw a bounding box around a picture with a 0.25 cm margin and output the resulting frame, use the command: @verbatim shipout(bbox(0.25cm)); @end verbatim A @code{picture} may be fit to a frame with the background color pen @code{p}, using the function @code{bbox(p,Fill)}. @cindex @code{pad} To pad a picture to a precise size in both directions, fit the picture to a frame using the function @verbatim frame pad(picture pic=currentpicture, real xsize=pic.xsize, real ysize=pic.ysize, filltype filltype=NoFill); @end verbatim The functions @verbatim pair min(picture pic, user=false); pair max(picture pic, user=false); pair size(picture pic, user=false); @end verbatim calculate the bounds that picture @code{pic} would have if it were currently fit to a frame using its default size specification. If @code{user} is @code{false} the returned value is in @code{PostScript} coordinates, otherwise it is in user coordinates. The function @verbatim pair point(picture pic=currentpicture, pair dir, bool user=true); @end verbatim is a convenient way of determining the point on the bounding box of @code{pic} in the direction @code{dir} relative to its center, ignoring the contributions from fixed-size objects (such as labels and arrowheads). If @code{user} is @code{true} the returned value is in user coordinates, otherwise it is in @code{PostScript} coordinates. The function @verbatim pair truepoint(picture pic=currentpicture, pair dir, bool user=true); @end verbatim is identical to @code{point}, except that it also accounts for fixed-size objects, using the scaling transform that picture @code{pic} would have if currently fit to a frame using its default size specification. If @code{user} is @code{true} the returned value is in user coordinates, otherwise it is in @code{PostScript} coordinates. @anchor{add} Sometimes it is useful to draw objects on separate pictures and add one picture to another using the @code{add} function: @cindex @code{add} @verbatim 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); @end verbatim @noindent The first example adds @code{src} to @code{currentpicture}; the second one adds @code{src} to @code{dest}. The @code{group} option specifies whether or not the graphical user interface should treat all of the elements of @code{src} as a single entity (@pxref{GUI}), @code{filltype} requests optional background filling or clipping, and @code{above} specifies whether to add @code{src} above or below existing objects. There are also routines to add a picture or frame @code{src} specified in postscript coordinates to another picture @code{dest} (or @code{currentpicture}) about the user coordinate @code{position}: @anchor{add about} @cindex @code{add} @cindex picture alignment @verbatim 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); @end verbatim The optional @code{align} argument in the last form specifies a direction to use for aligning the frame, in a manner analogous to the @code{align} argument of @code{label} (@pxref{label}). However, one key difference is that when @code{align} is not specified, labels are centered, whereas frames and pictures are aligned so that their origin is at @code{position}. Illustrations of frame alignment can be found in the examples @ref{errorbars} and @ref{image}. If you want to align three or more subpictures, group them two at a time: @verbatiminclude subpictures.asy @sp 1 @center @image{./subpictures} Alternatively, one can use @code{attach} to automatically increase the size of picture @code{dest} to accommodate adding a frame @code{src} about the user coordinate @code{position}: @cindex @code{attach} @verbatim 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); @end verbatim @cindex @code{erase} To erase the contents of a picture (but not the size specification), use the function @verbatim void erase(picture pic=currentpicture); @end verbatim @cindex @code{save} To save a snapshot of @code{currentpicture}, @code{currentpen}, and @code{currentprojection}, use the function @code{save()}. @cindex @code{restore} To restore a snapshot of @code{currentpicture}, @code{currentpen}, and @code{currentprojection}, use the function @code{restore()}. Many further examples of picture and frame operations are provided in the base module @code{plain}. @cindex verbatim @cindex @code{postscript} It is possible to insert verbatim @code{PostScript} commands in a picture with one of the routines @verbatim void postscript(picture pic=currentpicture, string s); void postscript(picture pic=currentpicture, string s, pair min, pair max) @end verbatim Here @code{min} and @code{max} can be used to specify explicit bounds associated with the resulting @code{PostScript} code. @anchor{tex} @cindex @code{tex} Verbatim @TeX{} commands can be inserted in the intermediate @code{LaTeX} output file with one of the functions @verbatim void tex(picture pic=currentpicture, string s); void tex(picture pic=currentpicture, string s, pair min, pair max) @end verbatim Here @code{min} and @code{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: @cindex @code{texpreamble} @verbatim void texpreamble(string s); @end verbatim The @TeX{} environment can be reset to its initial state, clearing all macro definitions, with the function @cindex @code{texreset} @verbatim void texreset(); @end verbatim @cindex @code{usepackage} The routine @verbatim void usepackage(string s, string options=""); @end verbatim provides a convenient abbreviation for @verbatim texpreamble("\usepackage["+options+"]{"+s+"}"); @end verbatim @noindent that can be used for importing @code{LaTeX} packages. @end table @node Files, Variable initializers, Frames and pictures, Programming @section Files @cindex @code{file} @code{Asymptote} can read and write text files (including comma-separated value) files and portable @acronym{XDR} (External Data Representation) binary files. @cindex @code{input} An input file can be opened with @verbatim input(string name="", bool check=true, string comment="#", string mode=""); @end verbatim reading is then done by assignment: @cindex open @cindex @code{input} @cindex reading @verbatim file fin=input("test.txt"); real a=fin; @end verbatim @cindex comment character @cindex @code{error} If the optional boolean argument @code{check} is @code{false}, no check will be made that the file exists. If the file does not exist or is not readable, the function @code{bool error(file)} will return @code{true}. The first character of the string @code{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 @code{Asymptote} is compiled with support for @code{libcurl}, @code{name} can be a @acronym{URL}. @anchor{cd} @cindex @code{cd} @cindex @code{noglobalread} @cindex directory Unless the @code{-noglobalread} command-line option is specified, one can change the current working directory for read operations to the contents of the string @code{s} with the function @code{string cd(string s)}, which returns the new working directory. If @code{string s} is empty, the path is reset to the value it had at program startup. @cindex @code{getc} 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, @code{Asymptote} provides the function @code{string getc(file)} to read the next character (treating the comment character as an ordinary character) and return it as a string. @cindex @code{output} @cindex @code{update} @cindex append A file named @code{name} can be open for output with @verbatim file output(string name="", bool update=false, string comment="#", string mode=""); @end verbatim @noindent @cindex @code{noglobalread} @cindex @code{globalwrite} If @code{update=false}, any existing data in the file will be erased and only write operations can be used on the file. If @code{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 @code{-globalwrite} (or @code{-nosafe}) command-line option is specified. Reading from files in other directories is allowed unless the @code{-noglobalread} command-line option is specified. @cindex @code{mktemp} The function @code{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 @code{s}. @cindex @code{stdin} @cindex @code{stdout} There are two special files: @code{stdin}, which reads from the keyboard, and @code{stdout}, which writes to the terminal. The implicit initializer for files is @code{null}. Data of a built-in type @code{T} can be written to an output file by calling one of the functions @cindex @code{write} @verbatim 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); @end verbatim @cindex @code{none} @cindex @code{flush} @cindex @code{endl} @cindex @code{newl} @cindex @code{DOSendl} @cindex @code{DOSnewl} @cindex @code{tab} @cindex @code{comma} If @code{file} is not specified, @code{stdout} is used and terminated by default with a newline. If specified, the optional identifying string @code{s} is written before the data @code{x}. An arbitrary number of data values may be listed when writing scalars or one-dimensional arrays. The @code{suffix} may be one of the following: @code{none} (do nothing), @code{flush} (output buffered data), @code{endl} (terminate with a newline and flush), @code{newl} (terminate with a newline), @code{DOSendl} (terminate with a DOS newline and flush), @code{DOSnewl} (terminate with a DOS newline), @code{tab} (terminate with a tab), or @code{comma} (terminate with a comma). Here are some simple examples of data output: @verbatim file fout=output("test.txt"); write(fout,1); // Writes "1" write(fout); // Writes a new line write(fout,"List: ",1,2,3); // Writes "List: 1 2 3" @end verbatim @noindent @cindex binary format @cindex single precision @cindex double precision @cindex @code{singlereal} @cindex @code{singleint} @cindex @code{signedint} @cindex @code{mode} @cindex @code{binary} @cindex @code{xdr} A file may be opened with @code{mode="xdr"}, to read or write double precision (64-bit) reals and single precision (32-bit) integers in Sun Microsystem's @acronym{XDR} (External Data Representation) portable binary format (available on all @code{UNIX} platforms). Alternatively, a file may also be opened with @code{mode="binary"} to read or write double precision reals and single precision integers in the native (nonportable) machine binary format, or to read the entire file into a string. The virtual member functions @code{file singlereal(bool b=true)} and @code{file singleint(bool b=true)} be used to change the precision of real and integer I/O operations, respectively, for an @acronym{XDR} or binary file @code{f}. Similarly, the function @code{file signedint(bool b=true)} can be used to modify the signedness of integer reads and writes for an @acronym{XDR} or binary file @code{f}. @cindex @code{name} @cindex @code{mode} @cindex @code{singlereal} @cindex @code{singleint} @cindex @code{signedint} The virtual members @code{name}, @code{mode}, @code{singlereal}, @code{singleint}, and @code{signedint} may be used to query the respective parameters for a given file. @cindex @code{eof} @cindex @code{eol} @cindex @code{error} @cindex @code{flush} @cindex @code{clear} @cindex @code{precision} @cindex @code{seek} @cindex @code{tell} @cindex rewind @cindex @code{seekeof} One can test a file for end-of-file with the boolean function @code{eof(file)}, end-of-line with @code{eol(file)}, and for I/O errors with @code{error(file)}. One can flush the output buffers with @code{flush(file)}, clear a previous I/O error with @code{clear(file)}, and close the file with @code{close(file)}. The function @code{int precision(file file=stdout, int digits=0)} sets the number of digits of output precision for @code{file} to @code{digits}, provided @code{digits} is nonzero, and returns the previous precision setting. The function @code{int tell(file)} returns the current position in a file relative to the beginning. The routine @code{seek(file file, int pos)} can be used to change this position, where a negative value for the position @code{pos} is interpreted as relative to the end-of-file. For example, one can rewind a file @code{file} with the command @code{seek(file,0)} and position to the final character in the file with @code{seek(file,-1)}. The command @code{seekeof(file)} sets the position to the end of the file. @cindex @code{scroll} @anchor{scroll} Assigning @code{settings.scroll=n} for a positive integer @code{n} requests a pause after every @code{n} output lines to @code{stdout}. One may then press @code{Enter} to continue to the next @code{n} output lines, @code{s} followed by @code{Enter} to scroll without further interruption, or @code{q} followed by @code{Enter} to quit the current output operation. If @code{n} is negative, the output scrolls a page at a time (i.e. by one less than the current number of display lines). The default value, @code{settings.scroll=0}, specifies continuous scrolling. The routines @cindex @code{getstring} @cindex @code{getreal} @cindex @code{getpair} @cindex @code{gettriple} @verbatim 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); @end verbatim @noindent defined in the module @code{plain} may be used to prompt for a value from @code{stdin} using the @acronym{GNU} @code{readline} library. If @code{store=true}, the history of values for @code{name} is stored in the file @code{".asy_history_"+name} (@pxref{history}). The most recent value in the history will be used to provide a default value for subsequent runs. The default value (initially @code{default}) is displayed after @code{prompt}. These functions are based on the internal routines @cindex @code{readline} @cindex @code{saveline} @verbatim string readline(string prompt="", string name="", bool tabcompletion=false); void saveline(string name, string value, bool store=true); @end verbatim Here, @code{readline} prompts the user with the default value formatted according to @code{prompt}, while @code{saveline} is used to save the string @code{value} in a local history named @code{name}, optionally storing the local history in a file @code{".asy_history_"+name}. @cindex @code{history} The routine @code{history(string name, int n=1)} can be used to look up the @code{n} most recent values (or all values up to @code{historylines} if @code{n=0}) entered for string @code{name}. The routine @code{history(int n=0)} returns the interactive history. For example, @verbatim write(output("transcript.asy"),history()); @end verbatim @noindent outputs the interactive history to the file @code{transcript.asy}. @cindex @code{delete} @cindex @code{globalwrite} The function @code{int delete(string s)} deletes the file named by the string @code{s}. Unless the @code{-globalwrite} (or @code{-nosafe}) option is enabled, the file must reside in the current directory. @cindex @code{rename} The function @code{int rename(string from, string to)} may be used to rename file @code{from} to file @code{to}. Unless the @code{-globalwrite} (or @code{-nosafe}) option is enabled, this operation is restricted to the current directory. @cindex @code{convert} @cindex @code{animate} The functions @verbatim int convert(string args="", string file="", string format=""); int animate(string args="", string file="", string format=""); @end verbatim @noindent call the @code{ImageMagick} commands @code{convert} and @code{animate}, respectively, with the arguments @code{args} and the file name constructed from the strings @code{file} and @code{format}. @node Variable initializers, Structures, Files, Programming @section Variable initializers @cindex variable initializers @cindex @code{operator init} @cindex initializers A variable can be assigned a value when it is declared, as in @code{int x=3;} where the variable @code{x} is assigned the value @code{3}. As well as literal constants such as @code{3}, arbitary expressions can be used as initializers, as in @code{real x=2*sin(pi/2);}. A variable is not added to the namespace until after the initializer is evaluated, so for example, in @verbatim int x=2; int x=5*x; @end verbatim @noindent the @code{x} in the initializer on the second line refers to the variable @code{x} declared on the first line. The second line, then, declares a variable @code{x} shadowing the original @code{x} and initializes it to the value @code{10}. Variables of most types can be declared without an explicit initializer and they will be initialized by the default initializer of that type: @itemize @item Variables of the numeric types @code{int}, @code{real}, and @code{pair} are all initialized to zero; variables of type @code{triple} are initialized to @code{O=(0,0,0)}. @item @code{boolean} variables are initialized to @code{false}. @item @code{string} variables are initialized to the empty string. @item @code{transform} variables are initialized to the identity transformation. @item @code{path} and @code{guide} variables are initialized to @code{nullpath}. @item @code{pen} variables are initialized to the default pen. @item @code{frame} and @code{picture} variables are initialized to empty frames and pictures, respectively. @item @code{file} variables are initialized to @code{null}. @end itemize The default initializers for user-defined array, structure, and function types are explained in their respective sections. Some types, such as @code{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 @code{T} can be redeclared by defining the function @code{T operator init()}. For instance, @code{int} variables are usually initialized to zero, but in @verbatim int operator init() { return 3; } int y; @end verbatim @noindent the variable @code{y} is initialized to @code{3}. This example was given for illustrative purposes; redeclaring the initializers of built-in types is not recommended. Typically, @code{operator init} is used to define sensible defaults for user-defined types. @cindex @code{var} The special type @code{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, @verbatim var x=5; var y=4.3; var reddash=red+dashed; @end verbatim @noindent is equivalent to @verbatim int x=5; real y=4.3; pen reddash=red+dashed; @end verbatim @code{var} may also be used with the extended @code{for} loop syntax. @verbatim int[] a = {1,2,3}; for (var x : a) write(x); @end verbatim @node Structures, Operators, Variable initializers, Programming @section Structures @cindex @code{struct} @cindex structures @cindex @code{public} @cindex @code{restricted} @cindex @code{private} @cindex @code{this} @cindex @code{new} @cindex @code{null} Users may also define their own data types as structures, along with user-defined operators, much as in C++. By default, structure members are @code{public} (may be read and modified anywhere in the code), but may be optionally declared @code{restricted} (readable anywhere but writeable only inside the structure where they are defined) or @code{private} (readable and writable only inside the structure). In a structure definition, the keyword @code{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: @verbatim struct T { int x; } T foo; T bar=foo; bar.x=5; @end verbatim The variable @code{foo} holds a reference to an instance of the structure @code{T}. When @code{bar} is assigned the value of @code{foo}, it too now holds a reference to the same instance as @code{foo} does. The assignment @code{bar.x=5} changes the value of the field @code{x} in that instance, so that @code{foo.x} will also be equal to @code{5}. The expression @code{new T} creates a new instance of the structure @code{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: @verbatim int Tcount=0; struct T { int x; ++Tcount; } T foo=new T; T foo; @end verbatim @noindent Here, @code{new T} produces a new instance of the class, which causes @code{Tcount} to be incremented, tracking the number of instances produced. The declarations @code{T foo=new T} and @code{T foo} are equivalent: the second form implicitly creates a new instance of @code{T}. That is, after the definition of a structure @code{T}, a variable of type @code{T} is initialized to a new instance (@code{new T}) by default. During the definition of the structure, however, variables of type @code{T} are initialized to @code{null} by default. This special behaviour is to avoid infinite recursion of creating new instances in code such as @verbatim struct tree { int value; tree left; tree right; } @end verbatim The expression @code{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. @cindex alias @cindex @code{==} @cindex @code{!=} The function @code{bool alias(T,T)} checks to see if two structure references refer to the same instance of the structure (or both to @code{null}). In example at the beginning of this section, @code{alias(foo,bar)} would return true, but @code{alias(foo,new T)} would return false, as @code{new T} creates a new instance of the structure @code{T}. The boolean operators @code{==} and @code{!=} are by default equivalent to @code{alias} and @code{!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: @verbatim 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 @end verbatim @cindex constructors It is often convenient to have functions that construct new instances of a structure. Say we have a @code{Person} structure: @verbatim struct Person { string firstname; string lastname; } Person joe; joe.firstname="Joe"; joe.lastname="Jones"; @end verbatim @noindent 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 a constructor function @code{Person(string,string)}: @verbatim struct Person { string firstname; string lastname; static Person Person(string firstname, string lastname) { Person p=new Person; p.firstname=firstname; p.lastname=lastname; return p; } } Person joe=Person.Person("Joe", "Jones"); @end verbatim While it is now easier than before to create a new instance, we still have to refer to the constructor by the qualified name @code{Person.Person}. If we add the line @verbatim from Person unravel Person; @end verbatim @noindent immediately after the structure definition, then the constructor can be used without qualification: @code{Person joe=Person("Joe", "Jones");}. The constructor is now easy to use, but it is quite a hassle to define. If you write a lot of constructors, you will find that you are repeating a lot of code in each of them. Fortunately, your friendly neighbourhood Asymptote developers have devised a way to automate much of the process. @cindex @code{operator init} If, in the body of a structure, Asymptote encounters the definition of a function of the form @code{void operator init(@var{args})}, it implicitly defines a constructor function of the arguments @code{@var{args}} that uses the @code{void operator init} function to initialize a new instance of the structure. That is, it essentially defines the following constructor (assuming the structure is called @code{Foo}): @example static Foo Foo(@var{args}) @{ Foo instance=new Foo; instance.operator init(@var{args}); return instance; @} @end example This constructor is also implicitly copied to the enclosing scope after the end of the structure definition, so that it can used subsequently without qualifying it by the structure name. Our @code{Person} example can thus be implemented as: @verbatim struct Person { string firstname; string lastname; void operator init(string firstname, string lastname) { this.firstname=firstname; this.lastname=lastname; } } Person joe=Person("Joe", "Jones"); @end verbatim The use of @code{operator init} to implicitly define constructors should not be confused with its use to define default values for variables (@pxref{Variable initializers}). Indeed, in the first case, the return type of the @code{operator init} must be @code{void} while in the second, it must be the (non-@code{void}) type of the variable. @cindex @code{cputime} The function @code{cputime()} returns a structure @code{cputime} with cumulative @acronym{CPU} times broken down into the fields @code{parent.user}, @code{parent.system}, @code{child.user}, and @code{child.system}, along with the cumulative wall clock time in @code{parent.clock}, all measured in seconds. For convenience, the incremental fields @code{change.user}, @code{change.system}, and @code{change.clock} indicate the change in the corresponding fields since the last call to @code{cputime()}. The function @verbatim void write(file file=stdout, string s="", cputime c, string format=cputimeformat, suffix suffix=none); @end verbatim @noindent 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''. @cindex inheritance @cindex virtual functions Much like in C++, casting (@pxref{Casts}) provides for an elegant implementation of structure inheritance, including virtual functions: @verbatim struct parent { real x; void operator init(int x) {this.x=x;} void virtual(int) {write(0);} void f() {virtual(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 virtual(int x) {write(x);} parent.virtual=virtual; 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; @end verbatim For further examples of structures, see @code{Legend} and @code{picture} in the @code{Asymptote} base module @code{plain}. @node Operators, Implicit scaling, Structures, Programming @section Operators @cindex operators @menu * Arithmetic & logical:: Basic mathematical operators * Self & prefix operators:: Increment and decrement * User-defined operators:: Overloading operators @end menu @node Arithmetic & logical, Self & prefix operators, Operators, Operators @subsection Arithmetic & logical operators @cindex arithmetic operators @cindex binary operators @cindex boolean operators @cindex logical operators @cindex @code{quotient} @code{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 @code{int quotient(int x, int y)}, which returns greatest integer less than or equal to @code{x/y}). In all other cases both operands are promoted to the same type, which will also be the type of the result: @table @code @cindex @code{+} @item + addition @cindex @code{-} @item - subtraction @cindex @code{*} @item * multiplication @cindex @code{/} @item / division @cindex integer division @cindex @code{#} @item # integer division; equivalent to @code{quotient(x,y)}. Noting that the @code{Python3} community adopted our comment symbol (@code{//}) for integer division, we decided to reciprocate and use their comment symbol for integer division in @code{Asymptote}! @cindex @code{%} @item % modulo; the result always has the same sign as the divisor. In particular, this makes @code{q*(p # q)+p % q == p} for all integers @code{p} and nonzero integers @code{q}. @cindex @code{^} @item ^ @cindex @code{**} power; if the exponent (second argument) is an int, recursive multiplication is used; otherwise, logarithms and exponentials are used (@code{**} is a synonym for @code{^}). @end table The usual boolean operators are also defined: @table @code @cindex @code{==} @item == equals @cindex @code{!=} @item != not equals @cindex @code{<} @item < less than @cindex @code{<=} @item <= less than or equals @cindex @code{>=} @item >= greater than or equals @cindex @code{>} @item > greater than @cindex @code{&&} @item && and (with conditional evaluation of right-hand argument) @cindex @code{&} @item & and @cindex @code{||} @item || or (with conditional evaluation of right-hand argument) @cindex @code{|} @item | or @cindex @code{^} @item ^ xor @cindex @code{!} @item ! not @end table @code{Asymptote} also supports the C-like conditional syntax: @cindex @code{:} @cindex @code{?} @cindex conditional @verbatim bool positive=(pi > 0) ? true : false; @end verbatim @cindex @code{interp} The function @code{T interp(T a, T b, real t)} returns @code{(1-t)*a+t*b} for nonintegral built-in arithmetic types @code{T}. If @code{a} and @code{b} are pens, they are first promoted to the same color space. @cindex @code{AND} @cindex @code{OR} @cindex @code{XOR} @cindex @code{NOT} @cindex @code{CLZ} @cindex @code{CTZ} @code{Asymptote} also defines bitwise functions @code{int AND(int,int)}, @code{int OR(int,int)}, @code{int XOR(int,int)}, @code{int NOT(int)}, @code{int CLZ(int)} (count leading zeros), @code{int CTZ(int)} (count trailing zeros), @code{int popcount(int)} (count bits populated by ones), and @code{int bitreverse(int a, int bits)} (reverse bits within a word of length bits). @node Self & prefix operators, User-defined operators, Arithmetic & logical, Operators @subsection Self & prefix operators @cindex self operators @cindex prefix operators @cindex @code{+=} @cindex @code{-=} @cindex @code{*=} @cindex @code{/=} @cindex @code{%=} @cindex @code{^=} @cindex @code{++} @cindex @code{--} As in C, each of the arithmetic operators @code{+}, @code{-}, @code{*}, @code{/}, @code{#}, @code{%}, and @code{^} can be used as a self operator. The prefix operators @code{++} (increment by one) and @code{--} (decrement by one) are also defined. For example, @verbatim int i=1; i += 2; int j=++i; @end verbatim @noindent is equivalent to the code @verbatim int i=1; i=i+2; int j=i=i+1; @end verbatim @cindex postfix operators However, postfix operators like @code{i++} and @code{i--} are not defined (because of the inherent ambiguities that would arise with the @code{--} path-joining operator). In the rare instances where @code{i++} and @code{i--} are really needed, one can substitute the expressions @code{(++i-1)} and @code{(--i+1)}, respectively. @node User-defined operators, , Self & prefix operators, Operators @subsection User-defined operators @cindex user-defined operators @cindex @code{operator} The following symbols may be used with @code{operator} to define or redefine operators on structures and built-in types: @verbatim - + * / % ^ ! < > == != <= >= & | ^^ .. :: -- --- ++ << >> $ $$ @ @@ <> @end verbatim @noindent The operators on the second line have precedence one higher than the boolean operators @code{<}, @code{>}, @code{<=}, and @code{>=}. Guide operators like @code{..} may be overloaded, say, to write a user function that produces a new guide from a given guide: @verbatim 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); @end verbatim @node Implicit scaling, Functions, Operators, Programming @section Implicit scaling @cindex implicit scaling If a numeric literal is in front of certain types of expressions, then the two are multiplied: @verbatim 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); @end verbatim This produces the output @verbatim 6 5 6 -3.204e-19 (1,1) 8 10 18 2.72789228047704 2.48046543129542 283.464008929116 @end verbatim @node Functions, Arrays, Implicit scaling, Programming @section Functions @cindex 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 @end menu @code{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 (@pxref{Structures}). Here are some significant features of @code{Asymptote} functions: @enumerate @item Variables with signatures (functions) and without signatures (nonfunction variables) are distinct: @verbatim int x, x(); x=5; x=new int() {return 17;}; x=x(); // calls x() and puts the result, 17, in the scalar x @end verbatim @item Traditional function definitions are allowed: @verbatim int sqr(int x) { return x*x; } sqr=null; // but the function is still just a variable. @end verbatim @item Casting can be used to resolve ambiguities: @verbatim 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. @end verbatim @item Anonymous (so-called "high-order") functions are also allowed: @cindex @code{typedef} @verbatim typedef int intop(int); intop adder(int m) { return new int(int n) {return m+n;}; } intop addby7=adder(7); write(addby7(1)); // Writes 8. @end verbatim @item @cindex overloading functions One may redefine a function @code{f}, even for calls to @code{f} in previously declared functions, by assigning another (anonymous or named) function to it. However, if @code{f} is overloaded by a new function definition, previous calls will still access the original version of @code{f}, as illustrated in this example: @verbatim 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" @end verbatim @cindex function declarations @item 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: @verbatim 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. @end verbatim @end enumerate @code{Asymptote} is the only language we know of that treats functions as variables, but allows overloading by distinguishing variables based on their signatures. @cindex @code{libsigsegv} @cindex stack overflow @anchor{stack overflow} @cindex recursion @cindex stack overflow Functions are allowed to call themselves recursively. As in C++, infinite nested recursion will generate a stack overflow (reported as a segmentation fault, unless a fully working version of the @acronym{GNU} library @code{libsigsegv} (e.g.@ 2.4 or later) is installed at configuration time). @node Default arguments, Named arguments, Functions, Functions @subsection Default arguments @cindex default arguments @cindex arguments @code{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 (@pxref{Casts}) one can often avoid ambiguities by ordering function arguments from the simplest to the most complicated. For example, given @verbatim real f(int a=1, real b=0) {return a+b;} @end verbatim @noindent then @code{f(1)} returns 1.0, but @code{f(1.0)} returns 2.0. The value of a default argument is determined by evaluating the given @code{Asymptote} expression in the scope where the called function is defined. @node Named arguments, Rest arguments, Default arguments, Functions @subsection Named arguments @cindex keywords @cindex 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, @emph{not within the local scope}. The command-line option @code{-d} may be used to check @code{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, @verbatim int f(int x, int y) { return 10x+y; } write(f(4,x=3)); @end verbatim @noindent outputs 34, as @code{x} is already matched when we try to match the unnamed argument @code{4}, so it gets matched to the next item, @code{y}. For the rare occasions where it is desirable to assign a value to local variable within a function argument (generally @emph{not} a good programming practice), simply enclose the assignment in parentheses. For example, given the definition of @code{f} in the previous example, @verbatim int x; write(f(4,(x=3))); @end verbatim @noindent is equivalent to the statements @verbatim int x; x=3; write(f(4,3)); @end verbatim @noindent and outputs 43. @cindex @code{keyword} @cindex keyword-only Parameters can be specified as ``keyword-only'' by putting @code{keyword} immediately before the parameter name, as in @code{int f(int keyword x)} or @code{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, @code{f(x=42)} is legal, but @code{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 @verbatim int f(int x, int x()) { return x+x(); } int seven() {return 7;} @end verbatim @noindent is legal in @code{Asymptote}, with @code{f(2,seven)} returning 9. A named argument matches the first unmatched formal of the same name, so @code{f(x=2,x=seven)} is an equivalent call, but @code{f(x=seven,2)} is not, as the first argument is matched to the first formal, and @code{int ()} cannot be implicitly cast to @code{int}. Default arguments do not affect which formal a named argument is matched to, so if @code{f} were defined as @verbatim int f(int x=3, int x()) { return x+x(); } @end verbatim @noindent then @code{f(x=seven)} would be illegal, even though @code{f(seven)} obviously would be allowed. @node Rest arguments, Mathematical functions, Named arguments, Functions @subsection Rest arguments @cindex rest arguments Rest arguments allow one to write functions that take a variable number of arguments: @verbatim // 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 @end verbatim @cindex packing Putting an argument into a rest array is called @emph{packing}. One can give an explicit list of arguments for the rest argument, so @code{subtract} could alternatively be implemented as @verbatim int subtract(int start ... int[] subs) { return start - sum(... subs); } @end verbatim One can even combine normal arguments with rest arguments: @verbatim sum(1,2,3 ... new int[] {4,5,6}); // returns 21 @end verbatim @noindent @cindex unpacking This builds a new six-element array that is passed to @code{sum} as @code{nums}. The opposite operation, @emph{unpacking}, is not allowed: @verbatim subtract(... new int[] {10, 1, 2}); @end verbatim @noindent is illegal, as the start formal is not matched. If no arguments are packed, then a zero-length array (as opposed to @code{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 @code{keyword} is to avoid @code{pnorm(1.0,2.0,0.3)} matching @code{1.0} to @code{p}. @verbatim real pnorm(real keyword p=2.0 ... real[] v) { return sum(v^p)^(1/p); } @end verbatim The overloading resolution in @code{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. @verbatim 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 @end verbatim @node Mathematical functions, , Rest arguments, Functions @subsection Mathematical functions @cindex mathematical functions @cindex functions @cindex @code{libm} routines @cindex @code{sin} @cindex @code{cos} @cindex @code{tan} @cindex @code{asin} @cindex @code{acos} @cindex @code{atan} @cindex @code{exp} @cindex @code{log} @cindex @code{pow10} @cindex @code{log10} @cindex @code{sinh} @cindex @code{cosh} @cindex @code{tanh} @cindex @code{asinh} @cindex @code{acosh} @cindex @code{atanh} @cindex @code{sqrt} @cindex @code{cbrt} @cindex @code{fabs} @cindex @code{expm1} @cindex @code{log1p} @cindex @code{identity} @cindex @code{J} @cindex @code{Y} @cindex @code{gamma} @cindex @code{erf} @cindex @code{erfc} @cindex @code{atan2} @cindex @code{hypot} @cindex @code{fmod} @cindex @code{remainder} @code{Asymptote} has built-in versions of the standard @code{libm} mathematical real(real) functions @code{sin}, @code{cos}, @code{tan}, @code{asin}, @code{acos}, @code{atan}, @code{exp}, @code{log}, @code{pow10}, @code{log10}, @code{sinh}, @code{cosh}, @code{tanh}, @code{asinh}, @code{acosh}, @code{atanh}, @code{sqrt}, @code{cbrt}, @code{fabs}, @code{expm1}, @code{log1p}, as well as the identity function @code{identity}. @code{Asymptote} also defines the order @code{n} Bessel functions of the first kind @code{Jn(int n, real)} and second kind @code{Yn(int n, real)}, as well as the gamma function @code{gamma}, the error function @code{erf}, and the complementary error function @code{erfc}. The standard real(real, real) functions @code{atan2}, @code{hypot}, @code{fmod}, @code{remainder} are also included. @cindex @code{degrees} @cindex @code{radians} @cindex @code{Degrees} The functions @code{degrees(real radians)} and @code{radians(real degrees)} can be used to convert between radians and degrees. The function @code{Degrees(real radians)} returns the angle in degrees in the interval [0,360). @cindex @code{Sin} @cindex @code{Cos} @cindex @code{Tan} @cindex @code{aSin} @cindex @code{aCos} @cindex @code{aTan} For convenience, @code{Asymptote} defines variants @code{Sin}, @code{Cos}, @code{Tan}, @code{aSin}, @code{aCos}, and @code{aTan} of the standard trigonometric functions that use degrees rather than radians. We also define complex versions of the @code{sqrt}, @code{sin}, @code{cos}, @code{exp}, @code{log}, and @code{gamma} functions. @cindex @code{floor} @cindex @code{ceil} @cindex @code{round} @cindex @code{sgn} The functions @code{floor}, @code{ceil}, and @code{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 @code{Floor}, @code{Ceil}, and @code{Round} are respectively similar, except that if the result cannot be converted to a valid int, they return @code{intMax} for positive arguments and @code{intMin} for negative arguments, rather than generating an integer overflow. We also define a function @code{sgn}, which returns the sign of its real argument as an integer (-1, 0, or 1). @cindex @code{abs} There is an @code{abs(int)} function, as well as an @code{abs(real)} function (equivalent to @code{fabs(real)}), an @code{abs(pair)} function (equivalent to @code{length(pair)}). @cindex @code{srand} @cindex @code{rand} @cindex @code{randMax} @cindex @code{unitrand} @cindex @code{Gaussrand} @cindex @code{histogram} @cindex @code{factorial} @cindex @code{choose} Random numbers can be seeded with @code{srand(int)} and generated with the @code{int rand()} function, which returns a random integer between 0 and the integer @code{randMax}. The @code{unitrand()} function returns a random number uniformly distributed in the interval [0,1]. A Gaussian random number generator @code{Gaussrand} and a collection of statistics routines, including @code{histogram}, are provided in the module @code{stats}. The functions @code{factorial(int n)}, which returns @math{n!}, and @code{choose(int n, int k)}, which returns @math{n!/(k!(n-k)!)}, are also defined. @cindex @acronym{GNU} Scientific Library @cindex @code{gsl} @cindex Airy @cindex Bessel @cindex Legendre @cindex elliptic functions @cindex exponential integral @cindex trigonometric integrals @cindex Riemann zeta function @cindex @code{Ai} @cindex @code{Bi} @cindex @code{Ai_deriv} @cindex @code{Bi_deriv} @cindex @code{zero_Ai} @cindex @code{zero_Bi} @cindex @code{zero_Ai_deriv} @cindex @code{zero_Bi_deriv} @cindex @code{J} @cindex @code{Y} @cindex @code{I} @cindex @code{K} @cindex @code{i_scaled} @cindex @code{k_scaled} @cindex @code{zero_J} @cindex @code{F} @cindex @code{E} @cindex @code{P} @cindex @code{sncndn} @cindex @code{Ei} @cindex @code{Si} @cindex @code{Ci} @cindex @code{Pl} @cindex @code{zeta} When configured with the @acronym{GNU} Scientific Library (GSL), available from @url{https://www.gnu.org/software/gsl/}, @code{Asymptote} contains an internal module @code{gsl} that defines the airy functions @code{Ai(real)}, @code{Bi(real)}, @code{Ai_deriv(real)}, @code{Bi_deriv(real)}, @code{zero_Ai(int)}, @code{zero_Bi(int)}, @code{zero_Ai_deriv(int)}, @code{zero_Bi_deriv(int)}, the Bessel functions @code{I(int, real)}, @code{K(int, real)}, @code{j(int, real)}, @code{y(int, real)}, @code{i_scaled(int, real)}, @code{k_scaled(int, real)}, @code{J(real, real)}, @code{Y(real, real)}, @code{I(real, real)}, @code{K(real, real)}, @code{zero_J(real, int)}, the elliptic functions @code{F(real, real)}, @code{E(real, real)}, and @code{P(real, real)}, the Jacobi elliptic functions @code{real[] sncndn(real,real)}, the exponential/trigonometric integrals @code{Ei}, @code{Si}, and @code{Ci}, the Legendre polynomials @code{Pl(int, real)}, and the Riemann zeta function @code{zeta(real)}. For example, to compute the sine integral @code{Si} of 1.0: @verbatim import gsl; write(Si(1.0)); @end verbatim @code{Asymptote} also provides a few general purpose numerical routines: @table @code @cindex @code{newton} @item @code{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 @code{f}, given its derivative @code{fprime} and an initial guess @code{x}. Diagnostics for each iteration are printed if @code{verbose=true}. If the iteration fails after the maximum allowed number of loops (@code{iterations}), @code{realMax} is returned. @cindex @code{newton} @item @code{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 @code{f} within an interval [@code{x1},@code{x2}] (on which the endpoint values of @code{f} have opposite signs), given its derivative @code{fprime}. Diagnostics for each iteration are printed if @code{verbose=true}. If the iteration fails after the maximum allowed number of loops (@code{iterations}), @code{realMax} is returned. @cindex integral @cindex integrate @cindex @code{simpson} @item @code{real simpson(real f(real), real a, real b, real acc=realEpsilon, real dxmax=b-a)} returns the integral of @code{f} from @code{a} to @code{b} using adaptive Simpson integration. @end table @node Arrays, Casts, Functions, Programming @section Arrays @cindex arrays @menu * Slices:: Python-style array slices @end menu Appending @code{[]} to a built-in or user-defined type yields an array. The array element @code{i} of an array @code{A} can be accessed as @code{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 @code{A} with an integer array @code{B}: the array @code{A[B]} is formed by indexing array @code{A} with successive elements of array @code{B}. A convenient Java-style shorthand exists for iterating over all elements of an array; see @ref{array iteration}. The declaration @verbatim real[] A; @end verbatim @noindent initializes @code{A} to be an empty (zero-length) array. Empty arrays should be distinguished from null arrays. If we say @verbatim real[] A=null; @end verbatim @noindent then @code{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: @verbatim real[] A={0,1,2}; @end verbatim Array assignment in @code{Asymptote} does a shallow copy: only the pointer is copied (if one copy if modified, the other will be too). The @code{copy} function listed below provides a deep copy of an array. @cindex @code{length} @cindex @code{cyclic} @cindex @code{keys} @cindex @code{push} @cindex @code{append} @cindex @code{pop} @cindex @code{insert} @cindex @code{delete} @cindex @code{initialized} Every array @code{A} of type @code{T[]} has the virtual members @itemize @item @code{int length}, @item @code{int cyclic}, @item @code{int[] keys}, @item @code{T push(T x)}, @item @code{void append(T[] a)}, @item @code{T pop()}, @item @code{void insert(int i ... T[] x)}, @item @code{void delete(int i, int j=i)}, @item @code{void delete()}, and @item @code{bool initialized(int n)}. @end itemize The member @code{A.length} evaluates to the length of the array. Setting @code{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 @code{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 @code{n} with all entries initialized, @code{A.keys} evaluates to @code{@{0,1,...,n-1@}}. A new keys array is produced each time @code{A.keys} is evaluated. The functions @code{A.push} and @code{A.append} append their arguments onto the end of the array, while @code{A.insert(int i ... T[] x)} inserts @code{x} into the array at index @code{i}. For convenience @code{A.push} returns the pushed item. The function @code{A.pop()} pops and returns the last element, while @code{A.delete(int i, int j=i)} deletes elements with indices in the range [@code{i},@code{j}], shifting the position of all higher-indexed elements down. If no arguments are given, @code{A.delete()} provides a convenient way of deleting all elements of @code{A}. The routine @code{A.initialized(int n)} can be used to examine whether the element at index @code{n} is initialized. Like all @code{Asymptote} functions, @code{push}, @code{append}, @code{pop}, @code{insert}, @code{delete}, and @code{initialized} can be "pulled off" of the array and used on their own. For example, @verbatim 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}. @end verbatim The @code{[]} suffix can also appear after the variable name; this is sometimes convenient for declaring a list of variables and arrays of the same type: @verbatim real a,A[]; @end verbatim @noindent This declares @code{a} to be @code{real} and implicitly declares @code{A} to be of type @code{real[]}. In the following list of built-in array functions, @code{T} represents a generic type. Note that the internal functions @code{alias}, @code{array}, @code{copy}, @code{concat}, @code{sequence}, @code{map}, and @code{transpose}, which depend on type @code{T[]}, are defined only after the first declaration of a variable of type @code{T[]}. @table @code @cindex @code{new} @item new T[] returns a new empty array of type @code{T[]}; @cindex @code{new} @item new T[] @{list@} returns a new array of type @code{T[]} initialized with @code{list} (a comma delimited list of elements); @item new T[n] returns a new array of @code{n} elements of type @code{T[]}. These @code{n} array elements are not initialized unless they are arrays themselves (in which case they are each initialized to empty arrays); @cindex @code{array} @item T[] array(int n, T value, int depth=intMax) returns an array consisting of @code{n} copies of @code{value}. If @code{value} is itself an array, a deep copy of @code{value} is made for each entry. If @code{depth} is specified, this deep copying only recurses to the specified number of levels; @cindex @code{sequence} @item int[] sequence(int n) if @code{n >= 1} returns the array @code{@{0,1,...,n-1@}} (otherwise returns a null array); @item int[] sequence(int n, int m) if @code{m >= n} returns an array @code{@{n,n+1,...,m@}} (otherwise returns a null array); @item int[] sequence(int n, int m, int skip) if @code{m >= n} returns an array @code{@{n,n+1,...,m@}} skipping by @code{skip} (otherwise returns a null array); @item T[] sequence(T f(int), int n) if @code{n >= 1} returns the sequence @code{@{f_i :i=0,1,...n-1@}} given a function @code{T f(int)} and integer @code{int n} (otherwise returns a null array); @cindex @code{map} @item T[] map(T f(T), T[] a) returns the array obtained by applying the function @code{f} to each element of the array @code{a}. This is equivalent to @code{sequence(new T(int i) @{return f(a[i]);@},a.length)}; @cindex @code{map} @item T2[] map(T2 f(T1), T1[] a) constructed by calling @code{mapArray("T1","T2")}, returns the array obtained by applying the function @code{f} to each element of the array @code{a}; @cindex @code{reverse} @item int[] reverse(int n) if @code{n >= 1} returns the array @code{@{n-1,n-2,...,0@}} (otherwise returns a null array); @cindex @code{complement} @item int[] complement(int[] a, int n) returns the complement of the integer array @code{a} in @code{@{0,1,2,...,n-1@}}, so that @code{b[complement(a,b.length)]} yields the complement of @code{b[a]}; @cindex @code{uniform} @item real[] uniform(real a, real b, int n) if @code{n >= 1} returns a uniform partition of @code{[a,b]} into @code{n} subintervals (otherwise returns a null array); @cindex @code{find} @item int find(bool[] a, int n=1) returns the index of the @code{n}th @code{true} value in the boolean array @code{a} or -1 if not found. If @code{n} is negative, search backwards from the end of the array for the @code{-n}th value; @cindex @code{findall} @item int[] findall(bool[] a) returns the indices of all @code{true} values in the boolean array @code{a}; @cindex @code{search} @item int search(T[] a, T key) For built-in ordered types @code{T}, searches a sorted array @code{a} of @code{n} elements for k, returning the index @code{i} if @code{a[i] <= key < a[i+1]}, @code{-1} if @code{key} is less than all elements of @code{a}, or @code{n-1} if @code{key} is greater than or equal to the last element of @code{a}; @cindex @code{search} @item int search(T[] a, T key, bool less(T i, T j)) searches an array @code{a} sorted in ascending order such that element @code{i} precedes element @code{j} if @code{less(i,j)} is true; @cindex @code{copy} @item T[] copy(T[] a) returns a deep copy of the array @code{a}; @cindex @code{concat} @item T[] concat(... T[][] a) returns a new array formed by concatenating the given one-dimensional arrays given as arguments; @cindex @code{alias} @item bool alias(T[] a, T[] b) returns @code{true} if the arrays @code{a} and @code{b} are identical; @cindex @code{sort} @item T[] sort(T[] a) For built-in ordered types @code{T}, returns a copy of @code{a} sorted in ascending order; @cindex @code{sort} @anchor{sort} @item T[][] sort(T[][] a) For built-in ordered types @code{T}, returns a copy of @code{a} with the rows sorted by the first column, breaking ties with successively higher columns. For example: @verbatim string[][] a={{"bob","9"},{"alice","5"},{"pete","7"}, {"alice","4"}}; // Row sort (by column 0, using column 1 to break ties): write(sort(a)); @end verbatim produces @verbatim alice 4 alice 5 bob 9 pete 7 @end verbatim @cindex @code{sort} @item T[] sort(T[] a, bool less(T i, T j), bool stable=true) returns a copy of @code{a} sorted in ascending order such that element @code{i} precedes element @code{j} if @code{less(i,j)} is true, subject to (if @code{stable} is @code{true}) the stability constraint that the original order of elements @code{i} and @code{j} is preserved if @code{less(i,j)} and @code{less(j,i)} are both @code{false}; @cindex @code{transpose} @item T[][] transpose(T[][] a) returns the transpose of @code{a}; @cindex @code{transpose} @item T[][][] transpose(T[][][] a, int[] perm) returns the 3D transpose of @code{a} obtained by applying the permutation @code{perm} of @code{new int[]@{0,1,2@}} to the indices of each entry; @cindex @code{sum} @item T sum(T[] a) for arithmetic types @code{T}, returns the sum of @code{a}. In the case where @code{T} is @code{bool}, the number of true elements in @code{a} is returned; @cindex @code{min} @item T min(T[] a) @item T min(T[][] a) @item T min(T[][][] a) for built-in ordered types @code{T}, returns the minimum element of @code{a}; @cindex @code{max} @item T max(T[] a) @item T max(T[][] a) @item T max(T[][][] a) for built-in ordered types @code{T}, returns the maximum element of @code{a}; @cindex @code{min} @item T[] min(T[] a, T[] b) for built-in ordered types @code{T}, and arrays @code{a} and @code{b} of the same length, returns an array composed of the minimum of the corresponding elements of @code{a} and @code{b}; @cindex @code{max} @item T[] max(T[] a, T[] b) for built-in ordered types @code{T}, and arrays @code{a} and @code{b} of the same length, returns an array composed of the maximum of the corresponding elements of @code{a} and @code{b}; @cindex @code{pairs} @item pair[] pairs(real[] x, real[] y); for arrays @code{x} and @code{y} of the same length, returns the pair array @code{sequence(new pair(int i) @{return (x[i],y[i]);@},x.length)}; @cindex @code{fft} @item pair[] fft(pair[] a, int sign=1) returns the unnormalized Fast Fourier Transform of @code{a} (if the optional @code{FFTW} package is installed), using the given @code{sign}. Here is a simple example: @verbatim 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); @end verbatim @cindex @code{fft} @item pair[][] fft(pair[][] a, int sign=1) returns the unnormalized two-dimensional Fourier transform of @code{a} using the given @code{sign}; @cindex @code{fft} @item pair[][][] fft(pair[][][] a, int sign=1) returns the unnormalized three-dimensional Fourier transform of @code{a} using the given @code{sign}; @cindex @code{dot} @item real dot(real[] a, real[] b) returns the dot product of the vectors @code{a} and @code{b}; @cindex @code{dot} @item pair dot(pair[] a, pair[] b) returns the complex dot product @code{sum(a*conj(b))} of the vectors @code{a} and @code{b}; @anchor{tridiagonal} @cindex @code{tridiagonal} @item real[] tridiagonal(real[] a, real[] b, real[] c, real[] f); Solve the periodic tridiagonal problem @math{L@code{x}=@code{f}} and return the solution @code{x}, where @code{f} is an @math{n} vector and @math{L} is the @math{n \times n} matrix @verbatim [ 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] ] @end verbatim For Dirichlet boundary conditions (denoted here by @code{u[-1]} and @code{u[n]}), replace @code{f[0]} by @code{f[0]-a[0]u[-1]} and @code{f[n-1]-c[n-1]u[n]}; then set @code{a[0]=c[n-1]=0}; @cindex @code{solve} @item real[] solve(real[][] a, real[] b, bool warn=true) Solve the linear equation @math{@code{a}x=@code{b}} by LU decomposition and return the solution @math{x}, where @code{a} is an @math{n \times n} matrix and @code{b} is an array of length @math{n}. For example: @verbatim 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); @end verbatim If @code{a} is a singular matrix and @code{warn} is @code{false}, return an empty array. If the matrix @code{a} is tridiagonal, the routine @code{tridiagonal} provides a more efficient algorithm (@pxref{tridiagonal}); @anchor{solve} @cindex @code{solve} @item real[][] solve(real[][] a, real[][] b, bool warn=true) Solve the linear equation @math{@code{a}x=@code{b}} and return the solution @math{x}, where @code{a} is an @math{n \times n} matrix and @code{b} is an @math{n \times m} matrix. If @code{a} is a singular matrix and @code{warn} is @code{false}, return an empty matrix; @cindex @code{identity} @item real[][] identity(int n); returns the @math{n \times n} identity matrix; @cindex @code{diagonal} @item real[][] diagonal(... real[] a) returns the diagonal matrix with diagonal entries given by a; @cindex @code{inverse} @item real[][] inverse(real[][] a) returns the inverse of a square matrix @code{a}; @cindex @code{quadraticroots} @item @code{real[] quadraticroots(real a, real b, real c);} This numerically robust solver returns the real roots of the quadratic equation @math{ax^2+bx+c=0}, in ascending order. Multiple roots are listed separately; @cindex @code{quadraticroots} @item @code{pair[] quadraticroots(explicit pair a, explicit pair b, explicit pair c);} This numerically robust solver returns the complex roots of the quadratic equation @math{ax^2+bx+c=0}; @cindex @code{cubicroots} @item @code{real[] cubicroots(real a, real b, real c, real d);} This numerically robust solver returns the real roots of the cubic equation @math{ax^3+bx^2+cx+d=0}. Multiple roots are listed separately. @end table @cindex vectorization @code{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 @verbatim real[] a={1,2}; real[] b={3,2}; @end verbatim @noindent then @code{a == b} and @code{a >= 2} both evaluate to the vector @code{@{false, true@}}. @cindex @code{all} To test whether all components of @code{a} and @code{b} agree, use the boolean function @code{all(a == b)}. One can also use conditionals like @code{(a >= 2) ? a : b}, which returns the array @code{@{3,2@}}, or @code{write((a >= 2) ? a : null}, which returns the array @code{@{2@}}. All of the standard built-in @code{libm} functions of signature @code{real(real)} also take a real array as an argument, effectively like an implicit call to @code{map}. As with other built-in types, arrays of the basic data types can be read in by assignment. In this example, the code @verbatim file fin=input("test.txt"); real[] A=fin; @end verbatim @cindex @code{eof} @cindex @code{eol} @cindex @code{line} @cindex line mode @noindent reads real values into @code{A} until the end-of-file is reached (or an I/O error occurs). The virtual members @code{dimension}, @code{line}, @code{csv}, @code{word}, and @code{read} of a file are useful for reading arrays. @cindex @code{line} For example, if line mode is set with @code{file line(bool b=true)}, then reading will stop once the end of the line is reached instead: @verbatim file fin=input("test.txt"); real[] A=fin.line(); @end verbatim @cindex reading string arrays @cindex @code{word} @cindex white-space string delimiter mode 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, @code{file word(bool b=true)}, which causes string reads to respect white-space delimiters, instead of the default end-of-line delimiter: @verbatim file fin=input("test.txt").line().word(); real[] A=fin; @end verbatim @cindex @code{csv} @cindex comma-separated-value mode Another useful mode is comma-separated-value mode, @code{file csv(bool b=true)}, which causes reads to respect comma delimiters: @verbatim file fin=input("test.txt").csv(); real[] A=fin; @end verbatim @cindex @code{dimension} To restrict the number of values read, use the @code{file dimension(int)} function: @verbatim file fin=input("test.txt"); real[] A=fin.dimension(10); @end verbatim 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: @verbatim file fin=input("test.txt"); real[][] A=fin.dimension(2,3); real[][][] B=fin.dimension(2,3,4); @end verbatim @noindent @cindex @code{read} 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 @code{read(1)}, @code{read(2)}, or @code{read(3)}, respectively: @verbatim file fin=input("test.txt"); real[] A=fin.read(1); real[][] B=fin.read(2); real[][][] C=fin.read(3); @end verbatim @cindex @code{write} One, two, and three-dimensional arrays of the basic data types can be output with the functions @code{write(file,T[])}, @code{write(file,T[][])}, @code{write(file,T[][][])}, respectively. @node Slices, , Arrays, Arrays @subsection Slices @cindex slices Asymptote allows a section of an array to be addressed as a slice using a Python-like syntax. If @code{A} is an array, the expression @code{A[m:n]} returns a new array consisting of the elements of @code{A} with indices from @code{m} up to but not including @code{n}. For example, @verbatim 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}; @end verbatim If the left index is omitted, it is taken be @code{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: @verbatim 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. @end verbatim 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 @code{A[m:n]} still consists of the cells with indices in the set [@code{m},@code{n}), but now negative values and values beyond the length of the array are allowed. The indices simply wrap around. For example: @verbatim 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} @end verbatim 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: @verbatim 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"} @end verbatim 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 @code{x[m:n]=x} is equivalent to @code{x[m:n]=copy(x)}. One can use the shorthand @code{x[m:m]=y} to insert the contents of the array @code{y} into the array @code{x} starting at the location just before @code{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 @code{A} is a cyclic array of length 10, @code{A[8:12]}, @code{A[-3:1]}, and @code{A[5:25]} are bridging slices whereas @code{A[3:7]}, @code{A[7:10]}, @code{A[-3:0]} and @code{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 @code{A[0]} and an error is reported. Non-bridging slices may be assigned an array of any length. For a cyclic array @code{A} an expression of the form @code{A[A.length:A.length]} is equivalent to the expression @code{A[0:0]} and so assigning to this slice will insert values at the start of the array. @code{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. @node Casts, Import, Arrays, Programming @section Casts @cindex casts @cindex implicit casts @cindex @code{explicit} @code{Asymptote} implicitly casts @code{int} to @code{real}, @code{int} to @code{pair}, @code{real} to @code{pair}, @code{pair} to @code{path}, @code{pair} to @code{guide}, @code{path} to @code{guide}, @code{guide} to @code{path}, @code{real} to @code{pen}, @code{pair[]} to @code{guide[]}, @code{pair[]} to @code{path[]}, @code{path} to @code{path[]}, and @code{guide} to @code{path[]}, along with various three-dimensional casts defined in module @code{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 @code{explicit} in the function signature, say to avoid an ambiguous function call in the following example, which outputs 0: @verbatim int f(pair a) {return 0;} int f(explicit real x) {return 1;} write(f(0)); @end verbatim @cindex explicit casts Other conversions, say @code{real} to @code{int} or @code{real} to @code{string}, require an explicit cast: @verbatim 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 @end verbatim In situations where casting from a string to a type @code{T} fails, an uninitialized variable is returned; this condition can be detected with the function @code{bool initialized(T);} @verbatim int i=(int) "2.5"; assert(initialized(i),"Invalid cast."); real x=(real) "2.5a"; assert(initialized(x),"Invalid cast."); @end verbatim @cindex @code{operator cast} Casting to user-defined types is also possible using @code{operator cast}: @verbatim 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) @end verbatim 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 @verbatim real operator cast(int x) {return x*100;} @end verbatim @noindent is equivalent to an infinite recursion, since the result @code{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: @verbatim real convert(int x) {return x*100;} real operator cast(int x)=convert; @end verbatim @cindex @code{operator ecast} Explicit casts are implemented similarly, with @code{operator ecast}. @node Import, Static, Casts, Programming @section Import @cindex @code{access} While @code{Asymptote} provides many features by default, some applications require specialized features contained in external @code{Asymptote} modules. For instance, the lines @verbatim access graph; graph.axes(); @end verbatim @noindent draw @math{x} and @math{y} axes on a two-dimensional graph. Here, the command looks up the module under the name @code{graph} in a global dictionary of modules and puts it in a new variable named @code{graph}. The module is a structure, and we can refer to its fields as we usually would with a structure. @cindex @code{from} Often, one wants to use module functions without having to specify the module name. The code @verbatim from graph access axes; @end verbatim @noindent adds the @code{axes} field of @code{graph} into the local name space, so that subsequently, one can just write @code{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: @verbatim from graph access axes, xaxis, yaxis; @end verbatim @noindent Wild card notation can be used to add all non-private fields and types of a module to the local name space: @verbatim from graph access *; @end verbatim @cindex @code{unravel} Similarly, one can add the non-private fields and types of a structure to the local environment with the @code{unravel} keyword: @verbatim struct matrix { real a,b,c,d; } real det(matrix m) { unravel m; return a*d-b*c; } @end verbatim Alternatively, one can unravel selective fields: @verbatim real det(matrix m) { from m unravel a,b,c as C,d; return a*d-b*C; } @end verbatim @cindex @code{import} @cindex @code{access} The command @verbatim import graph; @end verbatim is a convenient abbreviation for the commands @verbatim access graph; unravel graph; @end verbatim That is, @code{import graph} first loads a module into a structure called @code{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 @code{import} commands at the start of an @code{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. @cindex @code{as} To rename modules or fields when adding them to the local environment, use @code{as}: @verbatim access graph as graph2d; from graph access xaxis as xline, yaxis as yline; @end verbatim The command @verbatim import graph as graph2d; @end verbatim is a convenient abbreviation for the commands @verbatim access graph as graph2d; unravel graph2d; @end verbatim Except for a few built-in modules, such as @code{settings}, all modules are implemented as @code{Asymptote} files. When looking up a module that has not yet been loaded, @code{Asymptote} searches the standard search paths (@pxref{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 name contains nonalphanumeric characters, enclose it with quotation marks: @noindent @code{access "@value{Datadir}/asymptote/graph.asy" as graph;} @noindent @code{from "@value{Datadir}/asymptote/graph.asy" access axes;} @noindent @code{import "@value{Datadir}/asymptote/graph.asy" as graph;} @cindex @acronym{URL} @cindex @acronym{libcurl} If @code{Asymptote} is compiled with support for @code{libcurl}, the file name can even be a @acronym{URL}: @noindent @code{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. @cindex runtime imports @cindex @code{eval} However, you can import an @code{Asymptote} module determined by the string @code{s} at runtime like this: @verbatim eval("import "+s,true); @end verbatim @cindex @code{asy} To conditionally execute an array of asy files, use @verbatim void asy(string format, bool overwrite ... string[] s); @end verbatim The file will only be processed, using output format @code{format}, if overwrite is @code{true} or the output file is missing. One can evaluate an @code{Asymptote} expression (without any return value, however) contained in the string @code{s} with: @cindex @code{eval} @verbatim void eval(string s, bool embedded=false); @end verbatim It is not necessary to terminate the string @code{s} with a semicolon. If @code{embedded} is @code{true}, the string will be evaluated at the top level of the current environment. If @code{embedded} is @code{false} (the default), the string will be evaluated in an independent environment, sharing the same @code{settings} module (@pxref{settings}). @cindex @code{quote} One can evaluate arbitrary @code{Asymptote} code (which may contain unescaped quotation marks) with the command @verbatim void eval(code s, bool embedded=false); @end verbatim Here @code{code} is a special type used with @code{quote @{@}} to enclose @code{Asymptote code} like this: @verbatim real a=1; code s=quote { write(a); }; eval(s,true); // Outputs 1 @end verbatim One application of @code{eval} is the creation of parametrized Asymptote code, such as a map from one type to another (@pxref{map}). @cindex @code{include} To include the contents of an existing file @code{graph} verbatim (as if the contents of the file were inserted at that point), use one of the forms: @verbatim include graph; @end verbatim @noindent @code{include "@value{Datadir}/asymptote/graph.asy";} To list all global functions and variables defined in a module named by the contents of the string @code{s}, use the function @verbatim void list(string s, bool imports=false); @end verbatim @noindent Imported global functions and variables are also listed if @code{imports} is @code{true}. @node Static, , Import, Programming @section Static @cindex @code{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 @verbatim struct s { int count() { static int c=0; ++c; return c; } } @end verbatim @noindent there is one instance of the variable @code{c} for each object @code{s} (as opposed to each call of @code{count}). Similarly, in @verbatim int factorial(int n) { int helper(int k) { static int x=1; x *= k; return k == 1 ? x : helper(k-1); } return helper(n); } @end verbatim @noindent there is one instance of @code{x} for every call to @code{factorial} (and not for every call to @code{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, @verbatim struct A { struct B { static pair z; } } @end verbatim @noindent creates one object @code{z} for each object of type @code{A} created. In this example, @verbatim 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; } @end verbatim @noindent there is one instance of @code{x} for each call to @code{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: @verbatim void f(); for(int i=0; i < 10; ++i) { int x=i; if(x==5) { f=new void() {write(x);}; } } f(); @end verbatim Here, every iteration of the loop has its own variable @code{x}, so @code{f()} will write @code{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: @verbatim void f() { static int x; for(int i=0; i < 10; ++i) { static int y; } } @end verbatim @noindent both @code{x} and @code{y} will be allocated in the same place, which is also where @code{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: @verbatim struct A { static int x; } pair A; int y=A.x; @end verbatim @noindent does the @code{A} in @code{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. @node LaTeX usage, Base modules, Programming, Top @chapter @code{LaTeX} usage @cindex @code{LaTeX} usage @cindex @code{asymptote.sty} @code{Asymptote} comes with a convenient @code{LaTeX} style file @code{asymptote.sty} (v1.36 or later required) that makes @code{LaTeX} @code{Asymptote}-aware. Entering @code{Asymptote} code directly into the @code{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 @code{\usepackage@{asymptote@}} at the beginning of your file and enclose your @code{Asymptote} code within a @code{\begin@{asy@}...\end@{asy@}} environment. As with the @code{LaTeX} @code{comment} environment, the @code{\end@{asy@}} command must appear on a line by itself, with no trailing commands/comments. A blank line is not allowed after @code{\begin@{asy@}}. The sample @code{LaTeX} file below, named @code{latexusage.tex}, can be run as follows: @verbatim latex latexusage asy latexusage-*.asy latex latexusage @end verbatim @noindent or @verbatim pdflatex latexusage asy latexusage-*.asy pdflatex latexusage @end verbatim @noindent To switch between using inline Asymptote code with @code{latex} and @code{pdflatex} you may first need to remove the files @code{latexusage-*.tex}. @cindex @code{latexmk} @cindex @code{perl} An even better method for processing a @code{LaTeX} file with embedded @code{Asymptote} code is to use the @code{latexmk} utility from @quotation @url{http://mirror.ctan.org/support/latexmk/} @end quotation @noindent after putting the contents of @url{https://raw.githubusercontent.com/vectorgraphics/asymptote/HEAD/doc/latexmkrc} @noindent in a file @code{latexmkrc} in the same directory. The command @verbatim latexmk -pdf latexusage @end verbatim @noindent will then call @code{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 @code{asy}, one can define @verbatim \def\asydir{asy} @end verbatim in @code{latexusage.tex} and put the contents of @url{https://raw.githubusercontent.com/vectorgraphics/asymptote/HEAD/doc/latexmkrc_asydir} in a file @code{latexmkrc} in the same directory. @noindent External @code{Asymptote} code can be included with @cindex @code{asyinclude} @verbatim \asyinclude[]{} @end verbatim @noindent so that @code{latexmk} will recognize when the code is changed. Note that @code{latemk} requires @code{perl}, available from @url{https://www.perl.org/}. @cindex @code{width} @cindex @code{height} @cindex @code{keepAspect} @cindex @code{viewportwidth} @cindex @code{viewportheight} @cindex @code{attach} @cindex @code{inline} One can specify @code{width}, @code{height}, @code{keepAspect}, @code{viewportwidth}, @code{viewportheight}, @code{attach}, and @code{inline}. @code{keyval}-style options to the @code{asy} and @code{asyinclude} environments. Three-dimensional @acronym{PRC} files may either be embedded within the page (the default) or attached as annotated (but printable) attachments, using the @code{attach} option and the @code{attachfile2} (or older @code{attachfile}) @code{LaTeX} package. The @code{inline} option generates inline @code{LaTeX} code instead of @acronym{EPS} or @acronym{PDF} files. This makes 2D LaTeX symbols visible to the @code{\begin@{asy@}...\end@{asy@}} environment. In this mode, Asymptote correctly aligns 2D LaTeX symbols defined outside of @code{\begin@{asy@}...\end@{asy@}}, but treats their size as zero; an optional second string can be given to @code{Label} to provide an estimate of the unknown label size. Note that if the @code{latex} @TeX{} engine is used with the @code{inline} option, labels might not show up in @acronym{DVI} viewers that cannot handle raw @code{PostScript} code. One can use @code{dvips}/@code{dvipdf} to produce @code{PostScript}/@acronym{PDF} output (we recommend using the modified version of @code{dvipdf} in the @code{Asymptote} patches directory, which accepts the @code{dvips -z} hyperdvi option). Here now is @code{latexusage.tex}: @verbatiminclude latexusage.tex @page @image{./latexusage,,25cm} @node Base modules, Options, LaTeX usage, Top @chapter Base modules @cindex base modules @code{Asymptote} currently ships with the following base modules: @menu * plain:: Default @code{Asymptote} base file * simplex:: Linear programming: simplex method * math:: Extend @code{Asymptote}'s math capabilities * interpolate:: Interpolation routines * geometry:: Geometry routines * trembling:: Wavy lines * stats:: Statistics routines and histograms * patterns:: Custom fill and draw patterns * markers:: Custom path marker routines * map:: Map keys to values * tree:: Dynamic binary search tree * binarytree:: Binary tree drawing module * drawtree:: Tree drawing module * syzygy:: Syzygy and braid drawing module * feynman:: Feynman diagrams * roundedpath:: Round the sharp corners of paths * animation:: Embedded @acronym{PDF} and @acronym{MPEG} movies * embed:: Embedding movies, sounds, and 3D objects * slide:: Making presentations with @code{Asymptote} * MetaPost:: @code{MetaPost} compatibility routines * babel:: Interface to @code{LaTeX} @code{babel} package * labelpath:: Drawing curved labels * labelpath3:: Drawing curved labels in 3D * annotate:: Annotate your @acronym{PDF} files * CAD:: 2D CAD pen and measurement functions (DIN 15) * graph:: 2D linear & logarithmic graphs * palette:: Color density images and palettes * three:: 3D vector graphics * obj:: 3D obj files * graph3:: 3D linear & logarithmic graphs * grid3:: 3D grids * solids:: 3D solid geometry * tube:: 3D rotation minimizing tubes * flowchart:: Flowchart drawing routines * contour:: Contour lines * contour3:: Contour surfaces * smoothcontour3:: Smooth implicit surfaces * slopefield:: Slope fields * ode:: Ordinary differential equations @end menu @node plain, simplex, Base modules, Base modules @section @code{plain} @cindex @code{plain} This is the default @code{Asymptote} base file, which defines key parts of the drawing language (such as the @code{picture} structure). By default, an implicit @code{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 @code{plain}, of course). This means that the types and functions defined in @code{plain} are accessible in almost all @code{Asymptote} code. Use the @code{-noautoplain} command-line option to disable this feature. @node simplex, math, plain, Base modules @section @code{simplex} @cindex @code{simplex} @cindex @code{deferred drawing} This module solves the two-variable linear programming problem using the simplex method. It is used by the module @code{plain} for automatic sizing of pictures. @node math, interpolate, simplex, Base modules @section @code{math} @cindex @code{math} This module extends @code{Asymptote}'s mathematical capabilities with useful functions such as @table @code @cindex @code{drawline} @item void drawline(picture pic=currentpicture, pair P, pair Q, pen p=currentpen); draw the visible portion of the (infinite) line going through @code{P} and @code{Q}, without altering the size of picture @code{pic}, using pen @code{p}. @cindex @code{intersect} @item real intersect(triple P, triple Q, triple n, triple Z); returns the intersection time of the extension of the line segment @code{PQ} with the plane perpendicular to @code{n} and passing through @code{Z}. @cindex @code{intersectionpoint} @item triple intersectionpoint(triple n0, triple P0, triple n1, triple P1); Return any point on the intersection of the two planes with normals @code{n0} and @code{n1} passing through points @code{P0} and @code{P1}, respectively. If the planes are parallel, return @code{(infinity,infinity,infinity)}. @cindex @code{quarticroots} @item pair[] quarticroots(real a, real b, real c, real d, real e); returns the four complex roots of the quartic equation @math{ax^4+bx^3+cx^2+dx+e=0}. @cindex @code{time} @item real time(path g, real x, int n=0, real fuzz=-1) returns the @code{n}th intersection time of path @code{g} with the vertical line through x. @cindex @code{time} @item real time(path g, explicit pair z, int n=0, real fuzz=-1) returns the @code{n}th intersection time of path @code{g} with the horizontal line through @code{(0,z.y)}. @cindex @code{value} @item real value(path g, real x, int n=0, real fuzz=-1) returns the @code{n}th @code{y} value of @code{g} at @code{x}. @cindex @code{value} @item real value(path g, explicit pair z, int n=0, real fuzz=-1) returns the @code{n}th @code{x} value of @code{g} at @code{y=z.y}. @cindex @code{slope} @item real slope(path g, real x, int n=0, real fuzz=-1) returns the @code{n}th slope of @code{g} at @code{x}. @cindex @code{slope} @item real slope(path g, explicit pair z, int n=0, real fuzz=-1) returns the @code{n}th slope of @code{g} at @code{y=z.y}. @cindex @code{segment} int[][] segment(bool[] b) returns the indices of consecutive true-element segments of bool[] @code{b}. @cindex @code{partialsum} @item real[] partialsum(real[] a) returns the partial sums of a real array @code{a}. @cindex @code{partialsum} @item real[] partialsum(real[] a, real[] dx) returns the partial @code{dx}-weighted sums of a real array @code{a}. @cindex @code{increasing} @item bool increasing(real[] a, bool strict=false) returns, if @code{strict=false}, whether @code{i > j} implies @code{a[i] >= a[j]}, or if @code{strict=true}, whether @code{i > j} implies implies @code{a[i] > a[j]}. @cindex @code{unique} @item int unique(real[] a, real x) if the sorted array @code{a} does not contain @code{x}, insert it sequentially, returning the index of @code{x} in the resulting array. @cindex @code{lexorder} @item bool lexorder(pair a, pair b) returns the strict lexicographical partial order of @code{a} and @code{b}. @cindex @code{lexorder} @item bool lexorder(triple a, triple b) returns the strict lexicographical partial order of @code{a} and @code{b}. @end table @node interpolate, geometry, math, Base modules @section @code{interpolate} @cindex @code{interpolate} This module implements Lagrange, Hermite, and standard cubic spline interpolation in @code{Asymptote}, as illustrated in the example @code{interpolate1.asy}. @node geometry, trembling, interpolate, Base modules @section @code{geometry} @cindex @code{geometry} @cindex @code{triangle} @cindex @code{perpendicular} This module, written by Philippe Ivaldi, provides an extensive set of geometry routines, including @code{perpendicular} symbols and a @code{triangle} structure. Link to the documentation for the @code{geometry} module are posted here: @url{https://asymptote.sourceforge.io/links.html}, including an extensive set of examples, @url{http://www.piprime.fr/files/asymptote/geometry/}, and an index: @quotation @url{http://www.piprime.fr/files/asymptote/geometry/modules/geometry.asy.index.type.html} @end quotation @node trembling, stats, geometry, Base modules @section @code{trembling} @cindex @code{trembling} This module, written by Philippe Ivaldi and illustrated in the example @code{@uref{https://asymptote.sourceforge.io/gallery/floatingdisk.svg,,floatingdisk}@uref{https://asymptote.sourceforge.io/gallery/floatingdisk.asy,,.asy}}, allows one to draw wavy lines, as if drawn by hand. @node stats, patterns, trembling, Base modules @section @code{stats} @cindex @code{stats} @cindex @code{leastsquares} This module implements a Gaussian random number generator and a collection of statistics routines, including @code{histogram} and @code{leastsquares}. @node patterns, markers, stats, Base modules @section @code{patterns} @cindex @code{patterns} This module implements @code{PostScript} tiling patterns and includes several convenient pattern generation routines. @node markers, map, patterns, Base modules @section @code{markers} @cindex @code{markers} This module implements specialized routines for marking paths and angles. The principal mark routine provided by this module is @verbatim markroutine markinterval(int n=1, frame f, bool rotated=false); @end verbatim @noindent which centers @code{n} copies of frame @code{f} within uniformly space intervals in arclength along the path, optionally rotated by the angle of the local tangent. The @code{marker} (@pxref{marker}) routine can be used to construct new markers from these predefined frames: @cindex @code{stickframe} @verbatim frame stickframe(int n=1, real size=0, pair space=0, real angle=0, pair offset=0, pen p=currentpen); @end verbatim @cindex @code{circlebarframe} @verbatim 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); @end verbatim @cindex @code{crossframe} @verbatim frame crossframe(int n=3, real size=0, pair space=0, real angle=0, pair offset=0, pen p=currentpen); @end verbatim @cindex @code{tildeframe} @verbatim frame tildeframe(int n=1, real size=0, pair space=0, real angle=0, pair offset=0, pen p=currentpen); @end verbatim For convenience, this module also constructs the markers @code{StickIntervalMarker}, @code{CrossIntervalMarker}, @code{CircleBarIntervalMarker}, and @code{TildeIntervalMarker} from the above frames. The example @code{@uref{https://asymptote.sourceforge.io/gallery/markers1.svg,,markers1}@uref{https://asymptote.sourceforge.io/gallery/markers1.asy,,.asy}} illustrates the use of these markers: @sp 1 @center @image{./markers1} This module also provides a routine for marking an angle @math{AOB}: @cindex @code{markangle} @verbatim 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); @end verbatim @noindent as illustrated in the example @code{@uref{https://asymptote.sourceforge.io/gallery/markers2.svg,,markers2}@uref{https://asymptote.sourceforge.io/gallery/markers2.asy,,.asy}}. @sp 1 @center @image{./markers2} @node map, tree, markers, Base modules @section @code{map} @cindex @code{map} This module creates a struct parameterized by the types specified in strings @code{key} and @code{value}, mapping keys to values with a specified default: @verbatim import map; mapTemplate(name="map",key="string",value="int",default="-1"); map M; M.add("z",2); M.add("a",3); M.add("d",4); write(M.lookup("a")); write(M.lookup("y")); @end verbatim @node tree, binarytree, map, Base modules @section @code{tree} @cindex @code{tree} This module implements an example of a dynamic binary search tree. @node binarytree, drawtree, tree, Base modules @section @code{binarytree} @cindex @code{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 @code{@uref{https://asymptote.sourceforge.io/gallery/binarytreetest.svg,,binarytreetest}@uref{https://asymptote.sourceforge.io/gallery/binarytreetest.asy,,.asy}}: @verbatiminclude binarytreetest.asy @sp 1 @center @image{./binarytreetest} @node drawtree, syzygy, binarytree, Base modules @section @code{drawtree} @cindex @code{drawtree} This is a simple tree drawing module used by the example @code{@uref{https://asymptote.sourceforge.io/gallery/treetest.svg,,treetest}@uref{https://asymptote.sourceforge.io/gallery/treetest.asy,,.asy}}. @node syzygy, feynman, drawtree, Base modules @section @code{syzygy} @cindex @code{syzygy} This module automates the drawing of braids, relations, and syzygies, along with the corresponding equations, as illustrated in the example @code{@uref{https://asymptote.sourceforge.io/gallery/knots.svg,,knots}@uref{https://asymptote.sourceforge.io/gallery/knots.asy,,.asy}}. @node feynman, roundedpath, syzygy, Base modules @section @code{feynman} @cindex @code{feynman} This module, contributed by Martin Wiebusch, is useful for drawing Feynman diagrams, as illustrated by the examples @code{@uref{https://asymptote.sourceforge.io/gallery/eetomumu.svg,,eetomumu}@uref{https://asymptote.sourceforge.io/gallery/eetomumu.asy,,.asy}} and @code{@uref{https://asymptote.sourceforge.io/gallery/fermi.svg,,fermi}@uref{https://asymptote.sourceforge.io/gallery/fermi.asy,,.asy}}. @node roundedpath, animation, feynman, Base modules @section @code{roundedpath} @cindex @code{roundedpath} This module, contributed by Stefan Knorr, is useful for rounding the sharp corners of paths, as illustrated in the example file @code{@uref{https://asymptote.sourceforge.io/gallery/roundpath.svg,,roundpath}@uref{https://asymptote.sourceforge.io/gallery/roundpath.asy,,.asy}}. @node animation, embed, roundedpath, Base modules @section @code{animation} @cindex @code{animation} @cindex @code{convert} @cindex animation @cindex @code{ImageMagick} This module allows one to generate animations, as illustrated by the files @code{@uref{https://asymptote.sourceforge.io/gallery/animations/wheel.gif,,wheel}@uref{https://asymptote.sourceforge.io/gallery/animations/wheel.asy,,.asy}}, @code{@uref{https://asymptote.sourceforge.io/gallery/animations/wavepacket.gif,,wavepacket}@uref{https://asymptote.sourceforge.io/gallery/animations/wavepacket.asy,,.asy}}, and @code{@uref{https://asymptote.sourceforge.io/gallery/animations/cube.gif,,cube}@uref{https://asymptote.sourceforge.io/gallery/animations/cube.asy,,.asy}} in the @code{animations} subdirectory of the examples directory. These animations use the @code{ImageMagick} @code{convert} program to merge multiple images into a @acronym{GIF} or @acronym{MPEG} movie. @cindex @code{animate} @anchor{animate} The related @code{animate} module, derived from the @code{animation} module, generates higher-quality portable clickable @acronym{PDF} movies, with optional controls. This requires installing the module @quotation @url{http://mirror.ctan.org/macros/latex/contrib/animate/animate.sty} @noindent @end quotation @noindent (version 2007/11/30 or later) in a new directory @code{animate} in the local @code{LaTeX} directory (for example, in @code{/usr/local/share/texmf/tex/latex/animate}). On @code{UNIX} systems, one must then execute the command @code{texhash}. The example @code{@uref{https://asymptote.sourceforge.io/gallery/animations/pdfmovie.pdf,,pdfmovie}@uref{https://asymptote.sourceforge.io/gallery/animations/pdfmovie.asy,,.asy}} in the @code{animations} directory, along with the slide presentations @code{@uref{https://asymptote.sourceforge.io/gallery/animations/slidemovies.pdf,,slidemovies}@uref{https://asymptote.sourceforge.io/gallery/animations/slidemovies.asy,,.asy}} and @code{@uref{https://asymptote.sourceforge.io/intro.pdf,,intro}}, illustrate the use of embedded @acronym{PDF} movies. The examples @code{inlinemovie.tex} and @code{inlinemovie3.tex} show how to generate and embed @acronym{PDF} movies directly within a @code{LaTeX} file (@pxref{LaTeX usage}). The member function @verbatim string pdf(fit fit=NoBox, real delay=animationdelay, string options="", bool keep=settings.keep, bool multipage=true); @end verbatim @noindent of the @code{animate} structure accepts any of the @code{animate.sty} options, as described here: @quotation @url{http://mirror.ctan.org/macros/latex/contrib/animate/doc/animate.pdf} @end quotation @node embed, slide, animation, Base modules @section @code{embed} @cindex @code{embed} This module provides an interface to the @code{LaTeX} package (included with @code{MikTeX}) @quotation @url{http://mirror.ctan.org/macros/latex/contrib/media9} @end quotation @noindent for embedding movies, sounds, and 3D objects into a @acronym{PDF} document. @cindex @code{external} A more portable method for embedding movie files, which should work on any platform and does not require the @code{media9} package, is provided by using the @code{external} module instead of @code{embed}. Examples of the above two interfaces is provided in the file @code{embeddedmovie.asy} in the @code{animations} subdirectory of the examples directory and in @code{@uref{https://asymptote.sourceforge.io/gallery/animations/externalmovie.pdf,,externalmovie}@uref{https://asymptote.sourceforge.io/gallery/animations/externalmovie.asy,,.asy}}. For a higher quality embedded movie generated directly by @code{Asymptote}, use the @code{animate} module along with the @code{animate.sty} package to embed a portable @acronym{PDF} animation (@pxref{animate}). @cindex @code{U3D} An example of embedding @code{U3D} code is provided in the file @code{embeddedu3d}. @node slide, MetaPost, embed, Base modules @section @code{slide} @cindex @code{slide} This module provides a simple yet high-quality facility for making presentation slides, including portable embedded @acronym{PDF} animations (see the file @code{@uref{https://asymptote.sourceforge.io/gallery/animations/slidemovies.pdf,,slidemovies}@uref{https://asymptote.sourceforge.io/gallery/animations/slidemovies.asy,,.asy}}). A simple example is provided in @code{slidedemo.asy}. @node MetaPost, babel, slide, Base modules @section @code{MetaPost} @cindex @code{MetaPost} This module provides some useful routines to help @code{MetaPost} users migrate old @code{MetaPost} code to @code{Asymptote}. Further contributions here are welcome. @cindex @code{implicit linear solver} @cindex @code{MetaPost whatever} @cindex @code{extension} Unlike @code{MetaPost}, @code{Asymptote} does not implicitly solve linear equations and therefore does not have the notion of a @code{whatever} unknown. The routine @code{extension} (@pxref{extension}) provides a useful replacement for a common use of @code{whatever}: finding the intersection point of the lines through @code{P}, @code{Q} and @code{p}, @code{q}. For less common occurrences of @code{whatever}, one can use the built-in explicit linear equation solver @code{solve} instead. @node babel, labelpath, MetaPost, Base modules @section @code{babel} @cindex @code{babel} This module implements the @code{LaTeX} @code{babel} package in @code{Asymptote}. For example: @verbatim import babel; babel("german"); @end verbatim @node labelpath, labelpath3, babel, Base modules @section @code{labelpath} @cindex @code{labelpath} This module uses the @code{PSTricks} @code{pstextpath} macro to fit labels along a path (properly kerned, as illustrated in the example file @code{@uref{https://asymptote.sourceforge.io/gallery/curvedlabel.svg,,curvedlabel}@uref{https://asymptote.sourceforge.io/gallery/curvedlabel.asy,,.asy}}), using the command @verbatim void labelpath(picture pic=currentpicture, Label L, path g, string justify=Centered, pen p=currentpen); @end verbatim @noindent Here @code{justify} is one of @code{LeftJustified}, @code{Centered}, or @code{RightJustified}. The @math{x} component of a shift transform applied to the Label is interpreted as a shift along the curve, whereas the @math{y} component is interpreted as a shift away from the curve. All other Label transforms are ignored. This module requires the @code{latex} tex engine and inherits the limitations of the @code{PSTricks} @code{\pstextpath} macro. @node labelpath3, annotate, labelpath, Base modules @section @code{labelpath3} @cindex @code{labelpath3} This module, contributed by Jens Schwaiger, implements a 3D version of @code{labelpath} that does not require the @code{PSTricks} package. An example is provided in @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/curvedlabel3.html,,curvedlabel3}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/curvedlabel3.asy,,.asy}}. @node annotate, CAD, labelpath3, Base modules @section @code{annotate} @cindex @code{annotate} This module supports @acronym{PDF} annotations for viewing with @code{Adobe Reader}, via the function @verbatim void annotate(picture pic=currentpicture, string title, string text, pair position); @end verbatim @noindent Annotations are illustrated in the example file @code{@uref{https://asymptote.sourceforge.io/gallery/PDFs/annotation.pdf,,annotation}@uref{https://asymptote.sourceforge.io/gallery/PDFs/annotation.asy,,.asy}}. Currently, annotations are only implemented for the @code{latex} (default) and @code{tex} @TeX{} engines. @node CAD, graph, annotate, Base modules @section @code{CAD} @cindex @code{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 @code{CAD.pdf}. @node graph, palette, CAD, Base modules @section @code{graph} @cindex @code{graph} @cindex 2D graphs 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 @code{guide} (that can be drawn with the draw command, with an optional legend) constructed with one of the following routines: @itemize @item @verbatim 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 --); @end verbatim Returns a graph using the scaling information for picture @code{pic} (@pxref{automatic scaling}) of the function @code{f} on the interval [@code{T}(@code{a}),@code{T}(@code{b})], sampling at @code{n} points evenly spaced in [@code{a},@code{b}], optionally restricted by the bool3 function @code{cond} on [@code{a},@code{b}]. If @code{cond} is: @itemize @bullet @item @code{true}, the point is added to the existing guide; @item @code{default}, the point is added to a new guide; @item @code{false}, the point is omitted and a new guide is begun. @end itemize The points are connected using the interpolation specified by @code{join}: @itemize @bullet @cindex @code{operator --} @cindex @code{Straight} @item @code{operator --} (linear interpolation; the abbreviation @code{Straight} is also accepted); @cindex @code{operator ..} @cindex @code{Spline} @item @code{operator ..} (piecewise Bezier cubic spline interpolation; the abbreviation @code{Spline} is also accepted); @cindex @code{Hermite} @cindex @code{notaknot} @cindex @code{natural} @cindex @code{periodic} @cindex @code{clamped} @cindex @code{monotonic} @cindex @code{Hermite(splinetype splinetype} @item @code{Hermite} (standard cubic spline interpolation using boundary condition @code{notaknot}, @code{natural}, @code{periodic}, @code{clamped(real slopea, real slopeb)}), or @code{monotonic}. The abbreviation @code{Hermite} is equivalent to @code{Hermite(notaknot)} for nonperiodic data and @code{Hermite(periodic)} for periodic data). @end itemize @item @verbatim 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 --); @end verbatim Returns a graph using the scaling information for picture @code{pic} of the parametrized function (@code{x}(@math{t}),@code{y}(@math{t})) for @math{t} in the interval [@code{T}(@code{a}),@code{T}(@code{b})], sampling at @code{n} points evenly spaced in [@code{a},@code{b}], optionally restricted by the bool3 function @code{cond} on [@code{a},@code{b}], using the given interpolation type. @item @verbatim 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 --); @end verbatim Returns a graph using the scaling information for picture @code{pic} of the parametrized function @code{z}(@math{t}) for @math{t} in the interval [@code{T}(@code{a}),@code{T}(@code{b})], sampling at @code{n} points evenly spaced in [@code{a},@code{b}], optionally restricted by the bool3 function @code{cond} on [@code{a},@code{b}], using the given interpolation type. @item @verbatim guide graph(picture pic=currentpicture, pair[] z, interpolate join=operator --); guide[] graph(picture pic=currentpicture, pair[] z, bool3[] cond, interpolate join=operator --); @end verbatim Returns a graph using the scaling information for picture @code{pic} of the elements of the array @code{z}, optionally restricted to those indices for which the elements of the boolean array @code{cond} are @code{true}, using the given interpolation type. @item @verbatim 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 --); @end verbatim Returns a graph using the scaling information for picture @code{pic} of the elements of the arrays (@code{x},@code{y}), optionally restricted to those indices for which the elements of the boolean array @code{cond} are @code{true}, using the given interpolation type. @item @cindex @code{polargraph} @verbatim guide polargraph(picture pic=currentpicture, real f(real), real a, real b, int n=ngraph, interpolate join=operator --); @end verbatim Returns a polar-coordinate graph using the scaling information for picture @code{pic} of the function @code{f} on the interval [@code{a},@code{b}], sampling at @code{n} evenly spaced points, with the given interpolation type. @item @verbatim guide polargraph(picture pic=currentpicture, real[] r, real[] theta, interpolate join=operator--); @end verbatim Returns a polar-coordinate graph using the scaling information for picture @code{pic} of the elements of the arrays (@code{r},@code{theta}), using the given interpolation type. @end itemize @verbatim @end verbatim An axis can be drawn on a picture with one of the following commands: @itemize @item @verbatim 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); @end verbatim Draw an @math{x} axis on picture @code{pic} from @math{x}=@code{xmin} to @math{x}=@code{xmax} using pen @code{p}, optionally labelling it with Label @code{L}. The relative label location along the axis (a real number from [0,1]) defaults to 1 (@pxref{Label}), so that the label is drawn at the end of the axis. An infinite value of @code{xmin} or @code{xmax} specifies that the corresponding axis limit will be automatically determined from the picture limits. The optional @code{arrow} argument takes the same values as in the @code{draw} command (@pxref{arrows}). The axis is drawn before any existing objects in @code{pic} unless @code{above=true}. The axis placement is determined by one of the following @code{axis} types: @table @code @cindex @code{YZero} @item YZero(bool extend=true) Request an @math{x} axis at @math{y}=0 (or @math{y}=1 on a logarithmic axis) extending to the full dimensions of the picture, unless @code{extend}=false. @cindex @code{YEquals} @item YEquals(real Y, bool extend=true) Request an @math{x} axis at @math{y}=@code{Y} extending to the full dimensions of the picture, unless @code{extend}=false. @cindex @code{Bottom} @item Bottom(bool extend=false) Request a bottom axis. @cindex @code{Top} @item Top(bool extend=false) Request a top axis. @cindex @code{BottomTop} @item BottomTop(bool extend=false) Request a bottom and top axis. @end table @cindex custom axis types Custom axis types can be created by following the examples in the module @code{graph.asy}. One can easily override the default values for the standard axis types: @verbatim 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(); @end verbatim @anchor{ticks} @cindex @code{ticks} @cindex @code{NoTicks} @cindex @code{LeftTicks} @cindex @code{RightTicks} @cindex @code{Ticks} The default tick option is @code{NoTicks}. The options @code{LeftTicks}, @code{RightTicks}, or @code{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: @verbatim 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); @end verbatim If any of these parameters are omitted, reasonable defaults will be chosen: @table @code @item Label format @cindex @code{defaultformat} @cindex @code{trailingzero} override the default tick label format (@code{defaultformat}, initially "$%.4g$"), rotation, pen, and alignment (for example, @code{LeftSide}, @code{Center}, or @code{RightSide}) relative to the axis. To enable @code{LaTeX} math mode fonts, the format string should begin and end with @code{$} @pxref{format}. If the format string is @code{trailingzero}, trailing zeros will be added to the tick labels; if the format string is @code{"%"}, the tick label will be suppressed; @item ticklabel is a function @code{string(real x)} returning the label (by default, format(format.s,x)) for each major tick value @code{x}; @item bool beginlabel include the first label; @item bool endlabel include the last label; @item int N when automatic scaling is enabled (the default; @pxref{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; @item int n divide each interval into this many subintervals, separated by minor ticks; @item real Step the tick value spacing between major ticks (if @code{N}=@code{0}); @item real step the tick value spacing between minor ticks (if @code{n}=@code{0}); @item bool begin include the first major tick; @item bool end include the last major tick; @item tickmodifier modify; an optional function that takes and returns a @code{tickvalue} structure having real[] members @code{major} and @code{minor} consisting of the tick values (to allow modification of the automatically generated tick values); @item real Size the size of the major ticks (in @code{PostScript} coordinates); @item real size the size of the minor ticks (in @code{PostScript} coordinates); @item bool extend; extend the ticks between two axes (useful for drawing a grid on the graph); @item pen pTick an optional pen used to draw the major ticks; @item pen ptick an optional pen used to draw the minor ticks. @end table @cindex @code{OmitTick} @cindex @code{OmitTickInterval} @cindex @code{OmitTickIntervals} For convenience, the predefined tickmodifiers @code{OmitTick(... real[] x)}, @code{OmitTickInterval(real a, real b)}, and @code{OmitTickIntervals(real[] a, real[] b)} can be used to remove specific auto-generated ticks and their labels. The @code{OmitFormat(string s=defaultformat ... real[] x)} ticklabel can be used to remove specific tick labels but not the corresponding ticks. The tickmodifier @code{NoZero} is an abbreviation for @code{OmitTick(0)} and the ticklabel @code{NoZeroFormat} is an abbrevation for @code{OmitFormat(0)}. @cindex custom tick locations @cindex @code{LeftTicks} @cindex @code{RightTicks} @cindex @code{Ticks} It is also possible to specify custom tick locations with @code{LeftTicks}, @code{RightTicks}, and @code{Ticks} by passing explicit real arrays @code{Ticks} and (optionally) @code{ticks} containing the locations of the major and minor ticks, respectively: @verbatim 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) @end verbatim @item @verbatim 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); @end verbatim Draw a @math{y} axis on picture @code{pic} from @math{y}=@code{ymin} to @math{y}=@code{ymax} using pen @code{p}, optionally labelling it with a Label @code{L} that is autorotated unless @code{autorotate=false}. The relative location of the label (a real number from [0,1]) defaults to 1 (@pxref{Label}). An infinite value of @code{ymin} or @code{ymax} specifies that the corresponding axis limit will be automatically determined from the picture limits. The optional @code{arrow} argument takes the same values as in the @code{draw} command (@pxref{arrows}). The axis is drawn before any existing objects in @code{pic} unless @code{above=true}. The tick type is specified by @code{ticks} and the axis placement is determined by one of the following @code{axis} types: @table @code @cindex @code{XZero} @item XZero(bool extend=true) Request a @math{y} axis at @math{x}=0 (or @math{x}=1 on a logarithmic axis) extending to the full dimensions of the picture, unless @code{extend}=false. @cindex @code{XEquals} @item XEquals(real X, bool extend=true) Request a @math{y} axis at @math{x}=@code{X} extending to the full dimensions of the picture, unless @code{extend}=false. @cindex @code{Left} @item Left(bool extend=false) Request a left axis. @cindex @code{Right} @item Right(bool extend=false) Request a right axis. @cindex @code{LeftRight} @item LeftRight(bool extend=false) Request a left and right axis. @end table @item @cindex @code{xequals} @cindex @code{yequals} For convenience, the functions @verbatim 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); @end verbatim and @verbatim 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); @end verbatim can be respectively used to call @code{yaxis} and @code{xaxis} with the appropriate axis types @code{XEquals(x,extend)} and @code{YEquals(y,extend)}. This is the recommended way of drawing vertical or horizontal lines and axes at arbitrary locations. @item @verbatim 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); @end verbatim This convenience routine draws both @math{x} and @math{y} axes on picture @code{pic} from @code{min} to @code{max}, with optional labels @code{xlabel} and @code{ylabel} and any arrows specified by @code{arrow}. The axes are drawn on top of existing objects in @code{pic} only if @code{above=true}. @item @verbatim 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); @end verbatim This routine can be used to draw on picture @code{pic} a general axis based on an arbitrary path @code{g}, using pen @code{p}. One can optionally label the axis with Label @code{L} and add an arrow @code{arrow}. The tick type is given by @code{ticks}. The optional integer array @code{divisor} specifies what tick divisors to try in the attempt to produce uncrowded tick labels. A @code{true} value for the flag @code{opposite} identifies an unlabelled secondary axis (typically drawn opposite a primary axis). The axis is drawn before any existing objects in @code{pic} unless @code{above=true}. The tick locator @code{ticklocate} is constructed by the routine @verbatim ticklocate ticklocate(real a, real b, autoscaleT S=defaultS, real tickmin=-infinity, real tickmax=infinity, real time(real)=null, pair dir(real)=zero); @end verbatim @noindent where @code{a} and @code{b} specify the respective tick values at @code{point(g,0)} and @code{point(g,length(g))}, @code{S} specifies the autoscaling transformation, the function @code{real time(real v)} returns the time corresponding to the value @code{v}, and @code{pair dir(real t)} returns the absolute tick direction as a function of @code{t} (zero means draw the tick perpendicular to the axis). @item These routines are useful for manually putting ticks and labels on axes (if the variable @code{Label} is given as the @code{Label} argument, the @code{format} argument will be used to format a string based on the tick location): @cindex xtick @cindex ytick @cindex labelx @cindex labely @cindex tick @cindex Label @verbatim 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); @end verbatim @end itemize Here are some simple examples of two-dimensional graphs: @enumerate @cindex textbook graph @item This example draws a textbook-style graph of @math{y=} exp@math{(x)}, with the @math{y} axis starting at @math{y=0}: @verbatiminclude exp.asy @sp 1 @center @image{./exp} @item 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 (@pxref{GUI}). If an @code{UnFill(real xmargin=0, real ymargin=xmargin)} or @code{Fill(pen)} option is specified to @code{add}, the legend will obscure any underlying objects. Here we illustrate how to clip the portion of the picture covered by a label: @cindex scientific graph @verbatiminclude lineargraph0.asy @sp 1 @center @image{./lineargraph0} @cindex @code{attach} To specify a fixed size for the graph proper, use @code{attach}: @verbatiminclude lineargraph.asy @cindex @code{legend} A legend can have multiple entries per line: @verbatiminclude legend.asy @sp 1 @center @image{./legend} @item 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 @math{y} axis. @verbatiminclude datagraph.asy @sp 1 @center @image{./datagraph} @item This example shows how to graph columns of data read from a file. @verbatiminclude filegraph.asy @sp 1 @center @image{./filegraph} @cindex @code{polygon} @cindex @code{cross} @cindex @code{errorbars} @cindex @code{marker} @cindex @code{marknodes} @cindex @code{markuniform} @cindex @code{mark} @cindex path markers @anchor{pathmarkers} @item 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 @verbatim marker marker(path g, markroutine markroutine=marknodes, pen p=currentpen, filltype filltype=NoFill, bool above=true); @end verbatim using the path @code{unitcircle} (@pxref{filltype}), are drawn below each node. Any frame can be converted to a marker, using @anchor{marker} @verbatim marker marker(frame f, markroutine markroutine=marknodes, bool above=true); @end verbatim In the right-hand graph, the unit @math{n}-sided regular polygon @code{polygon(int n)} and the unit @math{n}-point cyclic cross @code{cross(int n, bool round=true, real r=0)} (where @code{r} is an optional ``inner'' radius) are used to build a custom marker frame. @anchor{markuniform} Here @code{markuniform(bool centered=false, int n, bool rotated=false)} adds this frame at @code{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 @code{n} evenly spaced arclength intervals). Alternatively, one can use markroutine @code{marknodes} to request that the marks be placed at each Bezier node of the path, or markroutine @code{markuniform(pair z(real t), real a, real b, int n)} to place marks at points @code{z(t)} for n evenly spaced values of @code{t} in @code{[a,b]}. These markers are predefined: @verbatim 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) }; @end verbatim The example also illustrates the @code{errorbar} routines: @verbatim 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); @end verbatim @noindent Here, the positive and negative extents of the error are given by the absolute values of the elements of the pair array @code{dp} and the optional pair array @code{dm}. If @code{dm} is not specified, the positive and negative extents of the error are assumed to be equal. @anchor{errorbars} @cindex error bars @verbatiminclude errorbars.asy @sp 1 @center @image{./errorbars} @cindex custom mark routine @item A custom mark routine can be also be specified: @verbatiminclude graphmarkers.asy @sp 1 @center @image{./graphmarkers} @item This example shows how to label an axis with arbitrary strings. @verbatiminclude monthaxis.asy @sp 1 @center @image{./monthaxis} @item The next example draws a graph of a parametrized curve. @cindex parametrized curve @cindex cropping graphs @cindex @code{xlimits} @cindex @code{ylimits} @cindex @code{limits} @cindex @code{crop} The calls to @verbatim xlimits(picture pic=currentpicture, real min=-infinity, real max=infinity, bool crop=NoCrop); @end verbatim @noindent and the analogous function @code{ylimits} can be uncommented to set the respective axes limits for picture @code{pic} to the specified @code{min} and @code{max} values. Alternatively, the function @verbatim void limits(picture pic=currentpicture, pair min, pair max, bool crop=NoCrop); @end verbatim can be used to limit the axes to the box having opposite vertices at the given pairs). Existing objects in picture @code{pic} will be cropped to lie within the given limits if @code{crop}=@code{Crop}. The function @code{crop(picture pic)} can be used to crop a graph to the current graph limits. @verbatiminclude parametricgraph.asy @sp 1 @center @image{./parametricgraph} @cindex @code{graphwithderiv} The function @verbatim guide graphwithderiv(pair f(real), pair fprime(real), real a, real b, int n=ngraph#10); @end verbatim can be used to construct the graph of the parametric function @code{f} on @code{[a,b]} with the control points of the @code{n} Bezier segments determined by the specified derivative @code{fprime}: @verbatiminclude graphwithderiv.asy @sp 1 @center @image{./graphwithderiv} @cindex scaled graph The next example illustrates how one can extract a common axis scaling factor. @verbatiminclude scaledgraph.asy @sp 1 @center @image{./scaledgraph} @anchor{automatic scaling} @cindex automatic scaling @cindex @code{scale} @cindex @code{Linear} @cindex @code{Log} @cindex automatic scaling Axis scaling can be requested and/or automatic selection of the axis limits can be inhibited with one of these @code{scale} routines: @verbatim void scale(picture pic=currentpicture, scaleT x, scaleT y); void scale(picture pic=currentpicture, bool xautoscale=true, bool yautoscale=xautoscale, bool zautoscale=yautoscale); @end verbatim This sets the scalings for picture @code{pic}. The @code{graph} routines accept an optional @code{picture} argument for determining the appropriate scalings to use; if none is given, it uses those set for @code{currentpicture}. Two frequently used scaling routines @code{Linear} and @code{Log} are predefined in @code{graph}. All picture coordinates (including those in paths and those given to the @code{label} and @code{limits} functions) are always treated as linear (post-scaled) coordinates. Use @cindex @code{Scale} @verbatim pair Scale(picture pic=currentpicture, pair z); @end verbatim to convert a graph coordinate into a scaled picture coordinate. The @math{x} and @math{y} components can be individually scaled using the analogous routines @verbatim real ScaleX(picture pic=currentpicture, real x); real ScaleY(picture pic=currentpicture, real y); @end verbatim The predefined scaling routines can be given two optional boolean arguments: @code{automin=false} and @code{automax=automin}. These default to @code{false} but can be respectively set to @code{true} to enable automatic selection of "nice" axis minimum and maximum values. The @code{Linear} scaling can also take as optional final arguments a multiplicative scaling factor and intercept (e.g.@ for a depth axis, @code{Linear(-1)} requests axis reversal). @cindex logarithmic graph @cindex log-log graph For example, to draw a log/log graph of a function, use @code{scale(Log,Log)}: @verbatiminclude loggraph.asy @sp 1 @center @image{./loggraph} @cindex grid By extending the ticks, one can easily produce a logarithmic grid: @verbatiminclude loggrid.asy @sp 1 @center @image{./loggrid} One can also specify custom tick locations and formats for logarithmic axes: @verbatiminclude logticks.asy @sp 1 @center @image{./logticks} @cindex @code{log2} graph It is easy to draw logarithmic graphs with respect to other bases: @verbatiminclude log2graph.asy @sp 1 @center @image{./log2graph} @cindex broken axis Here is an example of "broken" linear @math{x} and logarithmic @math{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. @verbatiminclude brokenaxis.asy @sp 1 @center @image{./brokenaxis} @cindex secondary axis @cindex @code{secondaryX} @cindex @code{secondaryY} @item @code{Asymptote} can draw secondary axes with the routines @verbatim picture secondaryX(picture primary=currentpicture, void f(picture)); picture secondaryY(picture primary=currentpicture, void f(picture)); @end verbatim In this example, @code{secondaryY} is used to draw a secondary linear @math{y} axis against a primary logarithmic @math{y} axis: @verbatiminclude Bode.asy @sp 1 @center @image{./Bode} A secondary logarithmic @math{y} axis can be drawn like this: @verbatiminclude secondaryaxis.asy @sp 1 @center @image{./secondaryaxis} @item Here is a histogram example, which uses the @code{stats} module. @cindex @code{axis} @verbatiminclude histogram.asy @sp 1 @center @image{./histogram} @item Here is an example of reading column data in from a file and a least-squares fit, using the @code{stats} module. @cindex @code{leastsquares} @verbatiminclude leastsquares.asy @sp 1 @center @image{./leastsquares} @item Here is an example that illustrates the general @code{axis} routine. @cindex @code{axis} @verbatiminclude generalaxis.asy @sp 1 @center @image{./generalaxis} @item To draw a vector field of @code{n} arrows evenly spaced along the arclength of a path, use the routine @cindex @code{vectorfield} @verbatim picture vectorfield(path vector(real), path g, int n, bool truesize=false, pen p=currentpen, arrowbar arrow=Arrow); @end verbatim as illustrated in this simple example of a flow field: @verbatiminclude flow.asy @sp 1 @center @image{./flow} @item To draw a vector field of @code{nx}@math{\times}@code{ny} arrows in @code{box(a,b)}, use the routine @cindex @code{vectorfield} @verbatim 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) @end verbatim as illustrated in this example: @verbatiminclude vectorfield.asy @sp 1 @center @image{./vectorfield} @item The following scientific graphs, which illustrate many features of @code{Asymptote}'s graphics routines, were generated from the examples @code{@uref{https://asymptote.sourceforge.io/gallery/2Dgraphs/diatom.svg,,diatom}@uref{https://asymptote.sourceforge.io/gallery/2Dgraphs/diatom.asy,,.asy}} and @code{@uref{https://asymptote.sourceforge.io/gallery/2Dgraphs/westnile.svg,,westnile}@uref{https://asymptote.sourceforge.io/gallery/2Dgraphs/westnile.asy,,.asy}}, using the comma-separated data in @code{@uref{https://asymptote.sourceforge.io/gallery/2Dgraphs/diatom.csv,,diatom.csv}} and @code{@uref{https://asymptote.sourceforge.io/gallery/2Dgraphs/westnile.csv,,westnile.csv}}. @page @sp 1 @center @image{./diatom} @sp 1 @center @image{./westnile,,7.5cm} @end enumerate @page @node palette, three, graph, Base modules @section @code{palette} @anchor{images} @cindex images @code{Asymptote} can also generate color density images and palettes. The following palettes are predefined in @code{palette.asy}: @table @code @cindex @code{Grayscale} @item pen[] Grayscale(int NColors=256) a grayscale palette; @cindex @code{Rainbow} @item pen[] Rainbow(int NColors=32766) a rainbow spectrum; @cindex @code{BWRainbow} @item pen[] BWRainbow(int NColors=32761) a rainbow spectrum tapering off to black/white at the ends; @cindex @code{BWRainbow2} @item pen[] BWRainbow2(int NColors=32761) a double rainbow palette tapering off to black/white at the ends, with a linearly scaled intensity. @cindex @code{Wheel} @item pen[] Wheel(int NColors=32766) a full color wheel palette; @cindex @code{Gradient} @item pen[] Gradient(int NColors=256 ... pen[] p) a palette varying linearly over the specified array of pens, using NColors in each interpolation interval; @end table The function @code{cmyk(pen[] Palette)} may be used to convert any of these palettes to the @acronym{CMYK} colorspace. A color density plot using palette @code{palette} can be generated from a function @code{f}(@math{x},@math{y}) and added to a picture @code{pic}: @cindex @code{image} @verbatim 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) @end verbatim The function @code{f} will be sampled at @code{nx} and @code{ny} evenly spaced points over a rectangle defined by the points @code{initial} and @code{final}, respecting the current graphical scaling of @code{pic}. The color space is scaled according to the @math{z} axis scaling (@pxref{automatic scaling}). If @math{@code{divs} > 1}, the palette is quantized to @math{@code{divs}-1} values. A @code{bounds} structure for the function values is returned: @cindex @code{bounds} @verbatim struct bounds { real min; real max; // Possible tick intervals: int[] divisor; } @end verbatim @noindent 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 @code{range}, which can be @code{Full}, @code{Automatic}, or an explicit range @code{Range(real min, real max)}. Here @code{Full} specifies a range varying from the minimum to maximum values of the function over the sampling interval, while @code{Automatic} selects "nice" limits. The examples @code{@uref{https://asymptote.sourceforge.io/gallery/2Dgraphs/fillcontour.svg,,fillcontour}@uref{https://asymptote.sourceforge.io/gallery/2Dgraphs/fillcontour.asy,,.asy}} and @code{@uref{https://asymptote.sourceforge.io/gallery/2Dgraphs/imagecontour.svg,,imagecontour}@uref{https://asymptote.sourceforge.io/gallery/2Dgraphs/imagecontour.asy,,.asy}} illustrate how level sets (contour lines) can be drawn on a color density plot (@pxref{contour}). A color density plot can also be generated from an explicit real[][] array @code{data}: @cindex @code{image} @verbatim 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); @end verbatim @noindent 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: @math{x}, second index: @math{y}) rather than the usual matrix convention (first index: @math{-y}, second index: @math{x}). To construct an image from an array of irregularly spaced points and an array of values @code{f} at these points, use one of the routines @verbatim 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) @end verbatim An optionally labelled palette bar may be generated with the routine @verbatim 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); @end verbatim The color space of @code{palette} is taken to be over bounds @code{bounds} with scaling given by the @math{z} scaling of @code{pic}. The palette orientation is specified by @code{axis}, which may be one of @code{Right}, @code{Left}, @code{Top}, or @code{Bottom}. The bar is drawn over the rectangle from @code{initial} to @code{final}. The argument @code{paletteticks} is a special tick type (@pxref{ticks}) that takes the following arguments: @verbatim 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); @end verbatim The image and palette bar can be fit to a frame and added and optionally aligned to a picture at the desired location: @anchor{image} @verbatiminclude image.asy @sp 1 @center @image{./image} Here is an example that uses logarithmic scaling of the function values: @anchor{logimage} @verbatiminclude logimage.asy @sp 1 @center @image{./logimage} One can also draw an image directly from a two-dimensional pen array or a function @code{pen f(int, int)}: @verbatim 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); @end verbatim @noindent as illustrated in the following examples: @anchor{penimage} @verbatiminclude penimage.asy @sp 1 @center @image{./penimage} @anchor{penfunctionimage} @verbatiminclude penfunctionimage.asy @sp 1 @center @image{./penfunctionimage} For convenience, the module @code{palette} also defines functions that may be used to construct a pen array from a given function and palette: @verbatim pen[] palette(real[] f, pen[] palette); pen[][] palette(real[][] f, pen[] palette); @end verbatim @node three, obj, palette, Base modules @section @code{three} @cindex @code{three} @cindex @code{guide3} @cindex @code{path3} @cindex @code{cycle} @cindex @code{curl} @cindex @code{tension} @cindex @code{controls} This module fully extends the notion of guides and paths in @code{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 @code{(x,y,z)} are used in place of pairs @code{(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 @code{Asymptote}, @code{MetaPost}, and @code{MetaFont} [cf.@ J. C. Bowman, Proceedings in Applied Mathematics and Mechanics, 7:1, 2010021-2010022 (2007)]. For example, a unit circle in the @math{XY} plane may be filled and drawn like this: @verbatiminclude unitcircle3.asy @sp 1 @center @image{./unitcircle3} @noindent and then distorted into a saddle: @verbatiminclude saddle.asy @sp 1 @center @image{./saddle} @noindent Module @code{three} provides constructors for converting two-dimensional paths to three-dimensional ones, and vice-versa: @cindex @code{path3} @cindex @code{path} @verbatim path3 path3(path p, triple plane(pair)=XYplane); path path(path3 p, pair P(triple)=xypart); @end verbatim @cindex @code{surface} @cindex @code{render} @cindex @code{defaultrender} A Bezier surface, the natural two-dimensional generalization of Bezier curves, is defined in @code{three_surface.asy} as a structure containing an array of Bezier patches. Surfaces may drawn with one of the routines @verbatim 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); @end verbatim The parameters @code{nu} and @code{nv} specify the number of subdivisions for drawing optional mesh lines for each Bezier patch. The optional @code{name} parameter is used as a prefix for naming the surface patches in the @acronym{PRC} model tree. Here material is a structure defined in @code{three_light.asy}: @cindex @code{material} @cindex @code{diffusepen} @cindex @code{emissivepen} @cindex @code{specularpen} @cindex @code{opacity} @cindex @code{shininess} @cindex @code{metallic} @cindex @code{freshnel0} @verbatim struct material { pen[] p; // diffusepen,emissivepen,specularpen real opacity; real shininess; real metallic; real fresnel0; } @end verbatim @noindent @cindex @code{PBR} @cindex @code{physically based rendering} These material properties are used to implement physically based rendering (PBR) using light properties defined in @code{plain_prethree.asy} and @code{three_light.asy}: @cindex @code{light} @cindex @code{diffuse} @cindex @code{specular} @cindex @code{background} @cindex @code{specularfactor} @cindex @code{position} @cindex @code{currentlight} @cindex @code{Viewport} @cindex @code{White} @cindex @code{Headlamp} @cindex @code{nolight} @verbatim 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; @end verbatim @cindex @code{background} @cindex @code{transparent} The @code{currentlight.background} (or @code{background} member of the specified @code{light}) can be used to set the background colour for 2D (or 3D) images. The default background is white for @code{HTML} images and transparent for all other formats. One can request a completely transparent background for 3D @code{WebGL} images with @code{currentlight.background=black+opacity(0.0);} @cindex image-based lighting @cindex @code{surface} @cindex @code{ibl} Asymptote also supports image-based lighting with the setting @code{settings.ibl=true}. This uses pre-rendered @acronym{EXR} images from the directory specified by @code{-imageDir} (which defaults to @code{ibl}) or, for @code{WebGL} rendering, the @acronym{URL} specified by @code{-imageURL} (which defaults to @url{https://vectorgraphics.gitlab.io/asymptote/ibl}). Additional rendered images can be generated on an @code{NVIDIA} @acronym{GPU} using the @code{reflect} program in the @code{cudareflect} subdirectory of the @code{Asymptote} source directory. Sample Bezier surfaces are contained in the example files @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/BezierSurface.html,,BezierSurface}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/BezierSurface.asy,,.asy}}, @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/teapot.html,,teapot}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/teapot.asy,,.asy}}, @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/teapotIBL.html,,teapotIBL}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/teapotIBL.asy,,.asy}}, and @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/parametricsurface.html,,parametricsurface}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/parametricsurface.asy,,.asy}}. The structure @code{render} contains specialized rendering options documented at the beginning of module @code{three}. @cindex patch-dependent colors @cindex vertex-dependent colors The examples @code{@uref{https://asymptote.sourceforge.io/gallery/3Dgraphs/elevation.html,,elevation}@uref{https://asymptote.sourceforge.io/gallery/3Dgraphs/elevation.asy,,.asy}} and @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/sphericalharmonic.html,,sphericalharmonic}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/sphericalharmonic.asy,,.asy}} illustrate how to draw a surface with patch-dependent colors. The examples @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/vertexshading.html,,vertexshading}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/vertexshading.asy,,.asy}} and @code{@uref{https://asymptote.sourceforge.io/gallery/3Dgraphs/smoothelevation.html,,smoothelevation}@uref{https://asymptote.sourceforge.io/gallery/3Dgraphs/smoothelevation.asy,,.asy}} illustrate vertex-dependent colors, which are supported by @code{Asymptote}'s native @code{OpenGL}/@code{WebGL} renderers and the two-dimensional vector output format (@code{settings.render=0}). Since the @acronym{PRC} output format does not currently support vertex shading of Bezier surfaces, @acronym{PRC} patches are shaded with the mean of the four vertex colors. @cindex @code{surface} @cindex @code{planar} @cindex @code{Bezier patch} @cindex @code{Bezier triangle} A surface can be constructed from a cyclic @code{path3} with the constructor @verbatim surface surface(path3 external, triple[] internal=new triple[], pen[] colors=new pen[], bool3 planar=default); @end verbatim @noindent and then filled: @verbatim 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); @end verbatim @noindent 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 @code{plane} can be constructed from a two-dimensional cyclic path @code{g} with the constructor @cindex @code{surface} @verbatim surface surface(path p, triple plane(pair)=XYplane); @end verbatim @noindent and then filled: @verbatim draw(surface((0,0)--E+2N--2E--E+N..0.2E..cycle),red); @end verbatim @noindent @cindex @code{bezulate} Planar Bezier surfaces patches are constructed using Orest Shardt's @code{bezulate} routine, which decomposes (possibly nonsimply connected) regions bounded (according to the @code{zerowinding} fill rule) by simple cyclic paths (intersecting only at the endpoints) into subregions bounded by cyclic paths of length @code{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: @cindex @code{draw} @cindex @code{triangles} @cindex @code{tessellation} @verbatim 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); @end verbatim Here, the triple array @code{v} lists the (typically distinct) vertices, while the array @code{vi} contains integer arrays of length 3 containing the indices of the elements in @code{v} that form the vertices of each triangle. Similarly, the arguments @code{n} and @code{ni} contain optional normal data and @code{p} and @code{pi} contain optional pen vertex data. If more than one normal or pen is specified for a vertex, the last one specified is used. An example of this tessellation facility is given in @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/triangles.html,,triangles}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/triangles.asy,,.asy}}. @cindex @code{thin} @cindex @code{thick} @cindex @code{tube} Arbitrary thick three-dimensional curves and line caps (which the @code{OpenGL} standard does not require implementations to provide) are constructed with @verbatim tube tube(path3 p, real width, render render=defaultrender); @end verbatim @noindent this returns a tube structure representing a tube of diameter @code{width} centered approximately on @code{g}. The tube structure consists of a surface @code{s} and the actual tube center, path3 @code{center}. Drawing thick lines as tubes can be slow to render, especially with the @code{Adobe Reader} renderer. The setting @code{thick=false} can be used to disable this feature and force all lines to be drawn with @code{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 @code{thin} is set to @code{false}. The pens @code{thin()} and @code{thick()} defined in @code{plain_pens.asy} can also be used to override these defaults for specific draw commands. @noindent There are six choices for viewing 3D @code{Asymptote} output: @enumerate @cindex @code{OpenGL} @cindex @code{render} @cindex @code{outformat} @cindex @code{multisample} @cindex @code{devicepixelratio} @cindex @code{position} @item Use the native @code{Asymptote} adaptive @code{OpenGL}-based renderer (with the command-line option @code{-V} and the default settings @code{outformat=""} and @code{render=-1}). On @code{UNIX} systems with graphics support for multisampling, the sample width can be controlled with the setting @code{multisample}. The ratio of physical to logical screen pixels can be specified with the setting @code{devicepixelratio}. An initial screen position can be specified with the pair setting @code{position}, where negative values are interpreted as relative to the corresponding maximum screen dimension. The default settings @cindex mouse bindings @verbatim 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"}; @end verbatim bind the mouse buttons as follows: @itemize @item Left: rotate @item Shift Left: zoom @item Ctrl Left: shift viewport @item Alt Left: pan @item Wheel Up: zoom in @item Wheel Down: zoom out @item Right: zoom @item Shift Right: rotate about the X axis @item Ctrl Right: rotate about the Y axis @item Alt Right: rotate about the Z axis @end itemize The keyboard shortcuts are: @cindex keyboard bindings: @itemize @item h: home @item f: toggle fitscreen @item x: spin about the X axis @item y: spin about the Y axis @item z: spin about the Z axis @item s: stop spinning @item m: rendering mode (solid/patch/mesh) @item e: export @item c: show camera parameters @item p: play animation @item r: reverse animation @item : step animation @item +: expand @item =: expand @item >: expand @item -: shrink @item _: shrink @item <: shrink @item q: exit @item Ctrl-q: exit @end itemize @cindex @code{WebGL} @cindex @code{HTML5} @cindex @code{mobile browser} @item Generate @code{WebGL} interactive vector graphics output with the the command-line option and @code{-f html} (or the setting @code{outformat="html"}). The resulting 3D @acronym{HTML} file can then be viewed directly in any modern desktop or mobile browser, or even embedded within another web page: @verbatim @end verbatim @cindex @code{absolute} Normally, @code{WebGL} files generated by @code{Asymptote} are dynamically remeshed to fit the browser window dimensions. However, the setting @code{absolute=true} can be used to force the image to be rendered at its designed size (accounting for multiple device pixels per @code{css} pixel). The interactive @code{WebGL} files produced by @code{Asymptote} use the default mouse and (many of the same) key bindings as the @code{OpenGL} renderer. Zooming via the mouse wheel of a @code{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 @code{ESC} key is pressed. By default, viewing the 3D @acronym{HTML} files generated by Asymptote requires network access to download the @code{AsyGL} rendering library, which is normally cached by the browser for future use. However, the setting @code{offline=true} can be used to embed this small (about 48kB) library within a stand-alone @acronym{HTML} file that can be viewed offline. @cindex @code{antialias} @cindex @code{maxviewport} @cindex @code{maxtile} @cindex @code{glOptions} @cindex @code{iconify} @cindex @code{black stripes} @item Render the scene to a specified rasterized format @code{outformat} at the resolution of @code{n} pixels per @code{bp}, as specified by the setting @code{render=n}. A negative value of @code{n} is interpreted as @code{|2n|} for @acronym{EPS} and @acronym{PDF} formats and @code{|n|} for other formats. The default value of @code{render} is -1. By default, the scene is internally rendered at twice the specified resolution; this can be disabled by setting @code{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 @code{maxtile} to your screen dimensions (indicated by @code{maxtile=(0,0)}. If your video card generates unwanted black stripes in the output, try setting the horizontal and vertical components of @code{maxtiles} to something less than your screen dimensions. The tile size is also limited by the setting @code{maxviewport}, which restricts the maximum width and height of the viewport. Some graphics drivers support batch mode (@code{-noV}) rendering in an iconified window; this can be enabled with the setting @code{iconify=true}. @cindex @code{prc} @cindex @code{views} @item Embed the 3D @acronym{PRC} format in a @acronym{PDF} file and view the resulting @acronym{PDF} file with version @code{9.0} or later of @code{Adobe Reader}. This requires @code{settings.outformat="pdf"} and @code{settings.prc=true}, which can be specified by the command-line options @code{-f pdf} and @code{-f prc}, put in the @code{Asymptote} configuration file (@pxref{configuration file}), or specified in the script before module @code{three} (or @code{graph3}) is imported. The @code{media9} LaTeX package is also required (@pxref{embed}). The example @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/100d.html,,100d}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/pdb.asy,,.asy}} illustrates how one can generate a list of predefined views (see @code{100d.views}). A stationary preview image with a resolution of @code{n} pixels per @code{bp} can be embedded with the setting @code{render=n}; this allows the file to be viewed with other @code{PDF} viewers. Alternatively, the file @code{externalprc.tex} illustrates how the resulting @acronym{PRC} and rendered image files can be extracted and processed in a separate @code{LaTeX} file. However, see @ref{LaTeX usage} for an easier way to embed three-dimensional @code{Asymptote} pictures within @code{LaTeX}. For specialized applications where only the raw @acronym{PRC} file is required, specify @code{settings.outformat="prc"}. The @acronym{PRC} specification is available from @url{https://web.archive.org/web/20081204104459/http://livedocs.adobe.com/acrobat_sdk/9/Acrobat9_HTMLHelp/API_References/PRCReference/PRC_Format_Specification/} @cindex @code{v3d} @item Output a @code{V3D} portable compressed binary file for viewing with an external viewer or conversion to an alternate 3D format with the Python @code{pyv3d} library. The @code{V3D} specification and the @code{pyv3d} library are available at @url{https://gitlab.com/vectorgraphics/v3d}. A @code{V3D} file @code{file.v3d} may be imported and viewed by @code{Asymptote} either by specifying @code{file.v3d} on the command line @verbatim asy -V file.v3d @end verbatim or using the @code{v3d} module and @code{importv3d} function in interactive mode (or within an @code{Asymptote} file): @cindex @code{importv3d} @verbatim import v3d; importv3d("file.v3d"); @end verbatim @item Project the scene to a two-dimensional vector (@acronym{EPS} or @acronym{PDF}) format with @code{render=0}. Only limited support for hidden surface removal, lighting, and transparency is available with this approach (@pxref{PostScript3D}). @end enumerate @cindex @code{double deferred drawing} 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 @cindex @code{size3} @verbatim void size3(picture pic=currentpicture, real x, real y=x, real z=y, bool keepAspect=pic.keepAspect); @end verbatim @noindent @cindex margins @cindex @code{viewportmargin} @cindex @code{viewportsize} 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 @code{size} parameters. The global pair @code{viewportmargin} may be used to add horizontal and vertical margins to the viewport dimensions. Alternatively, a minimum @code{viewportsize} may be specified. A 3D picture @code{pic} can be explicitly fit to a 3D frame by calling @cindex @code{fit3} @verbatim frame pic.fit3(projection P=currentprojection); @end verbatim @noindent and then added to picture @code{dest} about @code{position} with @cindex @code{add} @verbatim void add(picture dest=currentpicture, frame src, triple position=(0,0,0)); @end verbatim @cindex @code{O} @cindex @code{X} @cindex @code{Y} @cindex @code{Z} @cindex @code{unitcircle} For convenience, the @code{three} module defines @code{O=(0,0,0)}, @code{X=(1,0,0)}, @code{Y=(0,1,0)}, and @code{Z=(0,0,1)}, along with a unitcircle in the XY plane: @verbatim path3 unitcircle3=X..Y..-X..-Y..cycle; @end verbatim @cindex @code{circle} A general (approximate) circle can be drawn perpendicular to the direction @code{normal} with the routine @verbatim path3 circle(triple c, real r, triple normal=Z); @end verbatim @cindex @code{arc} A circular arc centered at @code{c} with radius @code{r} from @code{c+r*dir(theta1,phi1)} to @code{c+r*dir(theta2,phi2)}, drawing counterclockwise relative to the normal vector @code{cross(dir(theta1,phi1),dir(theta2,phi2))} if @code{theta2 > theta1} or if @code{theta2 == theta1} and @code{phi2 >= phi1}, can be constructed with @verbatim path3 arc(triple c, real r, real theta1, real phi1, real theta2, real phi2, triple normal=O); @end verbatim The normal must be explicitly specified if @code{c} and the endpoints are colinear. If @code{r} < 0, the complementary arc of radius @code{|r|} is constructed. For convenience, an arc centered at @code{c} from triple @code{v1} to @code{v2} (assuming @code{|v2-c|=|v1-c|}) in the direction CCW (counter-clockwise) or CW (clockwise) may also be constructed with @verbatim path3 arc(triple c, triple v1, triple v2, triple normal=O, bool direction=CCW); @end verbatim @noindent When high accuracy is needed, the routines @code{Circle} and @code{Arc} defined in @code{graph3} may be used instead. See @ref{GaussianSurface} for an example of a three-dimensional circular arc. @cindex @code{plane} The representation @code{O--O+u--O+u+v--O+v--cycle} of the plane passing through point @code{O} with normal @code{cross(u,v)} is returned by @verbatim path3 plane(triple u, triple v, triple O=O); @end verbatim A three-dimensional box with opposite vertices at triples @code{v1} and @code{v2} may be drawn with the function @cindex @code{box} @verbatim path3[] box(triple v1, triple v2); @end verbatim @noindent For example, a unit box is predefined as @cindex @code{box} @cindex @code{unitbox} @verbatim path3[] unitbox=box(O,(1,1,1)); @end verbatim @code{Asymptote} also provides optimized definitions for the three-dimensional paths @code{unitsquare3} and @code{unitcircle3}, along with the surfaces @code{unitdisk}, @code{unitplane}, @code{unitcube}, @code{unitcylinder}, @code{unitcone}, @code{unitsolidcone}, @code{unitfrustum(real t1, real t2)}, @code{unitsphere}, and @code{unithemisphere}. @noindent These projections to two dimensions are predefined: @table @code @item oblique @item oblique(real angle) @cindex @code{oblique} @cindex @code{obliqueZ} The point @code{(x,y,z)} is projected to @code{(x-0.5z,y-0.5z)}. If an optional real argument is given, the negative @math{z} axis is drawn at this angle in degrees. The projection @code{obliqueZ} is a synonym for @code{oblique}. @item obliqueX @item obliqueX(real angle) @cindex @code{obliqueX} The point @code{(x,y,z)} is projected to @code{(y-0.5x,z-0.5x)}. If an optional real argument is given, the negative @math{x} axis is drawn at this angle in degrees. @item obliqueY @item obliqueY(real angle) @cindex @code{obliqueY} The point @code{(x,y,z)} is projected to @code{(x+0.5y,z+0.5y)}. If an optional real argument is given, the positive @math{y} axis is drawn at this angle in degrees. @cindex @code{orthographic} @cindex @code{up} @cindex @code{target} @cindex @code{showtarget} @cindex @code{center} @item orthographic(triple camera, triple up=Z, triple target=O, @*@ @ @ @ @ @ @ @ @ @ @ @ @ real zoom=1, pair viewportshift=0, bool showtarget=true, @*@ @ @ @ @ @ @ @ @ @ @ @ @ bool center=false) This projects from three to two dimensions using the view as seen at a point infinitely far away in the direction @code{unit(camera)}, orienting the camera so that, if possible, the vector @code{up} points upwards. Parallel lines are projected to parallel lines. The bounding volume is expanded to include @code{target} if @code{showtarget=true}. If @code{center=true}, the target will be adjusted to the center of the bounding volume. @item orthographic(real x, real y, real z, triple up=Z, triple target=O, @*@ @ @ @ @ @ @ @ @ @ @ @ @ real zoom=1, pair viewportshift=0, bool showtarget=true, @*@ @ @ @ @ @ @ @ @ @ @ @ @ bool center=false) This is equivalent to @verbatim orthographic((x,y,z),up,target,zoom,viewportshift,showtarget,center) @end verbatim The routine @cindex @code{camera} @verbatim triple camera(real alpha, real beta); @end verbatim can be used to compute the camera position with the @math{x} axis below the horizontal at angle @code{alpha}, the @math{y} axis below the horizontal at angle @code{beta}, and the @math{z} axis up. @cindex @code{autoadjust} @item 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) @cindex @code{perspective} @cindex @code{NURBS} This projects from three to two dimensions, taking account of perspective, as seen from the location @code{camera} looking at @code{target}, orienting the camera so that, if possible, the vector @code{up} points upwards. If @code{autoadjust=true}, the camera will automatically be adjusted to lie outside the bounding volume for all possible interactive rotations about @code{target}. If @code{center=true}, the target will be adjusted to the center of the bounding volume. @item 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 @verbatim perspective((x,y,z),up,target,zoom,angle,viewportshift,showtarget, autoadjust,center) @end verbatim @end table @cindex @code{currentprojection} @noindent The default projection, @code{currentprojection}, is initially set to @code{perspective(5,4,2)}. @cindex @code{LeftView} @cindex @code{RightView} @cindex @code{FrontView} @cindex @code{BackView} @cindex @code{BottomView} @cindex @code{TopView} We also define standard orthographic views used in technical drawing: @verbatim 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); @end verbatim @noindent The function @cindex @code{addViews} @verbatim void addViews(picture dest=currentpicture, picture src, projection[][] views=SixViewsUS, bool group=true, filltype filltype=NoFill); @end verbatim @noindent adds to picture @code{dest} an array of views of picture @code{src} using the layout projection[][] @code{views}. The default layout @code{SixViewsUS} aligns the projection @code{FrontView} below @code{TopView} and above @code{BottomView}, to the right of @code{LeftView} and left of @code{RightView} and @code{BackView}. The predefined layouts are: @cindex @code{ThreeViewsUS} @cindex @code{SixViewsUS} @cindex @code{ThreeViewsFR} @cindex @code{SixViewsFR} @cindex @code{ThreeViews} @cindex @code{SixViews} @verbatim 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}}; @end verbatim A triple or path3 can be projected to a pair or path, with @code{project(triple, projection P=currentprojection)} or @code{project(path3, projection P=currentprojection)}. It is occasionally useful to be able to invert a projection, sending a pair @code{z} onto the plane perpendicular to @code{normal} and passing through @code{point}: @cindex @code{invert} @verbatim triple invert(pair z, triple normal, triple point, projection P=currentprojection); @end verbatim @noindent A pair @code{z} on the projection plane can be inverted to a triple with the routine @verbatim triple invert(pair z, projection P=currentprojection); @end verbatim @noindent A pair direction @code{dir} on the projection plane can be inverted to a triple direction relative to a point @code{v} with the routine @verbatim triple invert(pair dir, triple v, projection P=currentprojection). @end verbatim @cindex @code{transform3} @cindex @code{identity4} Three-dimensional objects may be transformed with one of the following built-in transform3 types (the identity transformation is @code{identity4}): @table @code @item shift(triple v) @cindex @code{shift} translates by the triple @code{v}; @item xscale3(real x) @cindex @code{xscale3} scales by @code{x} in the @math{x} direction; @item yscale3(real y) @cindex @code{yscale3} scales by @code{y} in the @math{y} direction; @item zscale3(real z) @cindex @code{zscale3} scales by @code{z} in the @math{z} direction; @item scale3(real s) @cindex @code{scale3} scales by @code{s} in the @math{x}, @math{y}, and @math{z} directions; @item scale(real x, real y, real z) @cindex @code{scale} scales by @code{x} in the @math{x} direction, by @code{y} in the @math{y} direction, and by @code{z} in the @math{z} direction; @cindex @code{rotate} @item rotate(real angle, triple v) rotates by @code{angle} in degrees about an axis @code{v} through the origin; @item rotate(real angle, triple u, triple v) rotates by @code{angle} in degrees about the axis @code{u--v}; @item reflect(triple u, triple v, triple w) reflects about the plane through @code{u}, @code{v}, and @code{w}. @cindex @code{XY} @end table When not multiplied on the left by a transform3, three-dimensional @TeX{} Labels are drawn as Bezier surfaces directly on the projection plane: @cindex @code{label} @verbatim 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) @end verbatim @noindent @cindex @code{Billboard} @cindex @code{Embedded} The optional @code{name} parameter is used as a prefix for naming the label patches in the @acronym{PRC} model tree. The default interaction is @code{Billboard}, which means that labels are rotated interactively so that they always face the camera. The interaction @code{Embedded} means that the label interacts as a normal @code{3D} surface, as illustrated in the example @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/billboard.html,,billboard}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/billboard.asy,,.asy}}. @cindex @code{transform} @cindex @code{XY} @cindex @code{YZ} @cindex @code{ZX} @cindex @code{YX} @cindex @code{ZY} @cindex @code{ZX} Alternatively, a label can be transformed from the @code{XY} plane by an explicit transform3 or mapped to a specified two-dimensional plane with the predefined transform3 types @code{XY}, @code{YZ}, @code{ZX}, @code{YX}, @code{ZY}, @code{ZX}. There are also modified versions of these transforms that take an optional argument @code{projection P=currentprojection} that rotate and/or flip the label so that it is more readable from the initial viewpoint. @cindex @code{planeproject} A transform3 that projects in the direction @code{dir} onto the plane with normal @code{n} through point @code{O} is returned by @verbatim transform3 planeproject(triple n, triple O=O, triple dir=n); @end verbatim @noindent One can use @cindex @code{normal} @verbatim triple normal(path3 p); @end verbatim @noindent to find the unit normal vector to a planar three-dimensional path @code{p}. As illustrated in the example @code{@uref{https://asymptote.sourceforge.io/gallery/3Dgraphs/planeproject.html,,planeproject}@uref{https://asymptote.sourceforge.io/gallery/3Dgraphs/planeproject.asy,,.asy}}, a transform3 that projects in the direction @code{dir} onto the plane defined by a planar path @code{p} is returned by @verbatim transform3 planeproject(path3 p, triple dir=normal(p)); @end verbatim The functions @cindex @code{extrude} @verbatim surface extrude(path p, triple axis=Z); surface extrude(Label L, triple axis=Z); @end verbatim @noindent return the surface obtained by extruding path @code{p} or Label @code{L} along @code{axis}. @cindex @code{length} @cindex @code{size} @cindex @code{point} @cindex @code{dir} @cindex @code{accel} @cindex @code{radius} @cindex @code{precontrol} @cindex @code{postcontrol} @cindex @code{arclength} @cindex @code{arctime} @cindex @code{reverse} @cindex @code{subpath} @cindex @code{intersect} @cindex @code{intersections} @cindex @code{intersectionpoint} @cindex @code{intersectionpoints} @cindex @code{min} @cindex @code{max} @cindex @code{cyclic} @cindex @code{straight} Three-dimensional versions of the path functions @code{length}, @code{size}, @code{point}, @code{dir}, @code{accel}, @code{radius}, @code{precontrol}, @code{postcontrol}, @code{arclength}, @code{arctime}, @code{reverse}, @code{subpath}, @code{intersect}, @code{intersections}, @code{intersectionpoint}, @code{intersectionpoints}, @code{min}, @code{max}, @code{cyclic}, and @code{straight} are also defined. The routine @cindex @code{intersections} @verbatim real[] intersect(path3 p, surface s, real fuzz=-1); @end verbatim @noindent returns a real array of length 3 containing the intersection times, if any, of a path @code{p} with a surface @code{s}. The routine @verbatim real[][] intersections(path3 p, surface s, real fuzz=-1); @end verbatim @noindent returns all (unless there are infinitely many) intersection times of a path @code{p} with a surface @code{s} as a sorted array of real arrays of length 3, and @cindex @code{intersectionpoints} @verbatim triple[] intersectionpoints(path3 p, surface s, real fuzz=-1); @end verbatim @noindent returns the corresponding intersection points. Here, the computations are performed to the absolute error specified by @code{fuzz}, or if @code{fuzz < 0}, to machine precision. The routine @cindex @code{orient} @verbatim real orient(triple a, triple b, triple c, triple d); @end verbatim @noindent is a numerically robust computation of @code{dot(cross(a-d,b-d),c-d)}, which is the determinant @verbatim |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| @end verbatim The result is negative (positive) if @code{a}, @code{b}, @code{c} appear in counterclockwise (clockwise) order when viewed from @code{d} or zero if all four points are coplanar. The routine @cindex @code{insphere} @verbatim real insphere(triple a, triple b, triple c, triple d, triple e); @end verbatim @noindent returns a positive (negative) value if @code{e} lies inside (outside) the sphere passing through points @code{a,b,c,d} oriented so that @code{dot(cross(a-d,b-d),c-d)} is positive, or zero if all five points are cospherical. The value returned is the determinant @verbatim |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| @end verbatim Here is an example showing all five guide3 connectors: @verbatiminclude join3.asy @sp 1 @center @image{./join3} @cindex @code{BeginBar3} @cindex @code{EndBar3} @cindex @code{Bar3} @cindex @code{Bars3} @cindex @code{BeginArrow3} @cindex @code{MidArrow3} @cindex @code{EndArrow3} @cindex @code{Arrow3} @cindex @code{Arrows3} @cindex @code{BeginArcArrow3} @cindex @code{MidArcArrow3} @cindex @code{EndArcArrow3} @cindex @code{ArcArrow3} @cindex @code{ArcArrows3} @cindex @code{DefaultHead3} @cindex @code{HookHead3} @cindex @code{TeXHead3} Three-dimensional versions of bars or arrows can be drawn with one of the specifiers @code{None}, @code{Blank}, @code{BeginBar3}, @code{EndBar3} (or equivalently @code{Bar3}), @code{Bars3}, @code{BeginArrow3}, @code{MidArrow3}, @code{EndArrow3} (or equivalently @code{Arrow3}), @code{Arrows3}, @code{BeginArcArrow3}, @code{EndArcArrow3} (or equivalently @code{ArcArrow3}), @code{MidArcArrow3}, and @code{ArcArrows3}. Three-dimensional bars accept the optional arguments @code{(real size=0, triple dir=O)}. If @code{size=O}, the default bar length is used; if @code{dir=O}, the bar is drawn perpendicular to the path and the initial viewing direction. The predefined three-dimensional arrowhead styles are @code{DefaultHead3}, @code{HookHead3}, @code{TeXHead3}. Versions of the two-dimensional arrowheads lifted to three-dimensional space and aligned according to the initial viewpoint (or an optionally specified @code{normal} vector) are also defined: @code{DefaultHead2(triple normal=O)}, @code{HookHead2(triple normal=O)}, @code{TeXHead2(triple normal=O)}. These are illustrated in the example @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/arrows3.html,,arrows3}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/arrows3.asy,,.asy}}. @cindex @code{NoMargin3} @cindex @code{BeginMargin3} @cindex @code{EndMargin3} @cindex @code{Margin3} @cindex @code{Margins3} @cindex @code{BeginPenMargin2} @cindex @code{EndPenMargin2} @cindex @code{PenMargin2} @cindex @code{PenMargins2} @cindex @code{BeginPenMargin3} @cindex @code{EndPenMargin3} @cindex @code{PenMargin3} @cindex @code{PenMargins3} @cindex @code{BeginDotMargin3} @cindex @code{EndDotMargin3} @cindex @code{DotMargin3} @cindex @code{DotMargins3} @cindex @code{Margin3} @cindex @code{TrueMargin3} Module @code{three} also defines the three-dimensional margins @code{NoMargin3}, @code{BeginMargin3}, @code{EndMargin3}, @code{Margin3}, @code{Margins3}, @code{BeginPenMargin2}, @code{EndPenMargin2}, @code{PenMargin2}, @code{PenMargins2}, @code{BeginPenMargin3}, @code{EndPenMargin3}, @code{PenMargin3}, @code{PenMargins3}, @code{BeginDotMargin3}, @code{EndDotMargin3}, @code{DotMargin3}, @code{DotMargins3}, @code{Margin3}, and @code{TrueMargin3}. @cindex @code{pixel} The routine @verbatim void pixel(picture pic=currentpicture, triple v, pen p=currentpen, real width=1); @end verbatim @noindent can be used to draw on picture @code{pic} a pixel of width @code{width} at position @code{v} using pen @code{p}. Further three-dimensional examples are provided in the files @code{@uref{https://asymptote.sourceforge.io/gallery/3Dgraphs/near_earth.html,,near_earth}@uref{https://asymptote.sourceforge.io/gallery/3Dgraphs/near_earth.asy,,.asy}}, @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/conicurv.html,,conicurv}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/conicurv.asy,,.asy}}, and (in the @code{animations} subdirectory) @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/cube.html,,cube}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/cube.asy,,.asy}}. @anchor{PostScript3D} @cindex 3D @code{PostScript} Limited support for projected vector graphics (effectively three-dimensional nonrendered @code{PostScript}) is available with the setting @code{render=0}. This currently only works for piecewise planar surfaces, such as those produced by the parametric @code{surface} routines in the @code{graph3} module. Surfaces produced by the @code{solids} module will also be properly rendered if the parameter @code{nslices} is sufficiently large. @cindex hidden surface removal @cindex @code{face} In the module @code{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 @code{face} derived from @code{picture}. A @code{face} may be given to a two-dimensional drawing routine in place of any @code{picture} argument. An array of such faces may then be drawn, removing hidden surfaces: @verbatim void add(picture pic=currentpicture, face[] faces, projection P=currentprojection); @end verbatim Labels may be projected to two dimensions, using projection @code{P}, onto the plane passing through point @code{O} with normal @code{cross(u,v)} by multiplying it on the left by the transform @verbatim transform transform(triple u, triple v, triple O=O, projection P=currentprojection); @end verbatim 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: @verbatiminclude planes.asy @sp 1 @center @image{./planes} @node obj, graph3, three, Base modules @section @code{obj} @cindex @code{obj} This module allows one to construct surfaces from simple obj files, as illustrated in the example files @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/galleon.html,,galleon}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/galleon.asy,,.asy}} and @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/triceratops.html,,triceratops}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/triceratops.asy,,.asy}}. @node graph3, grid3, obj, Base modules @section @code{graph3} @cindex @code{graph3} @cindex 3D graphs This module implements three-dimensional versions of the functions in @code{graph.asy}. @cindex @code{xaxis3} @cindex @code{yaxis3} @cindex @code{zaxis3} @noindent To draw an @math{x} axis in three dimensions, use the routine @verbatim 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, bool above=false); @end verbatim @noindent Analogous routines @code{yaxis} and @code{zaxis} can be used to draw @math{y} and @math{z} axes in three dimensions. There is also a routine for drawing all three axis: @verbatim 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, arrowbar3 arrow=None); @end verbatim @cindex @code{YZEquals} @cindex @code{XZEquals} @cindex @code{XYEquals} @cindex @code{YZZero} @cindex @code{XZZero} @cindex @code{XYZero} @cindex @code{Bounds} @noindent The predefined three-dimensional axis types are @verbatim 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); @end verbatim @noindent The optional @code{align} parameter to these routines can be used to specify the default axis and tick label alignments. The @code{Bounds} axis accepts two type parameters, each of which must be one of @code{Min}, @code{Max}, or @code{Both}. These parameters specify which of the four possible three-dimensional bounding box edges should be drawn. @cindex @code{NoTicks3} @cindex @code{InTicks} @cindex @code{OutTicks} @cindex @code{InOutTicks} The three-dimensional tick options are @code{NoTicks3}, @code{InTicks}, @code{OutTicks}, and @code{InOutTicks}. These specify the tick directions for the @code{Bounds} axis type; other axis types inherit the direction that would be used for the @code{Bounds(Min,Min)} axis. Here is an example of a helix and bounding box axes with ticks and axis labels, using orthographic projection: @verbatiminclude helix.asy @sp 1 @center @image{./helix} The next example illustrates three-dimensional @math{x}, @math{y}, and @math{z} axes, without autoscaling of the axis limits: @cindex @code{axis} @verbatiminclude axis3.asy @sp 1 @center @image{./axis3} One can also place ticks along a general three-dimensional axis: @cindex @code{axis} @verbatiminclude generalaxis3.asy @sp 1 @center @image{./generalaxis3} @cindex @code{surface} @cindex @code{Spline} @cindex parametric surface Surface plots of matrices and functions over the region @code{box(a,b)} in the @math{XY} plane are also implemented: @verbatim 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); @end verbatim @noindent The final two versions draw parametric surfaces for a function @math{f(u,v)} over the parameter space @code{box(a,b)}, as illustrated in the example @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/parametricsurface.html,,parametricsurface}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/parametricsurface.asy,,.asy}}. An optional splinetype @code{Spline} may be specified. The boolean array or function @code{cond} can be used to control which surface mesh cells are actually drawn (by default all mesh cells over @code{box(a,b)} are drawn). Surface lighting is illustrated in the example files @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/parametricsurface.html,,parametricsurface}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/parametricsurface.asy,,.asy}} and @code{@uref{https://asymptote.sourceforge.io/gallery/3D graphs/sinc.html,,sinc}@uref{https://asymptote.sourceforge.io/gallery/3D graphs/sinc.asy,,.asy}}. Lighting can be disabled by setting @code{light=nolight}, as in this example of a Gaussian surface: @anchor{GaussianSurface} @verbatiminclude GaussianSurface.asy @sp 1 @center @image{./GaussianSurface} @noindent A mesh can be drawn without surface filling by specifying @code{nullpen} for the surfacepen. A vector field of @code{nu}@math{\times}@code{nv} arrows on a parametric surface @code{f} over @code{box(a,b)} can be drawn with the routine @cindex @code{vectorfield3} @verbatim 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) @end verbatim as illustrated in the examples @code{@uref{https://asymptote.sourceforge.io/gallery/3Dgraphs/vectorfield3.html,,vectorfield3}@uref{https://asymptote.sourceforge.io/gallery/3Dgraphs/vectorfield3.asy,,.asy}} and @code{@uref{https://asymptote.sourceforge.io/gallery/3Dgraphs/vectorfieldsphere.html,,vectorfieldsphere}@uref{https://asymptote.sourceforge.io/gallery/3Dgraphs/vectorfieldsphere.asy,,.asy}}. @node grid3, solids, graph3, Base modules @section @code{grid3} @cindex @code{grid3} @cindex 3D grids This module, contributed by Philippe Ivaldi, can be used for drawing 3D grids. Here is an example (further examples can be found in @code{grid3.asy} and at @url{http://www.piprime.fr/files/asymptote/grid3/}): @verbatiminclude grid3xyz.asy @sp 1 @center @image{./grid3xyz} @node solids, tube, grid3, Base modules @section @code{solids} @cindex @code{solids} This solid geometry module defines a structure @code{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 @code{O--1.5unit(Y+Z)} with perspective projection: @verbatiminclude cylinderskeleton.asy @sp 1 @center @image{./cylinderskeleton} Further illustrations are provided in the example files @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/cylinder.html,,cylinder}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/cylinder.asy,,.asy}}, @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/cones.html,,cones}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/cones.asy,,.asy}}, @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/hyperboloid.html,,hyperboloid}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/hyperboloid.asy,,.asy}}, and @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/torus.html,,torus}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/torus.asy,,.asy}}. The structure @code{skeleton} contains the three-dimensional wireframe used to visualize a volume of revolution: @verbatim 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; } @end verbatim @node tube, flowchart, solids, Base modules @section @code{tube} @cindex @code{tube} This module extends the @code{tube} surfaces constructed in @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/three_arrows.html,,three_arrows}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/three_arrows.asy,,.asy}} to arbitrary cross sections, colors, and spine transformations. The routine @verbatim surface tube(path3 g, coloredpath section, transform T(real)=new transform(real t) {return identity();}, real corner=1, real relstep=0); @end verbatim @noindent draws a tube along @code{g} with cross section @code{section}, after applying the transformation @code{T(t)} at @code{point(g,t)}. The parameter @code{corner} controls the number of elementary tubes at the angular points of @code{g}. A nonzero value of @code{relstep} specifies a fixed relative time step (in the sense of @code{relpoint(g,t)}) to use in constructing elementary tubes along @code{g}. The type @code{coloredpath} is a generalization of @code{path} to which a @code{path} can be cast: @cindex @code{coloredpath} @verbatim struct coloredpath { path p; pen[] pens(real); int colortype=coloredSegments; } @end verbatim @noindent @cindex @code{coloredSegments} @cindex @code{coloredNodes} Here @code{p} defines the cross section and the method @code{pens(real t)} returns an array of pens (interpreted as a cyclic array) used for shading the tube patches at @code{relpoint(g,t)}. If @code{colortype=coloredSegments}, the tube patches are filled as if each segment of the section was colored with the pen returned by @code{pens(t)}, whereas if @code{colortype=coloredNodes}, the tube components are vertex shaded as if the nodes of the section were colored. A @code{coloredpath} can be constructed with one of the routines: @verbatim 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)); @end verbatim @noindent 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 @code{path} to @code{coloredpath} allows the use of a @code{path} instead of a @code{coloredpath}; in this case the shading behaviour is the default shading behavior for a surface. An example of @code{tube} is provided in the file @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/trefoilknot.html,,trefoilknot}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/trefoilknot.asy,,.asy}}. Further examples can be found at @url{http://www.piprime.fr/files/asymptote/tube/}. @node flowchart, contour, tube, Base modules @section @code{flowchart} @cindex @code{flowchart} This module provides routines for drawing flowcharts. The primary structure is a @code{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 @code{t}: @verbatim 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()); @end verbatim @cindex @code{block.top} @cindex @code{block.left} @cindex @code{block.right} @cindex @code{block.bottom} @cindex @code{block.topleft} @cindex @code{block.topright} @cindex @code{block.bottomleft} @cindex @code{block.bottomright} @noindent To obtain an arbitrary position along the boundary of the block in user coordinates, use: @verbatim pair block.position(real x, transform t=identity()); @end verbatim @cindex @code{block.position} @noindent @cindex @code{block.center} The center of the block in user coordinates is stored in @code{block.center} and the block size in @code{PostScript} coordinates is given by @code{block.size}. @noindent A frame containing the block is returned by @verbatim frame block.draw(pen p=currentpen); @end verbatim @cindex @code{block.draw} The following block generation routines accept a Label, string, or frame for their object argument: @table @dfn @item rectangular block with an optional header (and padding @code{dx} around header and body): @cindex @code{rectangle} @verbatim 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); @end verbatim @item parallelogram block: @cindex @code{parallelogram} @verbatim 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); @end verbatim @item diamond-shaped block: @cindex @code{diamond} @verbatim 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); @end verbatim @item circular block: @cindex @code{circle} @verbatim block circle(object body, pair center=(0,0), pen fillpen=invisible, pen drawpen=currentpen, real dr=3, real mindiameter=mincirclediameter); @end verbatim @item rectangular block with rounded corners: @cindex @code{roundrectangle} @verbatim 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); @end verbatim @item rectangular block with beveled edges: @cindex @code{bevel} @verbatim 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); @end verbatim @end table To draw paths joining the pairs in @code{point} with right-angled lines, use the routine: @cindex @code{path} @cindex @code{Horizontal} @cindex @code{Vertical} @verbatim path path(pair point[] ... flowdir dir[]); @end verbatim @noindent The entries in @code{dir} identify whether successive segments between the pairs specified by @code{point} should be drawn in the @code{Horizontal} or @code{Vertical} direction. Here is a simple flowchart example (see also the example @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/controlsystem.html,,controlsystem}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/controlsystem.asy,,.asy}}): @verbatiminclude flowchartdemo.asy @sp 1 @center @image{./flowchartdemo} @node contour, contour3, flowchart, Base modules @section @code{contour} @cindex @code{contour} This module draws contour lines. To construct contours corresponding to the values in a real array @code{c} for a function @code{f} on @code{box(a,b)}, use the routine @verbatim guide[][] contour(real f(real, real), pair a, pair b, real[] c, int nx=ngraph, int ny=nx, interpolate join=operator --, int subsample=1); @end verbatim @noindent The integers @code{nx} and @code{ny} define the resolution. The default resolution, @code{ngraph x ngraph} (here @code{ngraph} defaults to @code{100}) can be increased for greater accuracy. The default interpolation operator is @code{operator --} (linear). Spline interpolation (@code{operator ..}) may produce smoother contours but it can also lead to overshooting. The @code{subsample} parameter indicates the number of interior points that should be used to sample contours within each @code{1 x 1} box; the default value of @code{1} is usually sufficient. To construct contours for an array of data values on a uniform two-dimensional lattice on @code{box(a,b)}, use @verbatim guide[][] contour(real[][] f, pair a, pair b, real[] c, interpolate join=operator --, int subsample=1); @end verbatim To construct contours for an array of data values on a nonoverlapping regular mesh specified by the two-dimensional array @code{z}, @verbatim guide[][] contour(pair[][] z, real[][] f, real[] c, interpolate join=operator --, int subsample=1); @end verbatim @noindent To construct contours for an array of values @code{f} specified at irregularly positioned points @code{z}, use the routine @verbatim guide[][] contour(pair[] z, real[] f, real[] c, interpolate join=operator --); @end verbatim @noindent The contours themselves can be drawn with one of the routines @verbatim 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); @end verbatim The following simple example draws the contour at value @code{1} for the function @math{z=x^2+y^2}, which is a unit circle: @verbatiminclude onecontour.asy @sp 1 @center @image{./onecontour} The next example draws and labels multiple contours for the function @math{z=x^2-y^2} with the resolution @code{100 x 100}, using a dashed pen for negative contours and a solid pen for positive (and zero) contours: @verbatiminclude multicontour.asy @sp 1 @center @image{./multicontour} The next examples illustrates how contour lines can be drawn on color density images, with and without palette quantization: @verbatiminclude fillcontour.asy @sp 1 @center @image{./fillcontour} @verbatiminclude imagecontour.asy @sp 1 @center @image{./imagecontour} Finally, here is an example that illustrates the construction of contours from irregularly spaced data: @verbatiminclude irregularcontour.asy @sp 1 @center @image{./irregularcontour} In the above example, the contours of irregularly spaced data are constructed by first creating a triangular mesh from an array @code{z} of pairs: @cindex @code{triangulate} @verbatim int[][] triangulate(pair[] z); @end verbatim @verbatiminclude triangulate.asy @sp 1 @center @image{./triangulate} The example @code{@uref{https://asymptote.sourceforge.io/gallery/PDFs/Gouraudcontour.pdf,,Gouraudcontour}@uref{https://asymptote.sourceforge.io/gallery/PDFs/Gouraudcontour.asy,,.asy}} illustrates how to produce color density images over such irregular triangular meshes. @code{Asymptote} uses a robust version of Paul Bourke's Delaunay triangulation algorithm based on the public-domain exact arithmetic predicates written by Jonathan Shewchuk. @node contour3, smoothcontour3, contour, Base modules @section @code{contour3} @cindex @code{contour3} This module draws surfaces described as the null space of real-valued functions of @math{(x,y,z)} or @code{real[][][]} matrices. Its usage is illustrated in the example file @code{@uref{https://asymptote.sourceforge.io/gallery/3Dgraphs/magnetic.html,,magnetic}@uref{https://asymptote.sourceforge.io/gallery/3Dgraphs/magnetic.asy,,.asy}}. @node smoothcontour3, slopefield, contour3, Base modules @section @code{smoothcontour3} @cindex @code{smoothcontour3} This module, written by Charles Staats, draws implicitly defined surfaces with smooth appearance. The purpose of this module is similar to that of @code{contour3}: given a real-valued function @math{f(x,y,z)}, construct the surface described by the equation @math{f(x,y,z) = 0}. The @code{smoothcontour3} module generally produces nicer results than @code{contour3}, but takes longer to compile. Additionally, the algorithm assumes that the function and the surface are both smooth; if they are not, then @code{contour3} may be a better choice. To construct the null surface of a function @code{f(triple)} or @code{ff(real,real,real)} over @code{box(a,b)}, use the routine @cindex @code{implicitsurface} @verbatim 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); @end verbatim @noindent The optional parameter @code{overlapedges} attempts to compensate for an artifact that can cause the renderer to ``see through'' the boundary between patches. Although it defaults to @code{false}, it should usually be set to @code{true}. The example @code{@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/genustwo.html,,genustwo}@uref{https://asymptote.sourceforge.io/gallery/3Dwebgl/genustwo.asy,,.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 @url{https://github.com/charlesstaats/smoothcontour3}. @node slopefield, ode, smoothcontour3, Base modules @section @code{slopefield} @cindex @code{slopefield} To draw a slope field for the differential equation @math{dy/dx=f(x,y)} (or @math{dy/dx=f(x)}), use: @verbatim 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); @end verbatim @noindent Here, the points @code{a} and @code{b} are the lower left and upper right corners of the rectangle in which the slope field is to be drawn, @code{nx} and @code{ny} are the respective number of ticks in the @math{x} and @math{y} directions, @code{tickfactor} is the fraction of the minimum cell dimension to use for drawing ticks, and @code{p} is the pen to use for drawing the slope fields. The return value is a picture that can be added to @code{currentpicture} via the @code{add(picture)} command. The function @cindex @code{curve} @verbatim path curve(pair c, real f(real,real), pair a, pair b); @end verbatim @noindent takes a point (@code{c}) and a slope field-defining function @code{f} and returns, as a path, the curve passing through that point. The points @code{a} and @code{b} represent the rectangular boundaries over which the curve is interpolated. Both @code{slopefield} and @code{curve} alternatively accept a function @code{real f(real)} that depends on @math{x} only, as seen in this example: @verbatiminclude slopefield1.asy @sp 1 @center @image{./slopefield1} @node ode, , slopefield, Base modules @section @code{ode} @cindex @code{ode} The @code{ode} module, illustrated in the example @code{@uref{https://raw.githubusercontent.com/vectorgraphics/asymptote/HEAD/examples/odetest.asy,,odetest.asy}}, implements a number of explicit numerical integration schemes for ordinary differential equations. @node Options, Interactive mode, Base modules, Top @chapter Command-line options @cindex options @cindex command-line options Type @code{asy -h} to see the full list of command-line options supported by @code{Asymptote}: @verbatiminclude options All boolean options can be negated by prepending @code{no} to the option name. If no arguments are given, @code{Asymptote} runs in interactive mode (@pxref{Interactive mode}). In this case, the default output file is @code{out.eps}. If @code{-} is given as the file argument, @code{Asymptote} reads from standard input. If multiple files are specified, they are treated as separate @code{Asymptote} runs. @cindex @code{autoimport} If the string @code{autoimport} is nonempty, a module with this name is automatically imported for each run as the final step in loading module @code{plain}. @anchor{configuration file} @cindex configuration file @cindex @code{ASYMPTOTE_CONFIG} @cindex @code{config} @cindex @code{settings} @anchor{settings} Default option values may be entered as @code{Asymptote} code in a configuration file named @code{config.asy} (or the file specified by the environment variable @code{ASYMPTOTE_CONFIG} or @code{-config} option). @code{Asymptote} will look for this file in its usual search path (@pxref{Search paths}). Typically the configuration file is placed in the @code{.asy} directory in the user's home directory (@code{%USERPROFILE%\.asy} under @code{MSDOS}). Configuration variables are accessed using the long form of the option names: @verbatim import settings; outformat="pdf"; batchView=false; interactiveView=true; batchMask=false; interactiveMask=true; @end verbatim Command-line options override these defaults. Most configuration variables may also be changed at runtime. @cindex @code{dvipsOptions} @cindex @code{hyperrefOptions} @cindex @code{convertOptions} @cindex @code{gsOptions} @cindex @code{htmlviewerOptions} @cindex @code{psviewerOptions} @cindex @code{pdfviewerOptions} @cindex @code{pdfreloadOptions} @cindex @code{glOptions} @cindex @code{dvisvgmOptions} The advanced configuration variables @code{dvipsOptions}, @code{hyperrefOptions}, @code{convertOptions}, @code{gsOptions}, @code{htmlviewerOptions}, @code{psviewerOptions}, @code{pdfviewerOptions}, @code{pdfreloadOptions}, @code{glOptions}, and @code{dvisvgmOptions} allow specialized options to be passed as a string to the respective applications or libraries. The default value of @code{hyperrefOptions} is @code{setpagesize=false,unicode,pdfborder=0 0 0}. If you insert @verbatim import plain; settings.autoplain=true; @end verbatim @noindent at the beginning of the configuration file, it can contain arbitrary @code{Asymptote} code. @cindex @code{convert} @cindex @code{output} @cindex @code{format} @cindex @code{ImageMagick} @cindex @code{render} @cindex @code{antialias} @cindex @code{size} @cindex @code{latex} @cindex @code{tex} @cindex @code{pdflatex} @cindex @code{xelatex} @cindex @code{context} @cindex @code{luatex} @cindex @code{lualatex} @cindex @code{EPS} @cindex @code{PDF} @anchor{texengines} @anchor{convert} The default output format is @acronym{EPS} for the (default) @code{latex} and @code{tex} tex engine and @acronym{PDF} for the @code{pdflatex}, @code{xelatex}, @code{context}, @code{luatex}, and @code{lualatex} tex engines. Alternative output formats may be produced using the @code{-f} option (or @code{outformat} setting). @cindex @code{SVG} @cindex @code{dvisvgm} @cindex @code{libgs} @cindex @code{graphic} To produce @acronym{SVG} output, you will need @code{dvisvgm} (version 2.6.3 or later) from @url{https://dvisvgm.de}. You might need to adjust the configuration variable @code{libgs} to point to the location of your @code{Ghostscript} library @code{libgs.so} (or to an empty string, depending on how @code{dvisvgm} was configured). The 2.10 version (or later) of @code{dvisvgm} can display @acronym{SVG} output (used by the @code{xasy} editor) for embedded @acronym{EPS}, @acronym{PDF}, @acronym{PNG}, and @acronym{JPEG} images included with the @code{graphic()} function. @code{Asymptote} can also produce any output format supported by the @code{ImageMagick} @code{convert} program (version 6.3.5 or later recommended; an @code{Invalid Parameter} error message indicates that the @code{MSDOS} utility @code{convert} is being used instead of the one that comes with @code{ImageMagick}). The optional setting @code{-render n} requests an output resolution of @code{n} pixels per @code{bp}. Antialiasing is controlled by the parameter @code{antialias}, which by default specifies a sampling width of 2 pixels. To give other options to @code{convert}, use the @code{convertOptions} setting or call convert manually. This example emulates how @code{Asymptote} produces antialiased @code{tiff} output at one pixel per @code{bp}: @verbatim asy -o - venn | convert -alpha Off -density 144x144 -geometry 50%x eps:- venn.tiff @end verbatim @cindex @code{nosafe} @cindex @code{safe} @cindex @code{system} If the option @code{-nosafe} is given, @code{Asymptote} runs in unsafe mode. This enables the @code{int system(string s)} and @code{int system(string[] s)} calls, allowing one to execute arbitrary shell commands. The default mode, @code{-safe}, disables this call. @cindex offset @cindex @code{aligndir} A @code{PostScript} offset may be specified as a pair (in @code{bp} units) with the @code{-O} option: @verbatim asy -O 0,0 file @end verbatim @noindent The default offset is zero. The pair @code{aligndir} specifies an optional direction on the boundary of the page (mapped to the rectangle [-1,1]@math{\times}[-1,1]) to which the picture should be aligned; the default value @code{(0,0)} species center alignment. @cindex @code{-c} The @code{-c} (@code{command}) option may be used to execute arbitrary @code{Asymptote} code on the command line as a string. It is not necessary to terminate the string with a semicolon. Multiple @code{-c} options are executed in the order they are given. For example @verbatim asy -c 2+2 -c "sin(1)" -c "size(100); draw(unitsquare)" @end verbatim @noindent produces the output @verbatim 4 0.841470984807897 @end verbatim @noindent and draws a unitsquare of size @code{100}. @cindex @code{-u} The @code{-u} (@code{user}) option may be used to specify arbitrary @code{Asymptote} settings on the command line as a string. It is not necessary to terminate the string with a semicolon. Multiple @code{-u} options are executed in the order they are given. Command-line code like @code{-u x=sqrt(2)} can be executed within a module like this: @verbatim real x; usersetting(); write(x); @end verbatim @cindex @code{-l} When the @code{-l} (@code{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 @code{-v} option: @table @code @item -v Display top-level module and final output file names. @item -vv Also display imported and included module names and final @code{LaTeX} and @code{dvips} processing information. @item -vvv Also output @code{LaTeX} bidirectional pipe diagnostics. @item -vvvv Also output knot guide solver diagnostics. @item -vvvvv Also output @code{Asymptote} traceback diagnostics. @end table @node Interactive mode, GUI, Options, Top @chapter Interactive mode @cindex interactive mode Interactive mode is entered by executing the command @code{asy} with no file arguments. When the @code{-multiline} option is disabled (the default), each line must be a complete @code{Asymptote} statement (unless explicitly continued by a final backslash character @code{\}); it is not necessary to terminate input lines with a semicolon. If one assigns @code{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 @code{Asymptote} code directly into the interactive input buffer. @cindex @code{%} Interactive mode can be conveniently used as a calculator: expressions entered at the interactive prompt (for which a corresponding @code{write} function exists) are automatically evaluated and written to @code{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 @code{%} in the next line input at the prompt. For example: @verbatim > 2+3 5 > %*4 20 > 1/% 0.05 > sin(%) 0.0499791692706783 > currentpicture > %.size(200,0) > @end verbatim @cindex @code{operator answer} The @code{%} symbol, when used as a variable, is shorthand for the identifier @code{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: @table @code @cindex @code{help} @item help view the manual; @item erase erase @code{currentpicture}; @cindex @code{input} @item reset reset the @code{Asymptote} environment to its initial state, except for changes to the settings module (@pxref{settings}), the current directory (@pxref{cd}), and breakpoints (@pxref{Debugger}); @cindex @code{input} @item input FILE does an interactive reset, followed by the command @code{include FILE}. If the file name @code{FILE} contains nonalphanumeric characters, enclose it with quotation marks. A trailing semi-colon followed by optional @code{Asymptote} commands may be entered on the same line. @cindex @code{quit} @cindex @code{exit} @cindex @code{history} @anchor{history} @item quit exit interactive mode (@code{exit} is a synonym; the abbreviation @code{q} is also accepted unless there exists a top-level variable named @code{q}). @cindex @code{historylines} A history of the most recent 1000 (this number can be changed with the @code{historylines} configuration variable) previous commands will be retained in the file @code{.asy/history} in the user's home directory (unless the command-line option @code{-localhistory} was specified, in which case the history will be stored in the file @code{.asy_history} in the current directory). @end table Typing @code{ctrl-C} interrupts the execution of @code{Asymptote} code and returns control to the interactive prompt. Interactive mode is implemented with the @acronym{GNU} @code{readline} library, with command history and auto-completion. To customize the key bindings, see: @url{https://tiswww.case.edu/php/chet/readline/readline.html} @cindex @code{Python} usage The file @code{asymptote.py} in the @code{Asymptote} system directory provides an alternative way of entering @code{Asymptote} commands interactively, coupled with the full power of @code{Python}. Copy this file to your @code{Python path} and then execute from within @code{Python 3} the commands @verbatim 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") @end verbatim @node GUI, Command-Line Interface, Interactive mode, Top @chapter Graphical User Interface @cindex graphical user interface @cindex @acronym{GUI} @cindex mouse @cindex wheel mouse @cindex @code{Button-1} @cindex @code{Button-2} @cindex @code{xasy} @menu * GUI installation:: Installing @code{xasy} * GUI usage:: Using @code{xasy} to edit objects @end menu In the event that adjustments to the final figure are required, the preliminary Graphical User Interface (@acronym{GUI}) @code{xasy} included with @code{Asymptote} allows you to move graphical objects and draw new ones. The modified figure can then be saved as a normal @code{Asymptote} file. @node GUI installation, GUI usage, GUI, GUI @section GUI installation @cindex GUI installation As @code{xasy} is written in the interactive scripting language @code{Python/Qt}, it requires @code{Python} (@url{https://www.python.org}), along with the @code{Python} packages @code{pyqt5}, @code{cson}, and @code{numpy}: @verbatim pip3 install cson numpy pyqt5 PyQt5.sip @end verbatim Pictures are deconstructed into the @acronym{SVG} image format. Since @code{Qt5} does not support @code{SVG} clipping, you will need the @code{rsvg-convert} utility, which is part of the @code{librsvg2-tools} package on @code{UNIX} systems and the @code{librsvg} package on @code{MacOS X}; under @code{Microsoft Windows}, it is available as @url{https://sourceforge.net/projects/tumagcc/files/rsvg-convert-2.40.20.7z} @node GUI usage, , GUI installation, GUI @section GUI usage @cindex GUI usage @cindex arrow keys @cindex mouse wheel @cindex @code{deconstruct} The arrow keys (or mouse wheel) are convenient for temporarily raising and lowering objects within @code{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 @verbatim void begingroup(picture pic=currentpicture); void endgroup(picture pic=currentpicture); @end verbatim 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 (@pxref{add}). @node Command-Line Interface, PostScript to Asymptote, GUI, Top @chapter Command-Line Interface @cindex command-line interface @code{Asymptote} code may be sent to the @url{http://asymptote.ualberta.ca} server directly from the command line, specifying any options directly in the @acronym{URL}: @itemize @bullet @item SVG output: @code{curl --data-binary 'import venn;' 'asymptote.ualberta.ca:10007?f=svg' | display -} @item HTML output: @code{curl --data-binary @@/usr/local/share/doc/asymptote/examples/Klein.asy 'asymptote.ualberta.ca:10007' -o Klein.html} @item V3D output: @code{curl --data-binary 'import teapot;' 'asymptote.ualberta.ca:10007?f=v3d' -o teapot.v3d} @item PDF output with rendered bitmap at 2 pixels per bp: @code{curl --data-binary 'import teapot;' 'asymptote.ualberta.ca:10007?f=pdf' -o teapot.pdf} @item PDF output with rendered bitmap at 4 pixels per bp: @code{curl --data-binary 'import teapot;' 'asymptote.ualberta.ca:10007?f=pdf&render=4' -o teapot.pdf} @item PRC output: @code{curl --data-binary 'import teapot;' 'asymptote.ualberta.ca:10007?f=pdf&prc' -o teapot.pdf} @item PRC output with rendered preview bitmap at 4 pixels per bp: @code{curl --data-binary 'import teapot;' 'asymptote.ualberta.ca:10007?f=pdf&prc&render=4' -o teapot.pdf} @end itemize The source code for the command-line interface is available at @url{https://github.com/vectorgraphics/asymptote-http-server}. @node Language server protocol, PostScript to Asymptote, Command-Line Interface, Top @chapter Language server protocol @cindex @acronym{LSP} @cindex language server protocol Under @code{UNIX} and @code{MacOS X}, @code{Asymptote} supports features of the @uref{https://en.wikipedia.org/wiki/Language_Server_Protocol,Language Server Protocol (@acronym{LSP})}, including function signature and variable matching. Under @code{MSWindows}, @code{Asymptote} currently supports @acronym{LSP} only when compiled within the @code{Windows Subsystem for Linux}. @code{Emacs} users can enable the @code{Asymptote} language server protocol by installing @code{lsp-mode} using the following procedure: @itemize @bullet @item Add to the @code{.emacs} initialization file: @verbatim (require 'package) (add-to-list 'package-archives '("melpa" . "https://melpa.org/packages/") t) (package-initialize) @end verbatim @item Launch emacs and execute @verbatim M-x package-refresh-contents M-x package-install @end verbatim and select @code{lsp-mode}. @item Add to the @code{.emacs} initialization file: @verbatim (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 ) ) @end verbatim @item Launch emacs and execute @verbatim M-x lsp @end verbatim @end itemize @node PostScript to Asymptote, Help, Command-Line Interface, Top @chapter @code{PostScript} to @code{Asymptote} @cindex @code{pstoedit} The excellent @code{PostScript} editor @code{pstoedit} (version 3.50 or later; available from @url{https://sourceforge.net/projects/pstoedit/}) includes an @code{Asymptote} backend. Unlike virtually all other @code{pstoedit} backends, this driver includes native clipping, even-odd fill rule, @code{PostScript} subpath, and full image support. Here is an example: @noindent @code{asy -V @value{Datadir}/doc/asymptote/examples/venn.asy} @noindent @verbatim pstoedit -f asy venn.eps test.asy asy -V test @end verbatim @noindent If the line widths aren't quite correct, try giving @code{pstoedit} the @code{-dis} option. If the fonts aren't typeset correctly, try giving @code{pstoedit} the @code{-dt} option. @node Help, Debugger, PostScript to Asymptote, Top @chapter Help @cindex help @cindex forum A list of frequently asked questions (@acronym{FAQ}) is maintained at @quotation @url{https://asymptote.sourceforge.io/FAQ} @end quotation @noindent Questions on installing and using @code{Asymptote} that are not addressed in the @acronym{FAQ} should be sent to the @code{Asymptote} forum: @quotation @url{https://sourceforge.net/p/asymptote/discussion/409349} @end quotation @noindent Including an example that illustrates what you are trying to do will help you get useful feedback. @code{LaTeX} problems can often be diagnosed with the @code{-vv} or @code{-vvv} command-line options. Contributions in the form of patches or @code{Asymptote} modules can be posted here: @quotation @url{https://sourceforge.net/p/asymptote/patches} @end quotation @noindent To receive announcements of upcoming releases, please subscribe to @code{Asymptote} at @quotation @url{https://sourceforge.net/projects/asymptote/} @end quotation @cindex bug reports @noindent If you find a bug in @code{Asymptote}, please check (if possible) whether the bug is still present in the latest @code{git} developmental code (@pxref{Git}) before submitting a bug report. New bugs can be reported at @quotation @url{https://github.com/vectorgraphics/asymptote/issues} @end quotation @noindent To see if the bug has already been fixed, check bugs with Status @code{Closed} and recent lines in @quotation @url{https://asymptote.sourceforge.io/ChangeLog} @end quotation @noindent @cindex stack overflow @cindex segmentation fault @cindex @code{libsigsegv} @code{Asymptote} can be configured with the optional @acronym{GNU} library @code{libsigsegv}, available from @url{https://www.gnu.org/software/libsigsegv/}, which allows one to distinguish user-generated @code{Asymptote} stack overflows (@pxref{stack overflow}) from true segmentation faults (due to internal C++ programming errors; please submit the @code{Asymptote} code that generates such segmentation faults along with your bug report). @node Debugger, Credits, Help, Top @chapter Debugger @cindex 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 @code{file} at line @code{line}, use the command @cindex @code{stop} @verbatim void stop(string file, int line, code s=quote{}); @end verbatim @noindent The optional argument @code{s} may be used to conditionally set the variable @code{ignore} in @code{plain_debugger.asy} to @code{true}. For example, the first 10 instances of this breakpoint will be ignored (the variable @code{int count=0} is defined in @code{plain_debugger.asy}): @verbatim stop("test",2,quote{ignore=(++count <= 10);}); @end verbatim To set a break point in file @code{file} at the first line containing the string @code{text}, use @verbatim void stop(string file, string text, code s=quote{}); @end verbatim @noindent To list all breakpoints, use: @cindex @code{breakpoints} @verbatim void breakpoints(); @end verbatim @noindent To clear a breakpoint, use: @cindex @code{clear} @verbatim void clear(string file, int line); @end verbatim @noindent To clear all breakpoints, use: @verbatim void clear(); @end verbatim The following commands may be entered at the debugging prompt: @table @code @cindex @code{help} @item @code{h} help; @cindex @code{continue} @item @code{c} continue execution; @cindex @code{inst} @item @code{i} step to the next instruction; @cindex @code{step} @item @code{s} step to the next executable line; @cindex @code{next} @item @code{n} step to the next executable line in the current file; @cindex @code{file} @item @code{f} step to the next file; @cindex @code{return} @item @code{r} return to the file associated with the most recent breakpoint; @cindex @code{trace} @item @code{t} toggle tracing (@code{-vvvvv}) mode; @cindex @code{quit} @item @code{q} quit debugging and end execution; @cindex @code{exit} @item @code{x} exit the debugger and run to completion. @end table @noindent Arbitrary @code{Asymptote} code may also be entered at the debugging prompt; however, since the debugger is implemented with @code{eval}, currently only top-level (global) variables can be displayed or modified. The debugging prompt may be entered manually with the call @verbatim void breakpoint(code s=quote{}); @end verbatim @node Credits, Index, Debugger, Top @chapter Acknowledgments @cindex acknowledgments Financial support for the development of @code{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 @code{MetaPost} that inspired the development of @code{Asymptote}, and Donald E. Knuth, author of @TeX{} and @code{MetaFont} (on which @code{MetaPost} is based). The authors of @code{Asymptote} are Andy Hammerlindl, John Bowman, and Tom Prince. Sean Healy designed the @code{Asymptote} logo. Other contributors include Orest Shardt, Jesse Frohlich, Michail Vidiassov, Charles Staats, Philippe Ivaldi, Olivier Guib@'e, Radoslav Marinov, Jeff Samuelson, Chris Savage, Jacques Pienaar, Mark Henning, Steve Melenchuk, Martin Wiebusch, Stefan Knorr, and Supakorn ``Jamie'' Rassameemasmuang. Pedram Emami developed the @code{Asymptote Web Application} hosted at @url{http://asymptote.ualberta.ca}: @url{https://github.com/vectorgraphics/asymptote-http-server} @node Index, , Credits, Top @unnumbered Index @printindex cp @bye @c LocalWords: randMax Gaussrand asy cindex indices resized LaTeX TK latin au @c LocalWords: latexusage tex bbox PostScript subdirectory gcc emacs ASYDIR @c LocalWords: documentclass usepackage subpath shipout sqrt xN Mx bw AcroRd @c LocalWords: xscale xaxis yaxis BeginBar GIF postprocessing fpu de rpair xy @c LocalWords: ImageMagick cd asymptote Hy 0pt 1filll 's 3D 2D 'asy @c LocalWords: startup natively xasy tkinter VxN yingyang currentpicture toc @c LocalWords: MetaPost MetaFont Hammerlindl Healy texinfo autoload setq setf @c LocalWords: printindex setfilename settitle dircategory direntry titlepage @c LocalWords: vskip filll insertcopying ifnottex detailmenu alist augroup PQ @c LocalWords: bool behaviour facto zxf login Debian dev filetypedetect @c LocalWords: FFTW bp readline gv eps args Boehm gc evenoddoverlap png joe @c LocalWords: boolean initializer expi dir xpart ypart STL substring rfind @c LocalWords: pos substr strftime typedef pxref unitcircle yscale Bezier iff @c LocalWords: postcontrol precontrol atleast nullpath arclength arctime rgb @c LocalWords: dirtime currentpen colorspaces grayscale cmyk defaultpen x cx @c LocalWords: linetype longdashed dashdotted longdashdotted linewidth y XP @c LocalWords: fontsize defaultfilename keepAspect IgnoreAspect ise flushleft @c LocalWords: src dest XDR txt getc fout stdin stdout endl eof js prc ni @c LocalWords: Microsystem's eol exponentials postfix sayhi th Ubuntu @c LocalWords: sqr intop addby libm asin acos atan sinh tanh asinh acosh cbrt @c LocalWords: atanh fabs hypot fmod ceil srand dereferenced alice pete sqrtx @c LocalWords: eval fft csv runtime nonalphanumeric labely LeftTicks NoTicks @c LocalWords: RightTicks BottomTop LeftRight Ticksize UTF BufNewFile BufRead @c LocalWords: ticksize subintervals xlimits filetype plugin setlocal makeprg @c LocalWords: ylimits uncommented automin automax cp uninstall reals ecast @c LocalWords: scaleT RightSide yx yy NoAlign legendmargin opic CCW @c LocalWords: arrowbar LeftSide EndBar BeginArrow lly feynman isi showtarget @c LocalWords: EndArrow BeginArcArrow EndArcArrow ArcArrow ArcArrows NoFill @c LocalWords: filldraw fillpen drawpen errorformat bigsquare bezier darkblue @c LocalWords: quartercircle darkgreen lightblue urx ury texpreamble sgn texi @c LocalWords: lineargraph datagraph vertices parametricgraph uncomment ggv @c LocalWords: loggraph generalaxis texhash arrowsize arrowangle arrowlength @c LocalWords: SuppressQuiet MoveQuiet LIBREADLINE config MacOS prebuilt @c LocalWords: ghostview gsview SIGHUP PDF acroread xpdf cutbefore strptime @c LocalWords: libsigsegv intersectionpoint dotfactor vv firstcut pq logticks @c LocalWords: Unisys dvips vvv vvvv vvvvv traceback lastcut cutafter infodir @c LocalWords: zxvf xargs cond polargraph xmin xmax plabel YZero labelling ln @c LocalWords: ymin ymax XZero xequals tickmin tickmax unlabelled se pq pena @c LocalWords: yequals Nobre Barbarosie Schwaiger nearearth conicurv Wiebusch @c LocalWords: unfill posterSize ngraph interpolatetype ctrl dt pic getint Ai @c LocalWords: NNE jxf linecap linejoin unitsquare shadedtiling ei nomarker @c LocalWords: westnile minipage ra penb paletteticks drawline nV FillDraw uv @c LocalWords: susceptibleM flushright secondaryX secondaryY secondaryaxis tt @c LocalWords: titlelabel columnlabel rb xtick ytick labelx XEquals YEquals @c LocalWords: treetest eetomumu fermi backend pstoedit drawtree xFF MSDOS gz @c LocalWords: vimrc CFLAGS verbatiminclude online noindent bezier superpath @c LocalWords: evenodd squarecap roundcap extendcap miterjoin roundjoin NFSS @c LocalWords: beveljoin fillrule zerowinding insideness lineskip cmr pcrr Hx @c LocalWords: AvantGarde Bookman Helvetica NewCenturySchoolBook minbound pdf @c LocalWords: Palatino TimesRoman ZapfChancery ZapfDingbats german basealign @c LocalWords: nondeconstructed backends usr venn labelsquare nobasealign dp @c LocalWords: NoMargin BeginMargin EndMargin BeginPenMargin EndPenMargin dm @c LocalWords: PenMargin PenMargins TrueMargin labelmargin errorbars errorbar @c LocalWords: dpx dpy dmx dmy barsize arrowsize BeginDotMargin DotMargin acc @c LocalWords: EndDotMargin DotMargins NColors BWRainbow colorspace labelled @c LocalWords: PaletteTicks defaultformat leastsquares bjam fprintf endgroup @c LocalWords: begingroup xmargin ymargin pbox box ellipse wget exe Gouraud @c LocalWords: multithreaded newframe init emph nums concat xline yline zpart @c LocalWords: colatitude zscale cosh nullpen MetaFontbook cyclicflag FreeBSD @c LocalWords: nodeps Ghostgum beginlabel endlabel pTick ptick loggrid SAS dy @c LocalWords: currentprojection latticeshading subpictures colinear unitcube @c LocalWords: Autoscaling solveQuadratic MidArrow MidArcArrow Prebuilt url @c LocalWords: pdftex comment getstring getstringprefix getreal defaultS hsv @c LocalWords: ticklocate autoscaleT autoscaling vectorfield autolimits dvi @c LocalWords: zlimits inline dvipdf hyperdvi autoconf gui zerowindingoverlap @c LocalWords: prepended intMax quadraticroots cubicroots filltype prepend dx @c LocalWords: ticklabel popup UnFill markroutine marknodes markuniform erf @c LocalWords: intersectpoint cyrillic mathtext russian brokenaxis Datadir ds @c LocalWords: resetdefaultpen latticeshade axialshade radialshade erfc det @c LocalWords: gouraudshade unescaped nmesh surfacepen getpair MikTeX dw YZ @c LocalWords: meshpen localhistory axisT roundedpath unitsize aSin accel pre @c LocalWords: fontcommand makepen aCos aTan Knorr roundpath BeginPoint nView @c LocalWords: MidPoint EndPoint nmask antialiasing autoplain batchMask libgc @c LocalWords: batchView clearGUI ignoreGUI interactiveMask interactiveView @c LocalWords: listvariables outformat parseonly prepending psviewer nCircle @c LocalWords: pdfviewer papertype tabcompletion noautoplain plugins Teixeira @c LocalWords: embeddedmovie historylines RadialShade penc penr CJK tgz GPL @c LocalWords: legendlinelength legendskip USERPROFILE LDFLAGS currentlight @c LocalWords: subsampled sinc kai AtBeginDocument GBK clearpage lasy texpath @c LocalWords: AtEndDocument zaxis maxbound truepoint paperwidth paperheight @c LocalWords: GSL deriv texcolors fixedscaling UpsideDown texreset slidedemo @c LocalWords: subitem newslide realMin realMax realEpsilon realDigits gsl dh @c LocalWords: obliqueX asycolors monthaxis xautoscale yautoscale zautoscale @c LocalWords: obliqueZ obliqueY cylinderskeleton block llcorner dr py nx CPU @c LocalWords: loc topleft topright bottomleft bottomright flowrectangle UTC @c LocalWords: chartblock flowdiamond flowcircle xlabel BezierSurface el xyz @c LocalWords: flowroundrectangle flowbevel flowpath drawflow blocks ny cpu @c LocalWords: multipleView usersetting mediumgray flowchartdemo ylabel nv xf @c LocalWords: zlabel slopefields cputime roundrectangle slopefield libgccpp @c LocalWords: tickfactor USERNAME writeable imagecontour logimage Dumoulin's @c LocalWords: NoCrop parametricsurface realmult SoftLight HardLight interp @c LocalWords: ColorDodge ColorBurn Ivaldi buildcycle autorotate mexicanhat @c LocalWords: Gouraudcontour pdflatex preconfigured perline linelength hskip @c LocalWords: penimage filloutside legendhskip legendvskip maxwidth CDlabel @c LocalWords: tensorshade MPEG framepoint nonfunction Radoslav Marinov Mepis @c LocalWords: Pienaar Melenchuk finalout Linspire Dpkg sudo dpkg dtx Tcount @c LocalWords: windingnumber clickable pdfmovie dfn du animationdelay fprime @c LocalWords: slidemovies ifdraft embeddedu externalmovie headerpen bodypen @c LocalWords: GaussianSurface multiline binarytree tridiagonal portably AIX @c LocalWords: binarytreetest Henning subsample breakpoint locator wireframe @c LocalWords: labelpath intersectionpoints PSTricks pstextpath curvedlabel @c LocalWords: LeftJustified RightJustified tickmodifier gunzip gmake IRIX dv @c LocalWords: texcommand RET SITEDIR filegraph pathmarkers POSIX binput AOB @c LocalWords: nonportable markinterval stickframe circlebarframe tix @c LocalWords: crossframe tildeframe markangle StickIntervalMarker gswin expm @c LocalWords: CrossIntervalMarker CircleBarIntervalMarker Ghostscript syzygy @c LocalWords: TildeIntervalMarker autoimport calculateTransform bitwise tk @c LocalWords: headersize bodysize minheaderwidth minheaderheight minwidth ZX @c LocalWords: minbodywidth minbodyheight minheight mindiameter reltime PNG @c LocalWords: relpoint Syzygy syzygies seekeof splinetype notaknot slopea ZY @c LocalWords: slopeb nonperiodic circlescale MarkFill ScaleX ScaleY xformat @c LocalWords: onecontour multicontour irregularcontour dvipsOptions saveline @c LocalWords: dirSpecifier controlSpecifier tensionSpecifier atleastflag bsp @c LocalWords: curlSpecifier cputimeformat initializers arbitary redeclaring @c LocalWords: firstname lastname multdiagonal Raphson OmitTick OmitFormat sp @c LocalWords: NoZero NoZeroFormat abbrevation gsOptions namespace redeclared @c LocalWords: atLeast intMin globalwrite quarticroots deconsruct substrings @c LocalWords: usleep currentpatterns trailingzero Orest Shardt DefaultHead @c LocalWords: SimpleHead HookHead TeXHead multipage NURBS inlinemovie dxmax @c LocalWords: simpson NoBox truesize autoscale shadestroke recurses mintimes @c LocalWords: nonoverlapping texengine maxtimes maxheight pdb TEXMFCONFIG Jn @c LocalWords: piecewisestraight unitrand graphmarkers antialias nolight newl @c LocalWords: Delaunay Shewchuk convertOptions APPDATA pdfreload tempFile Yn @c LocalWords: pdfreloadOptions deferred OpenGL renderer unitbox 's @c LocalWords: bezulate Shardt's rasterized viewport unitdisk unitplane devel @c LocalWords: unitcylinder unitcone solidcone unitfrustum unitsphere nslices @c LocalWords: DPostScript YZZero externalprc nonrendered nosafe KDE @c LocalWords: unithemisphere versa XYplane xypart unitsolidcone YZEquals xml @c LocalWords: XZEquals XYEquals XZZero XYZero InTicks OutTicks InOutTicks @c LocalWords: fitscreen planeproject strokepath meshlight nullpens arrowdir @c LocalWords: diffusepen emissivepen specularpen arrowbarb keyval @c LocalWords: hstretch vstretch roundbox nonconvex miterlimit basealign cmd @c LocalWords: maxviewport maxtile antialiased sphericalharmonic attachfile @c LocalWords: vertexshading smoothelevation glOptions iconified iconify kate @c LocalWords: psviewerOptions pdfviewerOptions viewportmargin asyattach SVG @c LocalWords: multisampling autogen multisample coloredpath relstep flowdir @c LocalWords: colortype coloredSegments coloredNodes trefoilknot scaledgraph @c LocalWords: minblockwidth minblockheight mincirclediameter nonassociative @c LocalWords: nonintegral gettriple enablerepo hexadecimal XeLaTeX xelatex @c LocalWords: dvipdfmx autoadjust viewportsize viewportwidth viewportheight @c LocalWords: subregions nonsimply functionshade shader floatingdisk TopView @c LocalWords: functionshading maxlength LeftView odetest RadialShadeDraw CLZ @c LocalWords: vectorfieldsphere RightView FrontView BackView BottomView CTZ @c LocalWords: addViews outprefix addAllViews xsplinetype ysplinetype rotateX @c LocalWords: usplinetype vsplinetype leftbutton middlebutton rightbutton @c LocalWords: rotateY rotateZ wheelup zoomin wheeldown zoomout TeXLive pnorm @c LocalWords: viewportshift signedint signedness psview multiplatform nowarn @c LocalWords: singlereal singleint writeoverloaded dvisvg reddash lexorder @c LocalWords: bigdiagonal autobillboard dvisvgm maxtiles hyperrefOptions xdr @c LocalWords: setpagesize pdfborder controlsystem OmitTickInterval SixViews @c LocalWords: OmitTickIntervals tickmodifiers autorotated SixViewsUS latexmk @c LocalWords: ThreeViewsUS ThreeViewsFR SixViewsFR ThreeViews partialsum @c LocalWords: defaultrender Vidiassov latexmkrc mktemp DOSendl DOSnewl perl @c LocalWords: filename asyinclude latemk penfunctionimage Affine decrement @c LocalWords: affine Redisplay redisplay isnan radians defaultseparator Jens @c LocalWords: ascii piecewise arcpoint spacings tilings sncndn resizing @c LocalWords: differentiable vectorization vectorized asydir normals quartic @c LocalWords: wavepacket kerned parametrized specular hyperboloid Bourke's @c LocalWords: Michail 0pt 1filll 's 3D labelpath3 2D graph3 0pt 3D @c LocalWords: grid3 contour3 x86_64 psv a4 gsview32 freeglut 'load ' @c LocalWords: 'asy 'lasy 'auto 5bp 1cm sqrtx01 4g extenda extendb @c LocalWords: bb llx 2S 100pt 3t bezier2 bool3 x0 angle1 angle2 z1 @c LocalWords: z2 before' struct X11 x11colors type1cm 12pt OT1 5mm @c LocalWords: cmr12 x' y' xsize ysize 25cm s1 s2 neighbourhood u'' @c LocalWords: s'' 3x 5x 3y 602e 2x 2y 3sin 10cm 204e addby7 10x 's @c LocalWords: only'' pow10 log10 expm1 log1p atan2 0pt 1filll 's ' @c LocalWords: x1 x2 graph2d attachfile2 n0 P0 n1 P1 markers1 3D 2D @c LocalWords: interpolate1 markers2 inlinemovie3 media9 U3D T2A 5E @c LocalWords: embeddedu3d curvedlabel3 value2 tickvalue inner'' 2N @c LocalWords: lineargraph0 scalings log2 log2graph 5cm BWRainbow2 @c LocalWords: guide3 path3 unitcircle3 2E 2n noV 100d PostScript3D @c LocalWords: size3 fit3 theta1 phi1 theta2 phi2 v1 v2 unitsquare3 @c LocalWords: t1 t2 5z 5y transform3 identity4 xscale3 yscale3 0pt @c LocalWords: zscale3 scale3 join3 BeginBar3 EndBar3 Bar3 Bars3 's @c LocalWords: BeginArrow3 MidArrow3 EndArrow3 Arrow3 Arrows3 axes3 @c LocalWords: BeginArcArrow3 MidArcArrow3 EndArcArrow3 ArcArrow3 ' @c LocalWords: ArcArrows3 DefaultHead3 HookHead3 TeXHead3 HookHead2 @c LocalWords: DefaultHead2 TeXHead2 arrows3 NoMargin3 BeginMargin3 @c LocalWords: EndMargin3 Margin3 Margins3 BeginPenMargin2 xaxis3 ' @c LocalWords: EndPenMargin2 PenMargin2 PenMargins2 BeginPenMargin3 @c LocalWords: EndPenMargin3 PenMargin3 PenMargins3 BeginDotMargin3 @c LocalWords: EndDotMargin3 DotMargin3 DotMargins3 TrueMargin3 3D @c LocalWords: yaxis3 zaxis3 ticks3 NoTicks3 arrowbar3 type2 axis3 @c LocalWords: generalaxis3 vectorfield3 margin3 grid3xyz 5unit 2D @c LocalWords: slopefield1 144x144 1filll 'load 'asy 'lasy 'auto 4g @c LocalWords: libgs 'load 'asy 'lasy 'auto 5bp 1cm 2S 100pt 3t 5mm @c LocalWords: bracedefaultratio incircle 12pt 25cm 3x 5x 3y 602e ' @c LocalWords: 2x 2y 3sin 10cm 204e 10x 5E offaxis 'load 'lasy ' 3D @c LocalWords: 5cm 2N 2E 2n 100d 5z 5y 5unit dvisvgmOptions 144x144 @c LocalWords: 4g texengines coplanar 0pt 1filll 's 3D 2D 'load 5bp @c LocalWords: insphere cospherical 5unit luatex lualatex 'asy 1cm @c LocalWords: 'lasy 'auto 4g 2S 100pt 3t 12pt 5mm 25cm 3x 5x 3y 2x @c LocalWords: 602e 2y 3sin 10cm 204e 10x 0pt 1filll 2D DCMAKE CXX @c LocalWords: unnormalized 5E 5cm 2N 2E 2n 100d 5z 5y 0pt 1filll ' @c LocalWords: 5unit 144x144 aligndir smoothcontour3 's 3D 2D cmake @c LocalWords: 'load 'asy 'lasy 'auto 5bp 1cm 4g 2S 100pt 3t nan 3x @c LocalWords: 12pt 5mm 25cm 5x 3y 602e 2x 2y 3sin 10cm 204e 10x 4g @c LocalWords: 5E 5cm 2N 2E 2n 100d 5z 5y nz fcommon 'load 'asy 5bp @c LocalWords: 5unit Staats implicitsurface overlapedges maxdepth ' @c LocalWords: through'' genustwo 144x144 0pt 1filll 's 3D 2D 'load @c LocalWords: 'asy 'lasy 'auto 5bp 1cm 4g 2S 100pt 3t 12pt 5mm 3x @c LocalWords: 25cm 5x 3y 602e 2x 2y 3sin 10cm 204e 10x 'lasy 'auto @c LocalWords: 5E 5cm 2N 2E 2n 100d 5z 5y 5unit 144x144 1cm newpage @c LocalWords: Frohlich codequoteundirected center 0pt 1filll 's 3D @c LocalWords: acknowledgments Colors 2D Color WebGL uref x86 dnf @c LocalWords: htmlviewer asygl CPPFLAGS 'load 'asy 'lasy 'auto 5bp @c LocalWords: 1cm labeling dotfilltype 4g color colors centered 2S @c LocalWords: 100pt 3t forcemath gray colorless miter 12pt 5mm 3x @c LocalWords: zeroTransform 25cm Python3 popcount bitreverse 5x 3y @c LocalWords: 602e 2x 2y 3sin 10cm 204e 10x 2S 100pt 3t 12pt 5mm @c LocalWords: findall ax 5a centers 5E 5cm 2N 2E 2n HTML5 html 3x @c LocalWords: logo3 remeshed css 42kB 100d 5z 5y 5unit colored Qt5 @c LocalWords: behavior beveled usetriangles htmlviewerOptions cson @c LocalWords: 144x144 pyqt5 numpy pip3 PyQt5 rsvg librsvg2 1filll @c LocalWords: librsvg Supakorn Jamie'' Rassameemasmuang 2D Docdir @c LocalWords: microsoft 'load 'asy 'lasy 'auto dep 4g isometry 5x @c LocalWords: 5bp 1cm BezierPatch 2S 100pt 3t abs2 12pt cp1251 5mm @c LocalWords: anttor fontenc inputenc 25cm noglobalread 3x 25cm 3y @c LocalWords: 5x 3y 602e 2x 2y 3sin 10cm 204e 10x libcurl 602e 2x @c LocalWords: mapArray 5a parameterized mapTemplate 5E 2N 2y 3sin @c LocalWords: 5cm freshnel0 fresnel0 PBR prethree specularfactor @c LocalWords: renderers 2E ESC AsyGL 48kB 2n 100d 5z 5y 5unit 10cm @c LocalWords: unicode 144x144 Pedram Emami 204e 10x Ai Ai Ai Ai Ai @c LocalWords: Ai Ai Ai 5a 5E 5cm 2N 2E devicepixelratio 48kB 2n 5z @c LocalWords: 100d 5y 5unit 144x144