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diff --git a/Build/source/utils/asymptote/doc/png/asymptote.info-1 b/Build/source/utils/asymptote/doc/png/asymptote.info-1 new file mode 100644 index 00000000000..fb26ac9b3a1 --- /dev/null +++ b/Build/source/utils/asymptote/doc/png/asymptote.info-1 @@ -0,0 +1,7593 @@ +This is asymptote.info, produced by makeinfo version 4.13 from +../asymptote.texi. + +This file documents `Asymptote', version 1.75. + + `http://asymptote.sourceforge.net' + + Copyright (C) 2004-9 Andy Hammerlindl, John Bowman, and Tom Prince. + + Permission is granted to copy, distribute and/or modify this + document under the terms of the GNU Lesser General Public License + (see the file LICENSE in the top-level source directory). + + +INFO-DIR-SECTION Languages +START-INFO-DIR-ENTRY +* asymptote: (asymptote). Vector graphics language. +END-INFO-DIR-ENTRY + + +File: asymptote.info, Node: Top, Next: Description, Up: (dir) + +Asymptote +********* + +This file documents `Asymptote', version 1.75. + + `http://asymptote.sourceforge.net' + + Copyright (C) 2004-9 Andy Hammerlindl, John Bowman, and Tom Prince. + + Permission is granted to copy, distribute and/or modify this + document under the terms of the GNU Lesser General Public License + (see the file LICENSE in the top-level source directory). + + +* Menu: + +* Description:: What is `Asymptote'? +* Installation:: Downloading and installing +* Tutorial:: Getting started +* Drawing commands:: Four primitive graphics commands +* Programming:: The `Asymptote' vector graphics language +* LaTeX usage:: Embedding `Asymptote' commands within `LaTeX' +* Base modules:: Base modules shipped with `Asymptote' +* Options:: Command-line options +* Interactive mode:: Typing `Asymptote' commands interactively +* GUI:: Graphical user interface +* PostScript to Asymptote:: `Asymptote' backend to `pstoedit' +* Help:: Where to get help and submit bug reports +* Debugger:: Squish those bugs! +* Credits:: Contributions and acknowledgments +* Index:: General index + + --- The Detailed Node Listing --- + +Installation + +* UNIX binary distributions:: Prebuilt `UNIX' binaries +* MacOS X binary distributions:: Prebuilt `MacOS X' binaries +* Microsoft Windows:: Prebuilt `Microsoft Windows' binary +* Configuring:: Configuring `Asymptote' for your system +* Search paths:: Where `Asymptote' looks for your files +* Compiling from UNIX source:: Building `Asymptote' from scratch +* Editing modes:: Convenient `emacs' and `vim' modes +* Subversion:: +* Uninstall:: Goodbye, `Asymptote'! + +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:: +* 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 `Asymptote' packages +* 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 `Asymptote' base file +* simplex:: Linear programming: simplex method +* math:: Extend `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 +* 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 PDF and MPEG movies +* embed:: Embedding movies, sounds, and 3D objects +* slide:: Making presentations with `Asymptote' +* MetaPost:: `MetaPost' compatibility routines +* unicode:: Accept `unicode' (UTF-8) characters +* latin1:: Accept `ISO 8859-1' characters +* babel:: Interface to `LaTeX' `babel' package +* labelpath:: Drawing curved labels +* labelpath3:: Drawing curved labels in 3D +* annotate:: Annotate your 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 +* slopefield:: Slope fields + + +Graphical User Interface + +* GUI Installation:: Installing `xasy' +* GUI Usage:: + + +File: asymptote.info, Node: Description, Next: Installation, Prev: Top, Up: Top + +1 Description +************* + +`Asymptote' is a powerful descriptive vector graphics language that +provides a mathematical coordinate-based framework for technical +drawings. Labels and equations are typeset with `LaTeX', for overall +document consistency, yielding the same high-quality level of +typesetting that `LaTeX' provides for scientific text. By default it +produces `PostScript' output, but it can also generate any format that +the `ImageMagick' package can produce. + + A major advantage of `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 (GUI) methods for +producing figures. The rudimentary GUI `xasy' included with the package +allows one to move script-generated objects around. To make `Asymptote' +accessible to the average user, this GUI is currently being developed +into a full-fledged interface that can generate objects directly. +However, the script portion of the language is now ready for general +use by users who are willing to learn a few simple `Asymptote' graphics +commands (*note Drawing commands::). + + `Asymptote' is mathematically oriented (e.g. one can use complex +multiplication to rotate a vector) and uses `LaTeX' to do the +typesetting of labels. This is an important feature for scientific +applications. It was inspired by an earlier drawing program (with a +weaker syntax and capabilities) called `MetaPost'. + + The `Asymptote' vector graphics language provides: + + * a standard for typesetting mathematical figures, just as + TeX/`LaTeX' is the de-facto standard for typesetting equations. + + * `LaTeX' typesetting of labels, for overall document consistency; + + * a natural coordinate-based framework for technical drawings, + inspired by `MetaPost', with a much cleaner, powerful C++-like + programming syntax; + + * compilation of figures into virtual machine code for speed, without + sacrificing portability; + + * the power of a script-based language coupled to the convenience of + a GUI; + + * customization using its own C++-like graphics programming language; + + * sensible defaults for graphical features, with the ability to + override; + + * a high-level mathematically oriented interface to the `PostScript' + language for vector graphics, including affine transforms and + complex variables; + + * functions that can create new (anonymous) functions; + + * deferred drawing that uses the simplex method to solve overall size + constraint issues between fixed-sized objects (labels and + arrowheads) and objects that should scale with figure size; + + + Many of the features of `Asymptote' are written in the `Asymptote' +language itself. While the stock version of `Asymptote' is designed for +mathematics typesetting needs, one can write `Asymptote' modules that +tailor it to specific applications. A scientific graphing module has +already been written (*note graph::). Examples of `Asymptote' code and +output, including animations, are available at + + `http://asymptote.sourceforge.net/gallery/'. + Links to many external resources, including an excellent user-written +`Asymptote' tutorial can be found at + + `http://asymptote.sourceforge.net/links.html'. + + +File: asymptote.info, Node: Installation, Next: Tutorial, Prev: Description, Up: Top + +2 Installation +************** + +* Menu: + +* UNIX binary distributions:: Prebuilt `UNIX' binaries +* MacOS X binary distributions:: Prebuilt `MacOS X' binaries +* Microsoft Windows:: Prebuilt `Microsoft Windows' binary +* Configuring:: Configuring `Asymptote' for your system +* Search paths:: Where `Asymptote' looks for your files +* Compiling from UNIX source:: Building `Asymptote' from scratch +* Editing modes:: Convenient `emacs' and `vim' modes +* Subversion:: +* Uninstall:: Goodbye, `Asymptote'! + + After following the instructions for your specific distribution, +please see also *note Configuring::. + +We recommend subscribing to new release announcements at + + `http://freshmeat.net/projects/asy' + Users may also wish to monitor the `Asymptote' forum: + + `http://sourceforge.net/forum/monitor.php?forum_id=409349' + + +File: asymptote.info, Node: UNIX binary distributions, Next: MacOS X binary distributions, Up: Installation + +2.1 UNIX binary distributions +============================= + +We release both `tgz' and RPM binary distributions of `Asymptote'. The +root user can install the `Linux i386' `tgz' distribution of version +`x.xx' of `Asymptote' with the commands: +tar -C / -zxf asymptote-x.xx.i386.tgz +texhash + The `texhash' command, which installs LaTeX style files, is optional. +The executable file will be `/usr/local/bin/asy') and example code will +be installed by default in `/usr/local/share/doc/asymptote/examples'. + +Fedora users can easily install the most recent version of `Asymptote' +with the command +yum --enablerepo=rawhide install asymptote + +To install the latest version of `Asymptote' on a Debian-based +distribution (e.g. Ubuntu, Mepis, Linspire) follow the instructions for +compiling from `UNIX' source (*note Compiling from UNIX source::). +Alternatively, Debian users can install one of Hubert Chan's prebuilt +`Asymptote' binaries from + + `http://ftp.debian.org/debian/pool/main/a/asymptote' + + +File: asymptote.info, Node: MacOS X binary distributions, Next: Microsoft Windows, Prev: UNIX binary distributions, Up: Installation + +2.2 MacOS X binary distributions +================================ + +`MacOS X' users can either compile the `UNIX' source code (*note +Compiling from UNIX source::) or install the contributed `Asymptote' +binary available at + +`http://www.hmug.org/pub/MacOS_X/X/Applications/Publishing/asymptote/' + +Because these preconfigured binary distributions have strict +architecture and library dependencies that many installations do not +satisfy, we recommend installing `Asymptote' directly from the official +source: + + `http://sourceforge.net/project/showfiles.php?group_id=120000' + +Note that many `MacOS X' (and FreeBSD) systems inexplicably ship with +an extremely old GNU `readline' version (4.1, dated 21 March 2000). For +full interactive functionality, `readline' version 4.2 or later (16 +April 2001) is required. + + +File: asymptote.info, Node: Microsoft Windows, Next: Configuring, Prev: MacOS X binary distributions, Up: Installation + +2.3 Microsoft Windows +===================== + +Users of the `Microsoft Windows' operating system can install the +self-extracting `Asymptote' executable `asymptote-x.xx-setup.exe', +where `x.xx' denotes the latest version. + + A working TeX implementation (such as the one available at +`http://www.miktex.org') will be required to typeset labels. You will +also need to install `GPL Ghostscript' from +`http://sourceforge.net/projects/ghostscript/'. To view the default +`PostScript' output, you can install the program `gsview' available from +`http://www.cs.wisc.edu/~ghost/gsview/'. + + The `ImageMagick' package from + + `http://www.imagemagick.org/script/binary-releases.php' + +is required to support output formats other than EPS and PDF (*note +convert::). The `Python' interpreter from `http://www.python.org' is +only required if you wish to try out the graphical user interface +(*note GUI::). + +Example code will be installed by default in the `examples' +subdirectory of the installation directory (by default, `C:\Program +Files\Asymptote'). + + +File: asymptote.info, Node: Configuring, Next: Search paths, Prev: Microsoft Windows, Up: Installation + +2.4 Configuring +=============== + +In interactive mode, or when given the `-V' option (the default when +running `Asymptote' on a single file under `MSDOS'), `Asymptote' will +automatically invoke the `PostScript' viewer `gv' (under `UNIX') or +`gsview' (under `MSDOS' to display graphical output. These defaults may +be overridden with the configuration variable `psviewer'. The +`PostScript' viewer should be capable of automatically redrawing +whenever the output file is updated. The default `UNIX' `PostScript' +viewer `gv' supports this (via a `SIGHUP' signal). Version `gv-3.6.3' +or later (from `http://ftp.gnu.org/gnu/gv/') is required for +interactive mode to work properly. Users of `ggv' will need to enable +`Watch file' under `Edit/Postscript Viewer Preferences'. Users of +`gsview' will need to enable `Options/Auto Redisplay' (however, under +`MSDOS' it is still necessary to click on the `gsview' window; under +`UNIX' one must manually redisplay by pressing the `r' key). + + Configuration variables are most easily set as `Asymptote' variables +in the configuration file (by default, `.asy/config.asy' in the user's +home directory or `%USERPROFILE%\.asy\config.asy' under `MSDOS'); *note +configuration file::. Here are the default values of several important +configuration variables under `UNIX': + + +import settings; +psviewer="gv"; +pdfviewer="acroread"; +gs="gs"; +python=""; + +The (installation-dependent) default values of these configuration +variables under `MSDOS' are determined automatically from the +`Microsoft Windows' registry. + + For PDF format output, the `gs' setting specifies the location of +the `PostScript'-to-PDF processor `Ghostscript', available from +`http://sourceforge.net/projects/ghostscript/'. + + The setting `pdfviewer' specifies the location of the PDF viewer. On +`UNIX' systems, to support automatic document reloading in `Adobe +Reader', we recommend copying the file `reload.js' from the `Asymptote' +system directory (by default, `/usr/local/share/asymptote' under `UNIX' +to `~/.adobe/Acrobat/x.x/JavaScripts/', where `x.x' represents the +appropriate `Adobe Reader' version number. The automatic document +reload feature must then be explicitly enabled by putting +import settings; +pdfreload=true; +pdfreloadOptions="-tempFile"; + in the `Asymptote' configuration file. This reload feature is not +useful under `MSDOS' since the document cannot be updated anyway on +that operating system until it is first closed by `Adobe Reader'. + + The graphical user interface may also require setting the variable +`python' if `Python' is installed in a nonstandard location. + + The configuration variable `dir' can be used to adjust the search +path (*note Search paths::). + + By default, `Asymptote' attempts to center the figure on the page, +assuming that the paper type is `letter'. The default paper type may be +changed to `a4' with the configuration variable `papertype'. Alignment +to other paper sizes can be obtained by setting the configuration +variables `paperwidth' and `paperheight'. + + The following configuration variables normally do not require +adjustment: +texpath +texcommand +texdvicommand +dvips +convert +display +animate +xasy + The `texdvicommand' is used for `3D' label typesetting, which requires +`dvips' output. An empty string indicates the default setting of +`latex'/`tex', depending on the setting of `texengine'. + + Configuration variables may also be set or overwritten with a +command-line option: +asy -psviewer=gsview -V venn + + Alternatively, system environment versions of the above configuration +variables may be set in the conventional way. The corresponding +environment variable name is obtained by converting the configuration +variable name to upper case and prepending `ASYMPTOTE_': for example, +to set the environment variable +ASYMPTOTE_PSVIEWER="C:\Program Files\Ghostgum\gsview\gsview32.exe"; + under `Microsoft Windows XP': + 1. Click on the `Start' button; + + 2. Right-click on `My Computer'; + + 3. Choose `Properties' from the popup menu; + + 4. Click the `Advanced' tab; + + 5. Click the `Environment Variables' button. + + +File: asymptote.info, Node: Search paths, Next: Compiling from UNIX source, Prev: Configuring, Up: Installation + +2.5 Search paths +================ + +In looking for `Asymptote' system files, `asy' will search the +following paths, in the order listed: + 1. The current directory; + + 2. A list of one or more directories specified by the configuration + variable `dir' (separated by `:' under UNIX and `;' under `MSDOS'); + + 3. The directory `.asy' in the user's home directory + (`%USERPROFILE%\.asy' under `MSDOS'); + + 4. The `Asymptote' system directory (by default, + `/usr/local/share/asymptote' under `UNIX' and `C:\Program + Files\Asymptote' under `MSDOS'). + + +File: asymptote.info, Node: Compiling from UNIX source, Next: Editing modes, Prev: Search paths, Up: Installation + +2.6 Compiling from UNIX source +============================== + +To compile and install a `UNIX' executable from a source release +`x.xx', first execute the commands: +gunzip asymptote-x.xx.src.tgz +tar -xf asymptote-x.xx.src.tar +cd asymptote-x.xx + By default the system version of the Boehm garbage collector will be +used; if it is old we recommend first putting +`http://www.hpl.hp.com/personal/Hans_Boehm/gc/gc_source/gc-7.1.tar.gz' +in the `Asymptote' source directory. + +If your graphics card supports multisampling, we recommend using SVN +revision 761 (or later) of `freeglut' to support antialiasing in +`Asymptote''s adaptive `OpenGL' 3D renderer (`MacOS X' users can skip +this step since `Asymptote' is configured to use the native glut +library on that platform). Download + + `http://freeglut.svn.sourceforge.net/viewvc/freeglut/trunk/freeglut/freeglut.tar.gz' + and type (as the root user): +tar -zxf freeglut.tar.gz +cd freeglut +sh autogen.sh +./configure --prefix=/usr +make install +cd .. + Then compile `Asymptote' with the commands +./configure +make all +make install + Be sure to use GNU `make' (on non-GNU systems this command may be +called `gmake'). To build the documentation, you may need to install +the `texinfo-tex' package. If you get errors from a broken `texinfo' or +`pdftex' installation, simply put + + `http://asymptote.sourceforge.net/asymptote.pdf' + in the directory `doc' and repeat the command `make all'. + +For a (default) system-wide installation, the last command should be +done as the root user. To install without root privileges, change the +`./configure' command to +./configure --prefix=$HOME/asymptote + One can disable use of the Boehm garbage collector by configuring with +`./configure --disable-gc'. For a list of other configuration options, +say `./configure --help'. For example, one can tell configure to look +for header files and libraries in nonstandard locations: +./configure CFLAGS=-I/opt/usr/include LDFLAGS=-L/opt/usr/lib + + If you are compiling `Asymptote' with `gcc', you will need a +relatively recent version (e.g. 3.4.4 or later). For full interactive +functionality, you will need version 4.2 or later of the GNU `readline' +library. The file `gcc3.3.2curses.patch' in the `patches' directory can +be used to patch the broken curses.h header file (or a local copy +thereof in the current directory) on some `AIX' and `IRIX' systems. + + The `FFTW' library is only required if you want `Asymptote' to be +able to take Fourier transforms of data (say, to compute an audio power +spectrum). The `GSL' library is only required if you require the +special functions that it supports. + + If you don't want to install `Asymptote' system wide, just make sure +the compiled binary `asy' and GUI script `xasy' are in your path and +set the configuration variable `dir' to point to the directory `base' +(in the top level directory of the `Asymptote' source code). + + +File: asymptote.info, Node: Editing modes, Next: Subversion, Prev: Compiling from UNIX source, Up: Installation + +2.7 Editing modes +================= + +Users of `emacs' can edit `Asymptote' code with the mode `asy-mode', +after enabling it by putting the following lines in their `.emacs' +initialization file, replacing `ASYDIR' with the location of the +`Asymptote' system directory (by default, `/usr/local/share/asymptote' +or `C:\Program Files\Asymptote' under `MSDOS'): +(add-to-list 'load-path "ASYDIR") +(autoload 'asy-mode "asy-mode.el" "Asymptote major mode." t) +(autoload 'lasy-mode "asy-mode.el" "hybrid Asymptote/Latex major mode." t) +(autoload 'asy-insinuate-latex "asy-mode.el" "Asymptote insinuate LaTeX." t) +(add-to-list 'auto-mode-alist '("\\.asy$" . asy-mode)) + + Particularly useful key bindings in this mode are `C-c C-c', which +compiles and displays the current buffer, and the key binding `C-c ?', +which shows the available function prototypes for the command at the +cursor. For full functionality you should also install the Apache +Software Foundation package `two-mode-mode': + + `http://www.dedasys.com/freesoftware/files/two-mode-mode.el' + Once installed, you can use the hybrid mode `lasy-mode' to edit a +LaTeX file containing embedded `Asymptote' code (*note LaTeX usage::). +This mode can be enabled within `latex-mode' with the key sequence `M-x +lasy-mode <RET>'. On `UNIX' systems, additional keywords will be +generated from all `asy' files in the space-separated list of +directories specified by the environment variable `ASYMPTOTE_SITEDIR'. +Further documentation of `asy-mode' is available within `emacs' by +pressing the sequence keys `C-h f asy-mode <RET>'. + + Fans of `vim' can customize `vim' for `Asymptote' with + +`cp /usr/local/share/asymptote/asy.vim ~/.vim/syntax/asy.vim' + +and add the following to their `~/.vimrc' file: +augroup filetypedetect +au BufNewFile,BufRead *.asy setf asy +augroup END +filetype plugin on + + If any of these directories or files don't exist, just create them. +To set `vim' up to run the current asymptote script using `:make' just +add to `~/.vim/ftplugin/asy.vim': +setlocal makeprg=asy\ % +setlocal errorformat=%f:\ %l.%c:\ %m + + +File: asymptote.info, Node: Subversion, Next: Uninstall, Prev: Editing modes, Up: Installation + +2.8 Subversion (SVN) +==================== + +The following commands are needed to install the latest development +version of `Asymptote' using Subversion: +svn co http://asymptote.svn.sourceforge.net/svnroot/asymptote/trunk/asymptote +cd asymptote +./autogen.sh +./configure +make all +make install + +To compile without optimization, use the command `make CFLAGS=-g'. + + +File: asymptote.info, Node: Uninstall, Prev: Subversion, Up: Installation + +2.9 Uninstall +============= + +To uninstall an `Linux i386' binary distribution, use the commands +tar -zxvf asymptote-x.xx.i386.tgz | xargs --replace=% rm /% +texhash + +To uninstall all `Asymptote' files installed from a source +distribution, use the command +make uninstall + + +File: asymptote.info, Node: Tutorial, Next: Drawing commands, Prev: Installation, Up: Top + +3 Tutorial +********** + +_An excellent user-written `Asymptote' tutorial is also available from_ + + `http://www.artofproblemsolving.com/Wiki/index.php/Asymptote:_Basics' + +To draw a line from coordinate (0,0) to coordinate (100,100) using +`Asymptote''s interactive mode, type at the command prompt: +asy +draw((0,0)--(100,100)); + + + +The units here are `PostScript' "big points" (1 `bp' = 1/72 `inch'); +`--' means join with a linear segment. In `Asymptote' coordinates like +`(0,0)' and `(1000,100)' are called _pairs_. + + At this point you can type in further draw commands, which will be +added to the displayed figure, or type `quit' to exit interactive mode. +You can use the arrow keys in interactive mode to edit previous lines +(assuming that you have support for the GNU `readline' library +enabled). The tab key will automatically complete unambiguous words; +otherwise, hitting tab again will show the possible choices. Further +commands specific to interactive mode are described in *note +Interactive mode::. + + In batch mode, `Asymptote' reads commands directly from a file. To +try this out, type + +draw((0,0)--(100,100)); + into a file, say test.asy. Then execute this file with the `MSDOS' or +`UNIX' command +asy -V test + `MSDOS' users can drag and drop the file onto the Desktop `asy' icon +or make `Asymptote' the default application for files with the +extension `asy'. + +The `-V' option opens up a `PostScript' viewer window so you can +immediately view the encapsulated `PostScript' output. By default the +output will be written to the file `test.eps'; the prefix of the output +file may be changed with the `-o' command-line option. + + One can draw a line with more than two points and create a cyclic +path like this square: + +draw((0,0)--(100,0)--(100,100)--(0,100)--cycle); + + + +It is often inconvenient to work directly with `PostScript' coordinates. +The next example draws a unit square scaled to width 101 bp and height +101 bp. The output is identical to that of the previous example. +size(101,101); +draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); + + For convenience, the path `(0,0)--(1,0)--(1,1)--(0,1)--cycle' may be +replaced with the predefined variable `unitsquare', or equivalently, +`box((0,0),(1,1))'. + + One can also specify the size in `pt' (1 `pt' = 1/72.27 `inch'), +`cm', `mm', or `inches'. If 0 is given as a size argument, no +restriction is made in that direction; the overall scaling will be +determined by the other direction (*note size::): + +size(0,3cm); +draw(unitsquare); + + + +To make the user coordinates represent multiples of exactly `1cm': +unitsize(1cm); +draw(unitsquare); + + One can also specify different x and y unit sizes: +unitsize(1cm,2cm); +draw(unitsquare); + + Adding labels is easy in `Asymptote'; one specifies the label as a +double-quoted `LaTeX' string, a coordinate, and an optional alignment +direction: + +size(0,3cm); +draw(unitsquare); +label("$A$",(0,0),SW); +label("$B$",(1,0),SE); +label("$C$",(1,1),NE); +label("$D$",(0,1),NW); + + + +`Asymptote' uses the standard compass directions `E=(1,0)', `N=(0,1)', +`NE=unit(N+E)', and `ENE=unit(E+NE)', etc., which along with the +directions `up', `down', `right', and `left' are defined as pairs in +the `Asymptote' base module `plain'. A user who has a local variable +named `E' may access the compass direction `E' by prefixing it with the +name of the module where it is defined: `plain.E'. + + This example draws a path that approximates a quarter circle: + +size(100,0); +draw((1,0){up}..{left}(0,1)); + + + +In general, a path is specified as a list of pairs (or other paths) +interconnected with `--', which denotes a straight line segment, or +`..', which denotes a cubic spline. Specifying a final node `cycle' +creates a cyclic path that connects smoothly back to the initial node, +as in this approximation (accurate to within 0.06%) of a unit circle: +path unitcircle=E..N..W..S..cycle; + + Each interior node of a cubic spline may be given a direction prefix +or suffix `{dir}': the direction of the pair `dir' specifies the +direction of the incoming or outgoing tangent, respectively, to the +curve at that node. Exterior nodes may be given direction specifiers +only on their interior side. + + A cubic spline between the node z_0, with postcontrol point c_0, and +the node z_1, with precontrol point c_1, is computed as the Bezier curve + + + +As illustrated in the diagram below, the third-order midpoint (m_5) +constructed from two endpoints z_0 and z_1 and two control points c_0 +and c_1, is the point corresponding to t=1/2 on the Bezier curve formed +by the quadruple (z_0, c_0, c_1, z_1). This allows one to recursively +construct the desired curve, by using the newly extracted third-order +midpoint as an endpoint and the respective second- and first-order +midpoints as control points: + + + +Here m_0, m_1 and m_2 are the first-order midpoints, m_3 and m_4 are +the second-order midpoints, and m_5 is the third-order midpoint. The +curve is then constructed by recursively applying the algorithm to +(z_0, m_0, m_3, m_5) and (m_5, m_4, m_2, z_1). + + In fact, an analogous property holds for points located at any +fraction t in [0,1] of each segment, not just for midpoints (t=1/2). + + The Bezier curve constructed in this manner has the following +properties: + * It is entirely contained in the convex hull of the given four + points. + + * It starts heading from the first endpoint to the first control + point and finishes heading from the second control point to the + second endpoint. + + + The user can specify explicit control points between two nodes like +this: +draw((0,0)..controls (0,100) and (100,100)..(100,0)); + + However, it is usually more convenient to just use the `..' +operator, which tells `Asymptote' to choose its own control points +using the algorithms described in Donald Knuth's monograph, The +MetaFontbook, Chapter 14. The user can still customize the guide (or +path) by specifying direction, tension, and curl values. + + The higher the tension, the straighter the curve is, and the more it +approximates a straight line. One can change the spline tension from +its default value of 1 to any real value greater than or equal to 0.75 +(cf. John D. Hobby, Discrete and Computational Geometry 1, 1986): +draw((100,0)..tension 2 ..(100,100)..(0,100)); +draw((100,0)..tension 2 and 1 ..(100,100)..(0,100)); +draw((100,0)..tension atleast 1 ..(100,100)..(0,100)); + + The curl parameter specifies the curvature at the endpoints of a path +(0 means straight; the default value of 1 means approximately circular): +draw((100,0){curl 0}..(100,100)..{curl 0}(0,100)); + + The `MetaPost ...' path connector, which requests, when possible, an +inflection-free curve confined to a triangle defined by the endpoints +and directions, is implemented in `Asymptote' as the convenient +abbreviation `::' for `..tension atleast 1 ..' (the ellipsis `...' is +used in `Asymptote' to indicate a variable number of arguments; *note +Rest arguments::). For example, compare + +draw((0,0){up}..(100,25){right}..(200,0){down}); + + +with + +draw((0,0){up}::(100,25){right}::(200,0){down}); + + + +The `---' connector is an abbreviation for `..tension atleast +infinity..' and the `&' connector concatenates two paths, after first +stripping off the last node of the first path (which normally should +coincide with the first node of the second path). + + An `Asymptote' path, being connected, is equivalent to a `Postscript +subpath'. The `^^' binary operator, which requests that the pen be +moved (without drawing or affecting endpoint curvatures) from the final +point of the left-hand path to the initial point of the right-hand +path, may be used to group several `Asymptote' paths into a `path[]' +array (equivalent to a `PostScript' path): + +size(0,100); +path unitcircle=E..N..W..S..cycle; +path g=scale(2)*unitcircle; +filldraw(unitcircle^^g,evenodd+yellow,black); + + + +The `PostScript' even-odd fill rule here specifies that only the region +bounded between the two unit circles is filled (*note fillrule::). In +this example, the same effect can be achieved by using the default zero +winding number fill rule, if one is careful to alternate the +orientation of the paths: +filldraw(unitcircle^^reverse(g),yellow,black); + + The `^^' operator is used by the `box(triple, triple)' function in +`three.asy' to construct the edges of a cube `unitbox' without +retracing steps: + +import three; +dotgranularity=0; // Render dots as spheres. + +currentprojection=orthographic(5,4,2,center=true); + +size(5cm); +size3(3cm,5cm,8cm); + +draw(unitbox); + +dot(unitbox,red); + +label("$O$",(0,0,0),NW); +label("(1,0,0)",(1,0,0),S); +label("(0,1,0)",(0,1,0),E); +label("(0,0,1)",(0,0,1),Z); + + + +See section *note graph:: (or the online `Asymptote' gallery and +external links posted at `http://asymptote.sourceforge.net') for +further examples, including two and three-dimensional scientific +graphs. Additional examples have been posted by Philippe Ivaldi at +`http://piprim.tuxfamily.org/asymptote/'. + + +File: asymptote.info, Node: Drawing commands, Next: Programming, Prev: Tutorial, Up: Top + +4 Drawing commands +****************** + +All of `Asymptote''s graphical capabilities are based on four primitive +commands. The three `PostScript' drawing commands `draw', `fill', and +`clip' add objects to a picture in the order in which they are +executed, with the most recently drawn object appearing on top. The +labeling command `label' can be used to add text labels and external +EPS images, which will appear on top of the `PostScript' objects (since +this is normally what one wants), but again in the relative order in +which they were executed. After drawing objects on a picture, the +picture can be output with the `shipout' function (*note shipout::). + + If you wish to draw `PostScript' objects on top of labels (or +verbatim `tex' commands; *note tex::), the `layer' command may be used +to start a new `PostScript/LaTeX' layer: +void layer(picture pic=currentpicture); + + The `layer' function gives one full control over the order in which +objects are drawn. Layers are drawn sequentially, with the most recent +layer appearing on top. Within each layer, labels, images, and verbatim +`tex' commands are always drawn after the `PostScript' objects in that +layer. + + While some of these drawing commands take many options, they all +have sensible default values (for example, the picture argument +defaults to currentpicture). + +* Menu: + +* draw:: Draw a path on a picture or frame +* fill:: Fill a cyclic path on a picture or frame +* clip:: Clip a picture or frame to a cyclic path +* label:: Label a point on a picture + + +File: asymptote.info, Node: draw, Next: fill, Up: Drawing commands + +4.1 draw +======== + +void draw(picture pic=currentpicture, Label L="", path g, + align align=NoAlign, pen p=currentpen, + arrowbar arrow=None, arrowbar bar=None, margin margin=NoMargin, + Label legend="", marker marker=nomarker); + +Draw the path `g' on the picture `pic' using pen `p' for drawing, with +optional drawing attributes (Label `L', explicit label alignment +`align', arrows and bars `arrow' and `bar', margins `margin', legend, +and markers `marker'). Only one parameter, the path, is required. For +convenience, the arguments `arrow' and `bar' may be specified in either +order. The argument `legend' is a Label to use in constructing an +optional legend entry. + + Bars are useful for indicating dimensions. The possible values of +`bar' are `None', `BeginBar', `EndBar' (or equivalently `Bar'), and +`Bars' (which draws a bar at both ends of the path). Each of these bar +specifiers (except for `None') will accept an optional real argument +that denotes the length of the bar in `PostScript' coordinates. The +default bar length is `barsize(pen)'. + + The possible values of `arrow' are `None', `Blank' (which draws no +arrows or path), `BeginArrow', `MidArrow', `EndArrow' (or equivalently +`Arrow'), and `Arrows' (which draws an arrow at both ends of the path). +All of the arrow specifiers except for `None' and `Blank' may be given +the optional arguments arrowhead `arrowhead' (one of the predefined +arrowhead styles `DefaultHead', `SimpleHead', `HookHead', `TeXHead'), +real `size' (arrowhead size in `PostScript' coordinates), real `angle' +(arrowhead angle in degrees), filltype `filltype' (one of `FillDraw', +`Fill', `NoFill', `UnFill', `Draw') and (except for `MidArrow' and +`Arrows') a relative real `position' along the path (an `arctime') where +the tip of the arrow should be placed. The default arrowhead size when +drawn with a pen `p' is `arrowsize(p)'. There are also arrow versions +with slightly modified default values of `size' and `angle' suitable for +curved arrows: `BeginArcArrow', `EndArcArrow' (or equivalently +`ArcArrow'), `MidArcArrow', and `ArcArrows'. + + Margins can be used to shrink the visible portion of a path by +`labelmargin(p)' to avoid overlap with other drawn objects. Typical +values of `margin' are `NoMargin', `BeginMargin', `EndMargin' (or +equivalently `Margin'), and `Margins' (which leaves a margin at both +ends of the path). One may use `Margin(real begin, real end)' to +specify the size of the beginning and ending margin, respectively, in +multiples of the units `labelmargin(p)' used for aligning labels. +Alternatively, `BeginPenMargin', `EndPenMargin' (or equivalently +`PenMargin'), `PenMargins', `PenMargin(real begin, real end)' 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 `DotMargin', an +abbreviation for `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 `dotfactor*linewidth(p)'. The qualifiers `BeginDotMargin', +`EndDotMargin', and `DotMargins' work similarly. The qualifier +`TrueMargin(real begin, real end)' allows one to specify a margin +directly in `PostScript' units, independent of the pen line width. + + The use of arrows, bars, and margins is illustrated by the examples +`Pythagoras.asy', `sqrtx01.asy', and `triads.asy'. + + The legend for a picture `pic' can be fit and aligned to a frame +with the routine: +frame legend(picture pic=currentpicture, int perline=1, + real xmargin=legendmargin, real ymargin=xmargin, + real linelength=legendlinelength, + real hskip=legendhskip, real vskip=legendvskip, + real maxwidth=0, real maxheight=0, + bool hstretch=false, bool vstretch=false, pen p=currentpen); + Here `xmargin' and `ymargin' specify the surrounding x and y margins, +`perline' specifies the number of entries per line (default 1; 0 means +choose this number automatically), `linelength' specifies the length of +the path lines, `hskip' and `vskip' specify the line skip (as a +multiple of the legend entry size), `maxwidth' and `maxheight' specify +optional upper limits on the width and height of the resulting legend +(0 means unlimited), `hstretch' and `vstretch' allow the legend to +stretch horizontally or vertically, and `p' specifies the pen used to +draw the bounding box. The legend frame can then be added and aligned +about a point on a picture `dest' using `add' or `attach' (*note add +about::). + + To draw a dot, simply draw a path containing a single point. The +`dot' command defined in the module `plain' draws a dot having a +diameter equal to an explicit pen line width or the default line width +magnified by `dotfactor' (6 by default), using the specified filltype +(*note filltype::): +void dot(picture pic=currentpicture, pair z, pen p=currentpen, + filltype filltype=Fill); +void dot(picture pic=currentpicture, Label L, pair z, align align=NoAlign, + string format=defaultformat, pen p=currentpen, filltype filltype=Fill); +void dot(picture pic=currentpicture, Label[] L=new Label[], pair[] z, + align align=NoAlign, string format=defaultformat, pen p=currentpen, + filltype filltype=Fill) +void dot(picture pic=currentpicture, Label L, pen p=currentpen, + filltype filltype=Fill); + + If the variable `Label' is given as the `Label' argument to the +second routine, the `format' argument will be used to format a string +based on the dot location (here `defaultformat' is `"$%.4g$"'). The +third routine draws a dot at every point of a pair array `z'. One can +also draw a dot at every node of a path: +void dot(picture pic=currentpicture, Label[] L=new Label[], + path g, align align=RightSide, string format=defaultformat, + pen p=currentpen, filltype filltype=Fill); + See *note pathmarkers:: and *note markers:: for more general methods +for marking path nodes. + + To draw a fixed-sized object (in `PostScript' coordinates) about the +user coordinate `origin', use the routine +void draw(pair origin, picture pic=currentpicture, Label L="", path g, + align align=NoAlign, pen p=currentpen, arrowbar arrow=None, + arrowbar bar=None, margin margin=NoMargin, Label legend="", + marker marker=nomarker); + + +File: asymptote.info, Node: fill, Next: clip, Prev: draw, Up: Drawing commands + +4.2 fill +======== + +void fill(picture pic=currentpicture, path g, pen p=currentpen); + +Fill the interior region bounded by the cyclic path `g' on the picture +`pic', using the pen `p'. + + There is also a convenient `filldraw' command, which fills the path +and then draws in the boundary. One can specify separate pens for each +operation: +void filldraw(picture pic=currentpicture, path g, pen fillpen=currentpen, + pen drawpen=currentpen); + + This fixed-size version of `fill' allows one to fill an object +described in `PostScript' coordinates about the user coordinate +`origin': +void fill(pair origin, picture pic=currentpicture, path g, pen p=currentpen); + +This is just a convenient abbreviation for the commands: +picture opic; +fill(opic,g,p); +add(pic,opic,origin); + + The routine +void filloutside(picture pic=currentpicture, path g, pen p=currentpen); + fills the region exterior to the path `g', out to the current boundary +of picture `pic'. + + Lattice gradient shading varying smoothly over a two-dimensional +array of pens `p', using fill rule `fillrule', can be produced with +void latticeshade(picture pic=currentpicture, path g, bool stroke=false, + pen fillrule=currentpen, pen[][] p) + If `stroke=true', the region filled is the same as the region that +would be drawn by `draw(pic,g,fillrule+zerowinding)'; in this case the +path `g' need not be cyclic. The pens in `p' must belong to the same +color space. One can use the functions `rgb(pen)' or `cmyk(pen)' to +promote pens to a higher color space, as illustrated in the example file +`latticeshading.asy'. + + Axial gradient shading varying smoothly from `pena' to `penb' in the +direction of the line segment `a--b' can be achieved with +void axialshade(picture pic=currentpicture, path g, bool stroke=false, + pen pena, pair a, + pen penb, pair b); + + Radial gradient shading varying smoothly from `pena' on the circle +with center `a' and radius `ra' to `penb' on the circle with center `b' +and radius `rb' is similar: +void radialshade(picture pic=currentpicture, path g, bool stroke=false, + pen pena, pair a, real ra, + pen penb, pair b, real rb); + Illustrations of radial shading are provided in the example files +`shade.asy', `ring.asy', and `shadestroke.asy'. + + Gouraud shading using fill rule `fillrule' and the vertex colors in +the pen array `p' on a triangular lattice defined by the vertices `z' +and edge flags `edges' is implemented with +void gouraudshade(picture pic=currentpicture, path g, bool stroke=false, + pen fillrule=currentpen, pen[] p, pair[] z, + int[] edges); +void gouraudshade(picture pic=currentpicture, path g, bool stroke=false, + pen fillrule=currentpen, pen[] p, int[] edges); + In the second form, the elements of `z' are taken to be successive +nodes of path `g'. The pens in `p' must belong to the same color space. +Illustrations of Gouraud shading are provided in the example file +`Gouraud.asy' and in the solid geometry module `solids.asy'. The edge +flags used in Gouraud shading are documented here: + + `http://partners.adobe.com/public/developer/en/ps/sdk/TN5600.SmoothShading.pdf'. + + Tensor product shading using fill rule `fillrule' on patches bounded +by the n cyclic paths of length 4 in path array `b', using the vertex +colors specified in the n \times 4 pen array `p' and internal control +points in the n \times 4 array `z', is implemented with +void tensorshade(picture pic=currentpicture, path g, bool stroke=false, + pen fillrule=currentpen, pen[][] p, path[] b=g, + pair[][] z=new pair[][]); + If the array `z' is empty, Coons shading, in which the color control +points are calculated automatically, is used. The pens in `p' must +belong to the same color space. A simpler interface for the case of a +single patch (n=1) is also available: +void tensorshade(picture pic=currentpicture, path g, bool stroke=false, + pen fillrule=currentpen, pen[] p, path b=g, + pair[] z=new pair[]); + One can also smoothly shade the regions between consecutive paths of a +sequence using a given array of pens: +void draw(picture pic=currentpicture, path[] g, pen[] p); + Illustrations of tensor product and Coons shading are provided in the +example files `tensor.asy', `Coons.asy', `BezierSurface.asy', and +`rainbow.asy'. + + More general shading possibilities are available with the `pdflatex', +`context', and `pdftex' TeX engines: the routine +void functionshade(picture pic=currentpicture, path[] g, bool stroke=false, + pen fillrule=currentpen, string shader); + shades on picture `pic' the interior of path `g' according to fill +rule `fillrule' using the `PostScript' calculator routine specified by +the string `shader'; this routine takes 2 arguments, each in [0,1], and +returns `colors(fillrule).length' color components. Function shading +is illustrated in the example `functionshading.asy'. + + The following routine uses `evenodd' clipping together with the `^^' +operator to unfill a region: + +void unfill(picture pic=currentpicture, path g); + + +File: asymptote.info, Node: clip, Next: label, Prev: fill, Up: Drawing commands + +4.3 clip +======== + +void clip(picture pic=currentpicture, path g, stroke=false, + pen fillrule=currentpen); + +Clip the current contents of picture `pic' to the region bounded by the +path `g', using fill rule `fillrule' (*note fillrule::). If +`stroke=true', the clipped portion is the same as the region that would +be drawn with `draw(pic,g,fillrule+zerowinding)'; in this case the path +`g' need not be cyclic. For an illustration of picture clipping, see +the first example in *note LaTeX usage::. + + +File: asymptote.info, Node: label, Prev: clip, Up: Drawing commands + +4.4 label +========= + +void label(picture pic=currentpicture, Label L, pair position, + align align=NoAlign, pen p=nullpen, filltype filltype=NoFill) + +Draw Label `L' on picture `pic' using pen `p'. If `align' is `NoAlign', +the label will be centered at user coordinate `position'; otherwise it +will be aligned in the direction of `align' and displaced from +`position' by the `PostScript' offset `align*labelmargin(p)'. The +constant `Align' can be used to align the bottom-left corner of the +label at `position'. If `p' is `nullpen', the pen specified within the +Label, which defaults to `currentpen', will be used. The Label `L' can +either be a string or the structure obtained by calling one of the +functions +Label Label(string s="", pair position, align align=NoAlign, + pen p=nullpen, embed embed=Rotate, filltype filltype=NoFill); +Label Label(string s="", align align=NoAlign, + pen p=nullpen, embed embed=Rotate, filltype filltype=NoFill); +Label Label(Label L, pair position, align align=NoAlign, + pen p=nullpen, embed embed=L.embed, filltype filltype=NoFill); +Label Label(Label L, align align=NoAlign, + pen p=nullpen, embed embed=L.embed, filltype filltype=NoFill); + The text of a Label can be scaled, slanted, rotated, or shifted by +multiplying it on the left by an affine transform (*note Transforms::). +For example, `rotate(45)*xscale(2)*L' first scales `L' in the x +direction and then rotates it counterclockwise by 45 degrees. The final +position of a Label can also be shifted by a `PostScript' coordinate +translation: `shift(10,0)*L'. The `embed' argument determines how the +Label should transform with the embedding picture: +`Shift' + only shift with embedding picture; + +`Rotate' + only shift and rotate with embedding picture (default); + +`Rotate(pair z)' + rotate with (picture-transformed) vector `z'. + +`Slant' + only shift, rotate, slant, and reflect with embedding picture; + +`Scale' + shift, rotate, slant, reflect, and scale with embedding picture. + + + To add a label to a path, use +void label(picture pic=currentpicture, Label L, path g, align align=NoAlign, + pen p=nullpen, filltype filltype=NoFill); + By default the label will be positioned at the midpoint of the path. +An alternative label location (an `arctime' value between 0 and +`length(g)' *note arctime::) may be specified as real value for +`position' in constructing the Label. The position `Relative(real)' +specifies a location relative to the total arclength of the path. These +convenient abbreviations are predefined: +position BeginPoint=Relative(0); +position MidPoint=Relative(0.5); +position EndPoint=Relative(1); + + Path labels are aligned in the direction `align', which may be +specified as an absolute compass direction (pair) or a direction +`Relative(pair)' measured relative to a north axis in the local +direction of the path. For convenience `LeftSide', `Center', and +`RightSide' are defined as `Relative(W)', `Relative((0,0))', and +`Relative(E)', respectively. Multiplying `LeftSide', `Center', +`RightSide' on the left by a real scaling factor will move the label +further away from or closer to the path. + + A label with a fixed-size arrow of length `arrowlength' pointing to +`b' from direction `dir' can be produced with the routine +void arrow(picture pic=currentpicture, Label L="", pair b, pair dir, + real length=arrowlength, align align=NoAlign, + pen p=currentpen, arrowbar arrow=Arrow, margin margin=EndMargin); + If no alignment is specified (either in the Label or as an explicit +argument), the optional Label will be aligned in the direction `dir', +using margin `margin'. + + The function `string graphic(string name, string options="")' +returns a string that can be used to include an encapsulated +`PostScript' (EPS) file. Here, `name' is the name of the file to +include and `options' is a string containing a comma-separated list of +optional bounding box (`bb=llx lly urx ury'), width (`width=value'), +height (`height=value'), rotation (`angle=value'), scaling +(`scale=factor'), clipping (`clip=bool'), and draft mode (`draft=bool') +parameters. The `layer()' function can be used to force future objects +to be drawn on top of the included image: +label(graphic("file.eps","width=1cm"),(0,0),NE); +layer(); + + The `string baseline(string s, string template="\strut")' function +can be used to enlarge the bounding box of labels to match a given +template, so that their baselines will be typeset on a horizontal line. +See `Pythagoras.asy' for an example. + + One can prevent labels from overwriting one another with the +`overwrite' pen attribute (*note overwrite::). + + The structure `object' defined in `plain_Label.asy' allows Labels +and frames to be treated in a uniform manner. A group of objects may +be packed together into single frame with the routine +frame pack(pair align=2S ... object inset[]); + To draw or fill a box (or ellipse or other path) around a Label and +return the bounding object, use one of the routines +object draw(picture pic=currentpicture, Label L, envelope e, + real xmargin=0, real ymargin=xmargin, pen p=currentpen, + filltype filltype=NoFill, bool above=true); +object draw(picture pic=currentpicture, Label L, envelope e, pair position, + real xmargin=0, real ymargin=xmargin, pen p=currentpen, + filltype filltype=NoFill, bool above=true); + Here `envelope' is a boundary-drawing routine such as `box', +`roundbox', or `ellipse' defined in `plain_boxes.asy' (*note +envelope::). + + The function `path[] texpath(Label L)' returns the path array that +TeX would fill to draw the Label `L'. + + The `string minipage(string s, width=100pt)' function can be used to +format string `s' into a paragraph of width `width'. This example uses +`minipage', `clip', and `graphic' to produce a CD label: + + +size(11.7cm,11.7cm); +asy(nativeformat(),"logo"); +fill(unitcircle^^(scale(2/11.7)*unitcircle), + evenodd+rgb(124/255,205/255,124/255)); +label(scale(1.1)*minipage( +"\centering\scriptsize \textbf{\LARGE {\tt Asymptote}\\ +\smallskip +\small The Vector Graphics Language}\\ +\smallskip +\textsc{Andy Hammerlindl, John Bowman, and Tom Prince} +http://asymptote.sourceforge.net\\ +",8cm),(0,0.6)); +label(graphic("logo."+nativeformat(),"height=7cm"),(0,-0.22)); +clip(unitcircle^^(scale(2/11.7)*unitcircle),evenodd); + + +File: asymptote.info, Node: Programming, Next: LaTeX usage, Prev: Drawing commands, Up: Top + +5 Programming +************* + +Here is a short introductory example to the `Asymptote' programming +language that highlights the similarity of its control structures with +those of C, C++, and Java: +// This is a comment. + +// Declaration: Declare x to be a real variable; +real x; + +// Assignment: Assign the real variable x the value 1. +x=1.0; + +// Conditional: Test if x equals 1 or not. +if(x == 1.0) { + write("x equals 1.0"); +} else { + write("x is not equal to 1.0"); +} + +// Loop: iterate 10 times +for(int i=0; i < 10; ++i) { + write(i); +} + + `Asymptote' supports `while', `do', `break', and `continue' +statements just as in C/C++. It also supports the Java-style shorthand +for iterating over all elements of an array: + +// Iterate over an array +int[] array={1,1,2,3,5}; +for(int k : array) { + write(k); +} + In addition, it supports many features beyond the ones found in those +languages. + +* Menu: + +* Data types:: void, bool, int, real, pair, triple, string +* Paths and guides:: +* 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 `Asymptote' packages +* Static:: Where to allocate your variable? + + +File: asymptote.info, Node: Data types, Next: Paths and guides, Up: Programming + +5.1 Data types +============== + +`Asymptote' supports the following data types (in addition to +user-defined types): + +`void' + The void type is used only by functions that take or return no + arguments. + +`bool' + a boolean type that can only take on the values `true' or `false'. + For example: bool b=true; + + defines a boolean variable `b' and initializes it to the value + `true'. If no initializer is given: bool b; + + the value `false' is assumed. + +`bool3' + an extended boolean type that can take on the values `true', + `default', or `false'. A bool3 type can be cast to or from a bool. + The default initializer for bool3 is `default'. + +`int' + an integer type; if no initializer is given, the implicit value `0' + is assumed. The minimum allowed value of an integer is `intMin' + and the maximum value is `intMax'. + +`real' + a real number; this should be set to the highest-precision native + floating-point type on the architecture. The implicit initializer + for reals is `0.0'. Real numbers have precision `realEpsilon', + with `realDigits' significant digits. The smallest positive real + number is `realMin' and the largest positive real number is + `realMax'. + +`pair' + complex number, that is, an ordered pair of real components + `(x,y)'. The real and imaginary parts of a pair `z' can read as + `z.x' and `z.y'. We say that `x' and `y' are virtual members of + the data element pair; they cannot be directly modified, however. + The implicit initializer for pairs is `(0.0,0.0)'. + + There are a number of ways to take the complex conjugate of a pair: + pair z=(3,4); + z=(z.x,-z.y); + z=z.x-I*z.y; + z=conj(z); + + Here `I' is the pair `(0,1)'. A number of built-in functions are + defined for pairs: + + `pair conj(pair z)' + returns the conjugate of `z'; + + `real length(pair z)' + returns the complex modulus `|z|' of its argument `z'. For + example, + pair z=(3,4); + length(z); + returns the result 5. A synonym for `length(pair)' is + `abs(pair)'; + + `real angle(pair z, bool warn=true)' + returns the angle of `z' in radians in the interval + [-`pi',`pi'] or `0' if `warn' is `false' and `z=(0,0)' + (rather than producing an error); + + `real degrees(pair z, bool warn=true)' + returns the angle of `z' in degrees in the interval [0,360) + or `0' if `warn' is `false' and `z=(0,0)' (rather than + producing an error); + + `pair unit(pair z)' + returns a unit vector in the direction of the pair `z'; + + `pair expi(real angle)' + returns a unit vector in the direction `angle' measured in + radians; + + `pair dir(real degrees)' + returns a unit vector in the direction `degrees' measured in + degrees; + + `real xpart(pair z)' + returns `z.x'; + + `real ypart(pair z)' + returns `z.y'; + + `pair realmult(pair z, pair w)' + returns the element-by-element product `(z.x*w.x,z.y*w.y)'; + + `real dot(pair z, pair w)' + returns the dot product `z.x*w.x+z.y*w.y'; + + `pair minbound(pair z, pair w)' + returns `(min(z.x,w.x),min(z.y,w.y))'; + + `pair maxbound(pair z, pair w)' + returns `(max(z.x,w.x),max(z.y,w.y))'. + + +`triple' + an ordered triple of real components `(x,y,z)' used for + three-dimensional drawings. The respective components of a triple + `v' can read as `v.x', `v.y', and `v.z'. The implicit initializer + for triples is `(0.0,0.0,0.0)'. + + Here are the built-in functions for triples: + `real length(triple v)' + returns the length `|v|' of the vector `v'. A synonym for + `length(triple)' is `abs(triple)'; + + `real polar(triple v, bool warn=true)' + returns the colatitude of `v' measured from the z axis in + radians or `0' if `warn' is `false' and `v=O' (rather than + producing an error); + + `real azimuth(triple v, bool warn=true)' + returns the longitude of `v' measured from the x axis in + radians or `0' if `warn' is `false' and `v.x=v.y=0' (rather + than producing an error); + + `real colatitude(triple v, bool warn=true)' + returns the colatitude of `v' measured from the z axis in + degrees or `0' if `warn' is `false' and `v=O' (rather than + producing an error); + + `real latitude(triple v, bool warn=true)' + returns the latitude of `v' measured from the xy plane in + degrees or `0' if `warn' is `false' and `v=O' (rather than + producing an error); + + `real longitude(triple v, bool warn=true)' + returns the longitude of `v' measured from the x axis in + degrees or `0' if `warn' is `false' and `v.x=v.y=0' (rather + than producing an error); + + `triple unit(triple v)' + returns a unit triple in the direction of the triple `v'; + + `triple expi(real polar, real azimuth)' + returns a unit triple in the direction `(polar,azimuth)' + measured in radians; + + `triple dir(real colatitude, real longitude)' + returns a unit triple in the direction + `(colatitude,longitude)' measured in degrees; + + `real xpart(triple v)' + returns `v.x'; + + `real ypart(triple v)' + returns `v.y'; + + `real zpart(triple v)' + returns `v.z'; + + `real dot(triple u, triple v)' + returns the dot product `u.x*v.x+u.y*v.y+u.z*v.z'; + + `triple cross(triple u, triple v)' + returns the cross product + + `(u.y*v.z-u.z*v.y,u.z*v.x-u.x*v.z,u.x*v.y-v.x*u.y)'; + + `triple minbound(triple u, triple v)' + returns `(min(u.x,v.x),min(u.y,v.y),min(u.z,v.z))'; + + `triple maxbound(triple u, triple v)' + returns `(max(u.x,v.x),max(u.y,v.y),max(u.z,v.z)'). + + +`string' + a character string, implemented using the STL `string' class. + + Strings delimited by double quotes (`"') are subject to the + following mappings to allow the use of double quotes in TeX (e.g. + for using the `babel' package, *note babel::): + + * \" maps to " + + * \\ maps to \\ + + Strings delimited by single quotes (`'') have the same mappings as + character strings in ANSI `C': + + * \' maps to ' + + * \" maps to " + + * \? maps to ? + + * \\ maps to backslash + + * \a maps to alert + + * \b maps to backspace + + * \f maps to form feed + + * \n maps to newline + + * \r maps to carriage return + + * \t maps to tab + + * \v maps to vertical tab + + * \0-\377 map to corresponding octal byte + + * \x0-\xFF map to corresponding hexadecimal byte + + The implicit initializer for strings is the empty string `""'. + Strings may be concatenated with the `+' operator. In the following + string functions, position `0' denotes the start of the string: + `int length(string s)' + returns the length of the string `s'; + + `int find(string s, string t, int pos=0)' + returns the position of the first occurrence of string `t' in + string `s' at or after position `pos', or -1 if `t' is not a + substring of `s'; + + `int rfind(string s, string t, int pos=-1)' + returns the position of the last occurrence of string `t' in + string `s' at or before position `pos' (if `pos'=-1, at the + end of the string `s'), or -1 if `t' is not a substring of + `s'; + + `string insert(string s, int pos, string t)' + returns the string formed by inserting string `t' at position + `pos' in `s'; + + `string erase(string s, int pos, int n)' + returns the string formed by erasing the string of length `n' + (if `n'=-1, to the end of the string `s') at position `pos' + in `s'; + + `string substr(string s, int pos, int n=-1)' + returns the substring of `s' starting at position `pos' and + of length `n' (if `n'=-1, until the end of the string `s'); + + `string reverse(string s)' + returns the string formed by reversing string `s'; + + `string replace(string s, string before, string after)' + returns a string with all occurrences of the string `before' + in the string `s' changed to the string `after'; + + `string replace(string s, string[][] table)' + returns a string constructed by translating in string `s' all + occurrences of the string `before' in an array `table' of + string pairs {`before',`after'} to the corresponding string + `after'; + + `string[] split(string s, string delimiter)' + returns an array of strings obtained by splitting `s' into + substrings delimited by `delimiter'; + + `string format(string s, int n)' + returns a string containing `n' formatted according to the + C-style format string `s' using the current locale; + + `string format(string s, real x, string locale="")' + returns a string containing `x' formatted according to the + C-style format string `s' using locale `locale' (or the + current locale if an empty string is specified), following + the behaviour of the C function `fprintf'), except that only + one data field is allowed, trailing zeros are removed by + default (unless `#' is specified), and TeX is used to typeset + scientific notation; + + `int hex(string s);' + casts a hexidecimal string `s' to an integer. + + `string string(real x, int digits=realDigits)' + casts `x' to a string using precision `digits' and the C + locale; + + `string locale(string s="")' + sets the locale to the given string, if nonempty, and returns + the current locale. + + `string time(string format="%a %b %d %T %Z %Y")' + returns the current time formatted by the ANSI C routine + `strftime' according to the string `format' using the current + locale. Thus time(); + time("%a %b %d %H:%M:%S %Z %Y"); + + are equivalent ways of returning the current time in the + default format used by the `UNIX' `date' command; + + `int seconds(string t="", string format="")' + returns the time measured in seconds after the Epoch (Thu Jan + 01 00:00:00 UTC 1970) as determined by the ANSI C routine + `strptime' according to the string `format' using the current + locale, or the current time if `t' is the empty string. Note + that the `"%Z"' extension to the POSIX `strptime' + specification is ignored by the current GNU C Library. If an + error occurs, the value -1 is returned. Here are some + examples: seconds("Mar 02 11:12:36 AM PST 2007","%b %d %r PST %Y"); + seconds(time("%b %d %r %z %Y"),"%b %d %r %z %Y"); + seconds(time("%b %d %r %Z %Y"),"%b %d %r "+time("%Z")+" %Y"); + 1+(seconds()-seconds("Jan 1","%b %d"))/(24*60*60); + The last example returns today's ordinal date, measured from + the beginning of the year. + + `string time(int seconds, string format="%a %b %d %T %Z %Y")' + returns the time corresponding to `seconds' seconds after the + Epoch (Thu Jan 01 00:00:00 UTC 1970) formatted by the ANSI C + routine `strftime' according to the string `format' using the + current locale. For example, to return the date corresponding + to 24 hours ago: time(seconds()-24*60*60); + + `void abort(string s)' + aborts execution (with a non-zero return code in batch mode); + if string `s' is nonempty, a diagnostic message constructed + from the source file, line number, and `s' is printed; + + `void exit()' + exits with a zero error return code in batch mode; + + `void sleep(int seconds)' + pauses for the given number of seconds; + + `void usleep(int microseconds)' + pauses for the given number of microseconds; + + `void beep()' + produces a beep on the console; + + + + As in C/C++, complicated types may be abbreviated with `typedef' +(see the example in *note Functions::). + + +File: asymptote.info, Node: Paths and guides, Next: Pens, Prev: Data types, Up: Programming + +5.2 Paths and guides +==================== + +`path' + a cubic spline resolved into a fixed path. The implicit + initializer for paths is `nullpath'. + + For example, the routine `circle(pair c, real r)', which returns a + Bezier curve approximating a circle of radius `r' centered on `c', + is based on `unitcircle' (*note unitcircle::): path circle(pair c, real r) + { + return shift(c)*scale(r)*unitcircle; + } + If high accuracy is needed, a true circle may be produced with the + routine `Circle' defined in the module `graph.asy': import graph; + path Circle(pair c, real r, int n=nCircle); + + A circular arc consistent with `circle' centered on `c' with + radius `r' from `angle1' to `angle2' degrees, drawing + counterclockwise if `angle2 >= angle1', can be constructed with path arc(pair c, real r, real angle1, real angle2); + One may also specify the direction explicitly: path arc(pair c, real r, real angle1, real angle2, bool direction); + Here the direction can be specified as CCW (counter-clockwise) or + CW (clockwise). For convenience, an arc centered at `c' from pair + `z1' to `z2' (assuming `|z2-c|=|z1-c|') in the may also be + constructed with path arc(pair c, explicit pair z1, explicit pair z2, + bool direction=CCW) + + If high accuracy is needed, true arcs may be produced with routines + in the module `graph.asy' that produce Bezier curves with `n' + control points: import graph; + path Arc(pair c, real r, real angle1, real angle2, bool direction, + int n=nCircle); + path Arc(pair c, real r, real angle1, real angle2, int n=nCircle); + path Arc(pair c, explicit pair z1, explicit pair z2, + bool direction=CCW, int n=nCircle); + + An ellipse can be drawn with the routine @cindex @code{ellipse} + path ellipse(pair c, real a, real b) + { + return shift(c)*scale(a,b)*unitcircle; + } + + This example illustrates the use of all five guide connectors + discussed in *note Tutorial::: size(300,0); + pair[] z=new pair[10]; + + z[0]=(0,100); z[1]=(50,0); z[2]=(180,0); + + for(int n=3; n <= 9; ++n) + z[n]=z[n-3]+(200,0); + + path p=z[0]..z[1]---z[2]::{up}z[3] + &z[3]..z[4]--z[5]::{up}z[6] + &z[6]::z[7]---z[8]..{up}z[9]; + + draw(p,grey+linewidth(4mm)); + + dot(z); + + + + Here are some useful functions for paths: + + `int length(path p);' + This is the number of (linear or cubic) segments in path `p'. + If `p' is cyclic, this is the same as the number of nodes in + `p'. + + `int size(path p);' + This is the number of nodes in the path `p'. If `p' is + cyclic, this is the same as `length(p)'. + + `bool cyclic(path p);' + returns `true' iff path `p' is cyclic. + + `bool straight(path p, int i);' + returns `true' iff the segment of path `p' between node `i' + and node `i+1' is straight. + + `bool piecewisestraight(path p)' + returns `true' iff the path `p' is piecewise straight. + + `pair point(path p, int t);' + If `p' is cyclic, return the coordinates of node `t' mod + `length(p)'. Otherwise, return the coordinates of node `t', + unless `t' < 0 (in which case `point(0)' is returned) or `t' + > `length(p)' (in which case `point(length(p))' is returned). + + `pair point(path p, real t);' + This returns the coordinates of the point between node + `floor(t)' and `floor(t)+1' corresponding to the cubic spline + parameter `t-floor(t)' (*note Bezier::). If `t' lies outside + the range [0,`length(p)'], it is first reduced modulo + `length(p)' in the case where `p' is cyclic or else converted + to the corresponding endpoint of `p'. + + `pair dir(path p, int t, int sign=0, bool normalize=true);' + If `sign < 0', return the direction (as a pair) of the + incoming tangent to path `p' at node `t'; if `sign > 0', + return the direction of the outgoing tangent. If `sign=0', + the mean of these two directions is returned. + + `pair dir(path p, real t, bool normalize=true);' + returns the direction of the tangent to path `p' at the point + between node `floor(t)' and `floor(t)+1' corresponding to the + cubic spline parameter `t-floor(t)' (*note Bezier::). + + `pair accel(path p, int t, int sign=0);' + If `sign < 0', return the acceleration of the incoming path + `p' at node `t'; if `sign > 0', return the acceleration of + the outgoing path. If `sign=0', the mean of these two + accelerations is returned. + + `pair accel(path p, real t);' + returns the acceleration of the path `p' at the point `t'. + + `pair radius(path p, real t);' + returns the radius of curvature of the path `p' at the point + `t'. + + `pair precontrol(path p, int t);' + returns the precontrol point of `p' at node `t'. + + `pair precontrol(path p, real t);' + returns the effective precontrol point of `p' at parameter + `t'. + + `pair postcontrol(path p, int t);' + returns the postcontrol point of `p' at node `t'. + + `pair postcontrol(path p, real t);' + returns the effective postcontrol point of `p' at parameter + `t'. + + `real arclength(path p);' + returns the length (in user coordinates) of the piecewise + linear or cubic curve that path `p' represents. + + `real arctime(path p, real L);' + returns the path "time", a real number between 0 and the + length of the path in the sense of `point(path p, real t)', + at which the cumulative arclength (measured from the + beginning of the path) equals `L'. + + `real dirtime(path p, pair z);' + returns the first "time", a real number between 0 and the + length of the path in the sense of `point(path, real)', at + which the tangent to the path has the direction of pair `z', + or -1 if this never happens. + + `real reltime(path p, real l);' + returns the time on path `p' at the relative fraction `l' of + its arclength. + + `pair relpoint(path p, real l);' + returns the point on path `p' at the relative fraction `l' of + its arclength. + + `pair midpoint(path p);' + returns the point on path `p' at half of its arclength. + + `path reverse(path p);' + returns a path running backwards along `p'. + + `path subpath(path p, int a, int b);' + returns the subpath of `p' running from node `a' to node `b'. + If `a' < `b', the direction of the subpath is reversed. + + `path subpath(path p, real a, real b);' + returns the subpath of `p' running from path time `a' to path + time `b', in the sense of `point(path, real)'. If `a' < `b', + the direction of the subpath is reversed. + + `real[] intersect(path p, path q, real fuzz=-1);' + If `p' and `q' have at least one intersection point, return a + real array of length 2 containing the times representing the + respective path times along `p' and `q', in the sense of + `point(path, real)', for one such intersection point (as + chosen by the algorithm described on page 137 of `The + MetaFontbook'). The computations are performed to the + absolute error specified by `fuzz', or if `fuzz < 0', to + machine precision. If the paths do not intersect, return a + real array of length 0. + + `real[][] intersections(path p, path q, real fuzz=-1);' + Return all (unless there are infinitely many) intersection + times of paths `p' and `q' as a sorted array of real arrays + of length 2 (*note sort::). The computations are performed to + the absolute error specified by `fuzz', or if `fuzz < 0', to + machine precision. + + `real[] intersections(path p, explicit pair a, explicit pair b,' + real fuzz=-1); Return all (unless there are infinitely many) + intersection times of path `p' with the (infinite) line + through points `a' and `b' as a sorted array. The + intersections returned are guaranteed to be correct to within + the absolute error specified by `fuzz', or if `fuzz < 0', to + machine precision. + + `real[] times(path p, real x)' + returns all intersection times of path `p' with the vertical + line through `(x,0)'. + + `real[] times(path p, explicit pair z)' + returns all intersection times of path `p' with the + horizontal line through `(0,z.y)'. + + `real[] mintimes(path p)' + returns an array of length 2 containing times at which path + `p' reaches its minimal horizontal and vertical extents, + respectively. + + `real[] maxtimes(path p)' + returns an array of length 2 containing the times at which + path `p' reaches its maximal horizontal and vertical extents, + respectively. + + `pair intersectionpoint(path p, path q, real fuzz=-1);' + returns the intersection point + `point(p,intersect(p,q,fuzz)[0])'. + + `pair[] intersectionpoints(path p, path q, real fuzz=-1);' + returns an array containing all intersection points of the + paths `p' and `q'. + + `pair extension(pair P, pair Q, pair p, pair q);' + returns the intersection point of the extensions of the line + segments `P--Q' and `p--q', or if the lines are parallel, + `(infinity,infinity)'. + + `slice cut(path p, path knife, int n);' + returns the portions of path `p' before and after the `n'th + intersection of `p' with path `knife' as a structure `slice' + (if no intersection exist is found, the entire path is + considered to be `before' the intersection): struct slice { + path before,after; + } + The argument `n' is treated as modulo the number of + intersections. + + `slice firstcut(path p, path knife);' + equivalent to `cut(p,knife,0);' Note that `firstcut.after' + plays the role of the `MetaPost cutbefore' command. + + `slice lastcut(path p, path knife);' + equivalent to `cut(p,knife,-1);' Note that `lastcut.before' + plays the role of the `MetaPost cutafter' command. + + `path buildcycle(... path[] p);' + This returns the path surrounding a region bounded by a list + of two or more consecutively intersecting paths, following + the behaviour of the `MetaPost buildcycle' command. + + `pair min(path p);' + returns the pair (left,bottom) for the path bounding box of + path `p'. + + `pair max(path p);' + returns the pair (right,top) for the path bounding box of + path `p'. + + `int windingnumber(path p, pair z);' + returns the winding number of the cyclic path `p' relative to + the point `z'. The winding number is positive if the path + encircles `z' in the counterclockwise direction. If `z' lies + on `p' the constant `undefined' (defined to be the largest + odd integer) is returned. + + `bool inside(path p, pair z, pen fillrule=currentpen);' + returns `true' iff the point `z' is inside or on the edge of + the region bounded by the cyclic path `p' according to the + fill rule `fillrule' (*note fillrule::). + + `int inside(path p, path q, pen fillrule=currentpen);' + returns `1' if the cyclic path `p' strictly contains `q' + according to the fill rule `fillrule' (*note fillrule::), `-1' + if the cyclic path `q' strictly contains `p', and `0' + otherwise. + + `pair inside(path p, pen fillrule=currentpen);' + returns an arbitrary point strictly inside a cyclic path `p' + according to the fill rule `fillrule' (*note fillrule::). + + `path[] strokepath(path g, pen p=currentpen);' + returns the path array that `PostScript' would fill in + drawing path `g' with pen `p'. + + +`guide' + an unresolved cubic spline (list of cubic-spline nodes and control + points). The implicit initializer for a guide is `nullpath'; this + is useful for building up a guide within a loop. + + A guide is similar to a path except that the computation of the + cubic spline is deferred until drawing time (when it is resolved + into a path); this allows two guides with free endpoint conditions + to be joined together smoothly. The solid curve in the following + example is built up incrementally as a guide, but only resolved at + drawing time; the dashed curve is incrementally resolved at each + iteration, before the entire set of nodes (shown in red) is known: + + size(200); + + real mexican(real x) {return (1-8x^2)*exp(-(4x^2));} + + int n=30; + real a=1.5; + real width=2a/n; + + guide hat; + path solved; + + for(int i=0; i < n; ++i) { + real t=-a+i*width; + pair z=(t,mexican(t)); + hat=hat..z; + solved=solved..z; + } + + draw(hat); + dot(hat,red); + draw(solved,dashed); + + + + We point out an efficiency distinction in the use of guides and + paths: guide g; + for(int i=0; i < 10; ++i) + g=g--(i,i); + path p=g; + + runs in linear time, whereas path p; + for(int i=0; i < 10; ++i) + p=p--(i,i); + + runs in quadratic time, as the entire path up to that point is + copied at each step of the iteration. + + The following routines can be used to examine the individual + elements of a guide without actually resolving the guide to a + fixed path (except for internal cycles, which are resolved): + + `int size(guide g);' + Analogous to `size(path p)'. + + `int length(guide g);' + Analogous to `length(path p)'. + + `bool cyclic(path p);' + Analogous to `cyclic(path p)'. + + `pair point(guide g, int t);' + Analogous to `point(path p, int t)'. + + `guide reverse(guide g);' + Analogous to `reverse(path p)'. If `g' is cyclic and also + contains a secondary cycle, it is first solved to a path, + then reversed. If `g' is not cyclic but contains an internal + cycle, only the internal cycle is solved before reversal. If + there are no internal cycles, the guide is reversed but not + solved to a path. + + `pair[] dirSpecifier(guide g, int i);' + This returns a pair array of length 2 containing the outgoing + (in element 0) and incoming (in element 1) direction + specifiers (or `(0,0)' if none specified) for the segment of + guide `g' between nodes `i' and `i+1'. + + `pair[] controlSpecifier(guide g, int i);' + If the segment of guide `g' between nodes `i' and `i+1' has + explicit outgoing and incoming control points, they are + returned as elements 0 and 1, respectively, of a two-element + array. Otherwise, an empty array is returned. + + `tensionSpecifier tensionSpecifier(guide g, int i);' + This returns the tension specifier for the segment of guide + `g' between nodes `i' and `i+1'. The individual components of + the `tensionSpecifier' type can be accessed as the virtual + members `in', `out', and `atLeast'. + + `real[] curlSpecifier(guide g);' + This returns an array containing the initial curl specifier + (in element 0) and final curl specifier (in element 1) for + guide `g'. + + + As a technical detail we note that a direction specifier given to + `nullpath' modifies the node on the other side: the guides a..{up}nullpath..b; + c..nullpath{up}..d; + e..{up}nullpath{down}..f; + are respectively equivalent to a..nullpath..{up}b; + c{up}..nullpath..d; + e{down}..nullpath..{up}f; + + + +File: asymptote.info, Node: Pens, Next: Transforms, Prev: Paths and guides, Up: Programming + +5.3 Pens +======== + +In `Asymptote', pens provide a context for the four basic drawing +commands (*note Drawing commands::). They are used to specify the +following drawing attributes: color, line type, line width, line cap, +line join, fill rule, text alignment, font, font size, pattern, +overwrite mode, and calligraphic transforms on the pen nib. The default +pen used by the drawing routines is called `currentpen'. This provides +the same functionality as the `MetaPost' command `pickup'. The +implicit initializer for pens is `defaultpen'. + + Pens may be added together with the nonassociative binary operator +`+'. This will add the colors of the two pens. All other non-default +attributes of the rightmost pen will override those of the leftmost +pen. Thus, one can obtain a yellow dashed pen by saying +`dashed+red+green' or `red+green+dashed' or `red+dashed+green'. The +binary operator `*' can be used to scale the color of a pen by a real +number, until it saturates with one or more color components equal to 1. + + * Colors are specified using one of the following colorspaces: + `pen gray(real g);' + This produces a grayscale color, where the intensity `g' lies + in the interval [0,1], with 0.0 denoting black and 1.0 + denoting white. + + `pen rgb(real r, real g, real b);' + This produces an RGB color, where each of the red, green, and + blue intensities `r', `g', `b', lies in the interval [0,1]. + + `pen cmyk(real c, real m, real y, real k);' + This produces a CMYK color, where each of the cyan, magenta, + yellow, and black intensities `c', `m', `y', `k', lies in the + interval [0,1]. + + `pen invisible;' + This special pen writes in invisible ink, but adjusts the + bounding box as if something had been drawn (like the + `\phantom' command in TeX). The function `bool + invisible(pen)' can be used to test whether a pen is + invisible. + + + The default color is `black'; this may be changed with the routine + `defaultpen(pen)'. The function `colorspace(pen p)' returns the + colorspace of pen `p' as a string (`"gray"', `"rgb"', `"cmyk"', or + `""'). + + The function `real[] colors(pen)' returns the color components of + a pen. The functions `pen gray(pen)', `pen rgb(pen)', and `pen + cmyk(pen)' return new pens obtained by converting their arguments + to the respective color spaces. The function + `colorless(pen=currentpen)' returns a copy of its argument with + the color attributes stripped (to avoid color mixing). + + A 6-character RGB hexidecimal string can be converted to a pen with + the routine pen rgb(string s); + + Various shades and mixtures of the grayscale primary colors + `black' and `white', RGB primary colors `red', `green', and + `blue', and RGB secondary colors `cyan', `magenta', and `yellow' + are defined as named colors, along with the CMYK primary colors + `Cyan', `Magenta', `Yellow', and `Black', in the module `plain': + + + + The standard 140 RGB `X11' colors can be imported with the command import x11colors; + and the standard 68 CMYK TeX colors can be imported with the + command import texcolors; + Note that there is some overlap between these two standards and + the definitions of some colors (e.g. `Green') actually disagree. + + `Asymptote' also comes with a `asycolors.sty' `LaTeX' package that + defines to `LaTeX' CMYK versions of `Asymptote''s predefined + colors, so that they can be used directly within `LaTeX' strings. + Normally, such colors are passed to `LaTeX' via a pen argument; + however, to change the color of only a portion of a string, say + for a slide presentation, (*note slide::) it may be desirable to + specify the color directly to `LaTeX'. This file can be passed to + `LaTeX' with the `Asymptote' command usepackage("asycolors"); + + The structure `hsv' defined in `plain_pens.asy' may be used to + convert between HSV and RGB spaces, where the hue `h' is an angle + in [0,360) and the saturation `s' and value `v' lie in `[0,1]': pen p=hsv(180,0.5,0.75); + write(p); // ([default], red=0.375, green=0.75, blue=0.75) + hsv q=p; + write(q.h,q.s,q.v); // 180 0.5 0.75 + + * Line types are specified with the function `pen linetype(string s, + real offset=0, bool scale=true, bool adjust=true)', where `s' is a + string of integer or real numbers separated by spaces. The + optional parameter `offset' specifies where in the pattern to + begin. The first number specifies how far (if `scale' is `true', + in units of the pen line width; otherwise in `PostScript' units) + to draw with the pen on, the second number specifies how far to + draw with the pen off, and so on. If `adjust' is `true', these + spacings are automatically adjusted by `Asymptote' to fit the + arclength of the path. Here are the predefined line types: pen solid=linetype(""); + pen dotted=linetype("0 4"); + pen dashed=linetype("8 8"); + pen longdashed=linetype("24 8"); + pen dashdotted=linetype("8 8 0 8"); + pen longdashdotted=linetype("24 8 0 8"); + pen Dotted=dotted+1.0; + pen Dotted(pen p=currentpen) {return dotted+2*linewidth(p);} + + + + The default line type is `solid'; this may be changed with + `defaultpen(pen)'. The line type of a pen is returned by `int + linetype(pen p=currentpen)'. + + * The pen line width is specified in `PostScript' units with `pen + linewidth(real)'. The default line width is 0.5 bp; this value may + be changed with `defaultpen(pen)'. The line width of a pen is + returned by `real linewidth(pen p=currentpen)'. For convenience, + in the module `plain' we define static void defaultpen(real w) {defaultpen(linewidth(w));} + static pen operator +(pen p, real w) {return p+linewidth(w);} + static pen operator +(real w, pen p) {return linewidth(w)+p;} + so that one may set the line width like this: defaultpen(2); + pen p=red+0.5; + + * A pen with a specific `PostScript' line cap is returned on calling + `linecap' with an integer argument: pen squarecap=linecap(0); + pen roundcap=linecap(1); + pen extendcap=linecap(2); + + The default line cap, `roundcap', may be changed with + `defaultpen(pen)'. The line cap of a pen is returned by `int + linecap(pen p=currentpen)'. + + * A pen with a specific `PostScript' join style is returned on + calling `linejoin' with an integer argument: pen miterjoin=linejoin(0); + pen roundjoin=linejoin(1); + pen beveljoin=linejoin(2); + + The default join style, `roundjoin', may be changed with + `defaultpen(pen)'.The join style of a pen is returned by `int + linejoin(pen p=currentpen)'. + + * A pen with a specific `PostScript' miter limit is returned by + calling `miterlimit(real)'. The default miterlimit, `10.0', may + be changed with `defaultpen(pen)'. The miter limit of a pen is + returned by `real miterlimit(pen p=currentpen)'. + + * A pen with a specific `PostScript' fill rule is returned on + calling `fillrule' with an integer argument: pen zerowinding=fillrule(0); + pen evenodd=fillrule(1); + + The fill rule, which identifies the algorithm used to determine the + insideness of a path or array of paths, only affects the `clip', + `fill', and `inside' functions. For the `zerowinding' fill rule, a + point `z' is outside the region bounded by a path if the number of + upward intersections of the path with the horizontal line + `z--z+infinity' minus the number of downward intersections is + zero. For the `evenodd' fill rule, `z' is considered to be outside + the region if the total number of such intersections is even. The + default fill rule, `zerowinding', may be changed with + `defaultpen(pen)'. The fill rule of a pen is returned by `int + fillrule(pen p=currentpen)'. + + * A pen with a specific text alignment setting is returned on + calling `basealign' with an integer argument: pen nobasealign=basealign(0); + pen basealign=basealign(1); + + The default setting, `nobasealign',which may be changed with + `defaultpen(pen)', causes the label alignment routines to use the + full label bounding box for alignment. In contrast, `basealign' + requests that the TeX baseline be respected. The base align + setting of a pen is returned by `int basealigin(pen p=currentpen)'. + + * The font size is specified in TeX points (1 pt = 1/72.27 inches) + with the function `pen fontsize(real size, real + lineskip=1.2*size)'. The default font size, 12pt, may be changed + with `defaultpen(pen)'. Nonstandard font sizes may require + inserting import fontsize; + at the beginning of the file (this requires the `fix-cm' package + available from + + `http://www.ctan.org/tex-archive/help/Catalogue/entries/fix-cm' + and included in recent `LaTeX' distributions). The font size and + line skip of a pen can be examined with the routines `real + fontsize(pen p=currentpen)' and `real lineskip(pen p=currentpen)', + respectively. + + * A pen using a specific `LaTeX' `NFSS' font is returned by calling + the function `pen font(string encoding, string family, string + series, string shape)'. The default setting, + `font("OT1","cmr","m","n")', corresponds to 12pt Computer Modern + Roman; this may be changed with `defaultpen(pen)'. The font + setting of a pen is returned by `string font(pen p=currentpen)'. + Support for standardized international characters is provided by + the `unicode' package (*note unicode::). + + Alternatively, one may select a fixed-size TeX font (on which + `fontsize' has no effect) like `"cmr12"' (12pt Computer Modern + Roman) or `"pcrr"' (Courier) using the function `pen font(string + name)'. An optional size argument can also be given to scale the + font to the requested size: `pen font(string name, real size)'. + + A nonstandard font command can be generated with `pen + fontcommand(string)'. + + A convenient interface to the following standard `PostScript' + fonts is also provided: pen AvantGarde(string series="m", string shape="n"); + pen Bookman(string series="m", string shape="n"); + pen Courier(string series="m", string shape="n"); + pen Helvetica(string series="m", string shape="n"); + pen NewCenturySchoolBook(string series="m", string shape="n"); + pen Palatino(string series="m", string shape="n"); + pen TimesRoman(string series="m", string shape="n"); + pen ZapfChancery(string series="m", string shape="n"); + pen Symbol(string series="m", string shape="n"); + pen ZapfDingbats(string series="m", string shape="n"); + + * The transparency of a pen can be changed with the command: pen opacity(real opacity=1, string blend="Compatible"); + The opacity can be varied from `0' (fully transparent) to the + default value of `1' (opaque), and `blend' specifies one of the + following foreground-background blending operations: "Compatible","Normal","Multiply","Screen","Overlay","SoftLight", + "HardLight","ColorDodge","ColorBurn","Darken","Lighten","Difference", + "Exclusion","Hue","Saturation","Color","Luminosity", + as described in + + `http://partners.adobe.com/public/developer/en/pdf/PDFReference16.pdf'. + Since `PostScript' does not support transparency, this feature is + only effective with the `-f pdf' output format option; other + formats can be produced from the resulting PDF file with the + `ImageMagick' `convert' program. Labels are always drawn with an + `opacity' of 1. A simple example of transparent filling is + provided in the example file `transparency.asy'. + + * `PostScript' commands within a `picture' may be used to create a + tiling pattern, identified by the string `name', for `fill' and + `draw' operations by adding it to the global `PostScript' frame + `currentpatterns', with optional left-bottom margin `lb' and + right-top margin `rt'. import patterns; + void add(string name, picture pic, pair lb=0, pair rt=0); + + To `fill' or `draw' using pattern `name', use the pen + `pattern("name")'. For example, rectangular tilings can be + constructed using the routines `picture tile(real Hx=5mm, real + Hy=0, pen p=currentpen, filltype filltype=NoFill)', `picture + checker(real Hx=5mm, real Hy=0, pen p=currentpen)', and `picture + brick(real Hx=5mm, real Hy=0, pen p=currentpen)' defined in + `patterns.asy': size(0,90); + import patterns; + + add("tile",tile()); + add("filledtilewithmargin",tile(6mm,4mm,red,Fill),(1mm,1mm),(1mm,1mm)); + add("checker",checker()); + add("brick",brick()); + + real s=2.5; + filldraw(unitcircle,pattern("tile")); + filldraw(shift(s,0)*unitcircle,pattern("filledtilewithmargin")); + filldraw(shift(2s,0)*unitcircle,pattern("checker")); + filldraw(shift(3s,0)*unitcircle,pattern("brick")); + + + + Hatch patterns can be generated with the routines `picture + hatch(real H=5mm, pair dir=NE, pen p=currentpen)', `picture + crosshatch(real H=5mm, pen p=currentpen)': size(0,100); + import patterns; + + add("hatch",hatch()); + add("hatchback",hatch(NW)); + add("crosshatch",crosshatch(3mm)); + + real s=1.25; + filldraw(unitsquare,pattern("hatch")); + filldraw(shift(s,0)*unitsquare,pattern("hatchback")); + filldraw(shift(2s,0)*unitsquare,pattern("crosshatch")); + + + + You may need to turn off aliasing in your `PostScript' viewer for + patterns to appear correctly. Custom patterns can easily be + constructed, following the examples in `patterns.asy'. The tiled + pattern can even incorporate shading (*note gradient shading::), + as illustrated in this example (not included in the manual because + not all printers support `PostScript' 3): size(0,100); + import patterns; + + real d=4mm; + picture tiling; + path square=scale(d)*unitsquare; + axialshade(tiling,square,white,(0,0),black,(d,d)); + fill(tiling,shift(d,d)*square,blue); + add("shadedtiling",tiling); + + filldraw(unitcircle,pattern("shadedtiling")); + + + + * One can specify a custom pen nib as an arbitrary polygonal path + with `pen makepen(path)'; this path represents the mark to be + drawn for paths containing a single point. This pen nib path can be + recovered from a pen with `path nib(pen)'. Unlike in `MetaPost', + the path need not be convex: + + size(200); + pen convex=makepen(scale(10)*polygon(8))+grey; + draw((1,0.4),convex); + draw((0,0)---(1,1)..(2,0)--cycle,convex); + + pen nonconvex=scale(10)* + makepen((0,0)--(0.25,-1)--(0.5,0.25)--(1,0)--(0.5,1.25)--cycle)+red; + draw((0.5,-1.5),nonconvex); + draw((0,-1.5)..(1,-0.5)..(2,-1.5),nonconvex); + + + + The value `nullpath' represents a circular pen nib (the default); + an elliptical pen can be achieved simply by multiplying the pen by + a transform: `yscale(2)*currentpen'. + + * One can prevent labels from overwriting one another by using the + pen attribute `overwrite', which takes a single argument: + + `Allow' + Allow labels to overwrite one another. This is the default + behaviour (unless overridden with `defaultpen(pen)'. + + `Suppress' + Suppress, with a warning, each label that would overwrite + another label. + + `SuppressQuiet' + Suppress, without warning, each label that would overwrite + another label. + + `Move' + Move a label that would overwrite another out of the way and + issue a warning. As this adjustment is during the final + output phase (in `PostScript' coordinates) it could result in + a larger figure than requested. + + `MoveQuiet' + Move a label that would overwrite another out of the way, + without warning. As this adjustment is during the final + output phase (in `PostScript' coordinates) it could result in + a larger figure than requested. + + + + The routine `defaultpen()' returns the current default pen +attributes. Calling the routine `resetdefaultpen()' resets all pen +default attributes to their initial values. + + +File: asymptote.info, Node: Transforms, Next: Frames and pictures, Prev: Pens, Up: Programming + +5.4 Transforms +============== + +`Asymptote' makes extensive use of affine transforms. A pair `(x,y)' is +transformed by the transform `t=(t.x,t.y,t.xx,t.xy,t.yx,t.yy)' to +`(x',y')', where +x' = t.x + t.xx * x + t.xy * y +y' = t.y + t.yx * x + t.yy * y + This is equivalent to the `PostScript' transformation `[t.xx t.yx t.xy +t.yy t.x t.y]'. + + Transforms can be applied to pairs, guides, paths, pens, strings, +transforms, frames, and pictures by multiplication (via the binary +operator `*') on the left (*note circle:: for an example). Transforms +can be composed with one another and inverted with the function +`transform inverse(transform t)'; they can also be raised to any +integer power with the `^' operator. + + The built-in transforms are: + +`transform identity();' + the identity transform; + +`transform shift(pair z);' + translates by the pair `z'; + +`transform shift(real x, real y);' + translates by the pair `(x,y)'; + +`transform xscale(real x);' + scales by `x' in the x direction; + +`transform yscale(real y);' + scales by `y' in the y direction; + +`transform scale(real s);' + scale by `s' in both x and y directions; + +`transform scale(real x, real y);' + scale by `x' in the x direction and by `y' in the y direction; + +`transform slant(real s);' + maps `(x,y)' -> `(x+s*y,y)'; + +`transform rotate(real angle, pair z=(0,0));' + rotates by `angle' in degrees about `z'; + +`transform reflect(pair a, pair b);' + reflects about the line `a--b'. + + The implicit initializer for transforms is `identity()'. The +routines `shift(transform t)' and `shiftless(transform t)' return the +transforms `(t.x,t.y,0,0,0,0)' and `(0,0,t.xx,t.xy,t.yx,t.yy)' +respectively. + + +File: asymptote.info, Node: Frames and pictures, Next: Files, Prev: Transforms, Up: Programming + +5.5 Frames and pictures +======================= + +`frame' + Frames are canvases for drawing in `PostScript' coordinates. While + working with frames directly is occasionally necessary for + constructing deferred drawing routines, pictures are usually more + convenient to work with. The implicit initializer for frames is + `newframe'. The function `bool empty(frame f)' returns `true' only + if the frame `f' is empty. A frame may be erased with the + `erase(frame)' routine. The functions `pair min(frame)' and `pair + max(frame)' return the (left,bottom) and (right,top) coordinates + of the frame bounding box, respectively. The contents of frame + `src' may be appended to frame `dest' with the command void add(frame dest, frame src); + or prepended with void prepend(frame dest, frame src); + A frame obtained by aligning frame `f' in the direction `align', + in a manner analogous to the `align' argument of `label' (*note + label::), is returned by frame align(frame f, pair align); + + 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 + `envelope' routines 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); + +`picture' + Pictures are high-level structures (*note Structures::) defined in + the module `plain' that provide canvases for drawing in user + coordinates. The default picture is called `currentpicture'. A + new picture can be created like this: picture pic; + Anonymous pictures can be made by the expression `new picture'. + + The `size' routine specifies the dimensions of the desired picture: + + void size(picture pic=currentpicture, real x, real y=x, + bool keepAspect=Aspect); + + If the `x' and `y' sizes are both 0, user coordinates will be + interpreted as `PostScript' coordinates. In this case, the + transform mapping `pic' to the final output frame is `identity()'. + + If exactly one of `x' or `y' is 0, no size restriction is imposed + in that direction; it will be scaled the same as the other + direction. + + If `keepAspect' is set to `Aspect' or `true', the picture will be + scaled with its aspect ratio preserved such that the final width + is no more than `x' and the final height is no more than `y'. + + If `keepAspect' is set to `IgnoreAspect' or `false', the picture + will be scaled in both directions so that the final width is `x' + and the height is `y'. + + To make the user coordinates of picture `pic' represent multiples + of `x' units in the x direction and `y' units in the y direction, + use void unitsize(picture pic=currentpicture, real x, real y=x); + When nonzero, these `x' and `y' values override the corresponding + size parameters of picture `pic'. + + The routine void size(picture pic=currentpicture, real xsize, real ysize, + pair min, pair max); + forces the final picture scaling to map the user coordinates + `box(min,max)' to a region of width `xsize' and height `ysize' + (when these parameters are nonzero). + + Alternatively, calling the routine transform fixedscaling(picture pic=currentpicture, pair min, + pair max, pen p=nullpen, bool warn=false); + will cause picture `pic' to use a fixed scaling to map user + coordinates in `box(min,max)' to the (already specified) picture + size, taking account of the width of pen `p'. A warning will be + issued if the final picture exceeds the specified size. + + A picture `pic' can be fit to a frame and output to a file + `prefix'.`format' using image format `format' by calling the + `shipout' function: void shipout(string prefix=defaultfilename, picture pic=currentpicture, + orientation orientation=orientation, + string format="", bool wait=false, bool view=true, + string options="", string script="", + projection P=currentprojection); + The default output format, `PostScript', may be changed with the + `-f' or `-tex' command-line options. The `options', `script', and + `projection' parameters are only relevant for 3D pictures. + + A `shipout()' command is added implicitly at file exit if no + previous `shipout' commands have been executed. The default page + orientation is `Portrait'; this may be modified by changing the + variable `orientation'. To output in landscape mode, simply set + the variable `orientation=Landscape' or issue the command shipout(Landscape); + + To rotate the page by -90 degrees, use the orientation `Seascape'. The + orientation `UpsideDown' rotates the page by 180 degrees. + + A picture `pic' can be explicitly fit to a frame by calling frame pic.fit(real xsize=pic.xsize, real ysize=pic.ysize, + bool keepAspect=pic.keepAspect); + The default size and aspect ratio settings are those given to the + `size' command (which default to `0', `0', and `true', + respectively). The transformation that would currently be used to + fit a picture `pic' to a frame is returned by the member function + `pic.calculateTransform()'. + + In certain cases (e.g. 2D graphs) where only an approximate size + estimate for `pic' is available, the picture fitting routine frame pic.scale(real xsize=this.xsize, real ysize=this.ysize, + bool keepAspect=this.keepAspect); + (which scales the resulting frame, including labels and fixed-size + objects) will enforce perfect compliance with the requested size + specification, but should not normally be required. + + To draw a bounding box with margins around a picture, fit the + picture to a frame using the function frame bbox(picture pic=currentpicture, real xmargin=0, + real ymargin=xmargin, pen p=currentpen, + filltype filltype=NoFill); + Here `filltype' specifies one of the following fill types: + `FillDraw' + Fill the interior and draw the boundary. + + `FillDraw(real xmargin=0, real ymargin=xmargin, pen fillpen=nullpen,' + `pen drawpen=nullpen);' If `fillpen' is `nullpen', fill with + the drawing pen; otherwise fill with pen `fillpen'. If + `drawpen' is `nullpen', draw the boundary with `fillpen'; + otherwise with `drawpen'. An optional margin of `xmargin' and + `ymargin' can be specified. + + `Fill' + Fill the interior. + + `Fill(real xmargin=0, real ymargin=xmargin, pen p=nullpen)' + If `p' is `nullpen', fill with the drawing pen; otherwise + fill with pen `p'. An optional margin of `xmargin' and + `ymargin' can be specified. + + `NoFill' + Do not fill. + + `Draw' + Draw only the boundary. + + `Draw(real xmargin=0, real ymargin=xmargin, pen p=nullpen)' + If `p' is `nullpen', draw the boundary with the drawing pen; + otherwise draw with pen `p'. An optional margin of `xmargin' + and `ymargin' can be specified. + + `UnFill' + Clip the region. + + `UnFill(real xmargin=0, real ymargin=xmargin)' + Clip the region and surrounding margins `xmargin' and + `ymargin'. + + `RadialShade(pen penc, pen penr)' + Fill varying radially from `penc' at the center of the + bounding box to `penr' at the edge. + + + For example, to draw a bounding box around a picture with a 0.25 cm + margin and output the resulting frame, use the command: shipout(bbox(0.25cm)); + A `picture' may be fit to a frame with the background color of + pen `p' with the function `bbox(p,Fill)'. + + The functions pair min(picture pic, user=false); + pair max(picture pic, user=false); + pair size(picture pic, user=false); + calculate the `PostScript' bounds that picture `pic' would have + if it were currently fit to a frame using its default size + specification. If `user' is `false' the returned value is in + `PostScript' coordinates, otherwise it is in user coordinates. + + The function pair point(picture pic=currentpicture, pair dir, bool user=true); + is a convenient way of determining the point on the bounding box + of `pic' in the direction `dir' relative to its center, ignoring + the contributions from fixed-size objects (such as labels and + arrowheads). If `user' is `true' the returned value is in user + coordinates, otherwise it is in `PostScript' coordinates. + + The function pair truepoint(picture pic=currentpicture, pair dir, bool user=true); + is identical to `point', except that it also accounts for + fixed-size objects, using the scaling transform that picture `pic' + would have if currently fit to a frame using its default size + specification. If `user' is `true' the returned value is in user + coordinates, otherwise it is in `PostScript' coordinates. + + Sometimes it is useful to draw objects on separate pictures and + add one picture to another using the `add' function: void add(picture src, bool group=true, + filltype filltype=NoFill, bool above=true); + void add(picture dest, picture src, bool group=true, + filltype filltype=NoFill, bool above=true); + The first example adds `src' to `currentpicture'; the second one + adds `src' to `dest'. The `group' option specifies whether or not + the graphical user interface `xasy' should treat all of the + elements of `src' as a single entity (*note GUI::), `filltype' + requests optional background filling or clipping, and `above' + specifies whether to add `src' above or below existing objects. + + There are also routines to add a picture or frame `src' specified + in postscript coordinates to another picture `dest' (or + `currentpicture') about the user coordinate `position': void add(picture src, pair position, bool group=true, + filltype filltype=NoFill, bool above=true); + void add(picture dest, picture src, pair position, + bool group=true, filltype filltype=NoFill, bool above=true); + void add(picture dest=currentpicture, frame src, pair position=0, + bool group=true, filltype filltype=NoFill, bool above=true); + void add(picture dest=currentpicture, frame src, pair position, + pair align, bool group=true, filltype filltype=NoFill, + bool above=true); + + The optional `align' argument in the last three forms specifies a + direction to use for aligning the frame, in a manner analogous to + the `align' argument of `label' (*note label::). However, one key + difference is that when `align' is not specified, labels are + centered, whereas frames and pictures are aligned so that their + origin is at `position'. Illustrations of frame alignment can be + found in the examples *note errorbars:: and *note image::. If you + want to align three or more subpictures, group them two at a time: + + picture pic1; + real size=50; + size(pic1,size); + fill(pic1,(0,0)--(50,100)--(100,0)--cycle,red); + + picture pic2; + size(pic2,size); + fill(pic2,unitcircle,green); + + picture pic3; + size(pic3,size); + fill(pic3,unitsquare,blue); + + picture pic; + add(pic,pic1.fit(),(0,0),N); + add(pic,pic2.fit(),(0,0),10S); + + add(pic.fit(),(0,0),N); + add(pic3.fit(),(0,0),10S); + + + + Alternatively, one can use `attach' to automatically increase the + size of picture `dest' to accommodate adding a frame `src' about + the user coordinate `position': void attach(picture dest=currentpicture, frame src, + pair position=0, bool group=true, + filltype filltype=NoFill, bool above=true); + void attach(picture dest=currentpicture, frame src, + pair position, pair align, bool group=true, + filltype filltype=NoFill, bool above=true); + + To erase the contents of a picture (but not the size + specification), use the function void erase(picture pic=currentpicture); + + To save a snapshot of `currentpicture', `currentpen', and + `currentprojection', use the function `save()'. + + To restore a snapshot of `currentpicture', `currentpen', and + `currentprojection', use the function `restore()'. + + Many further examples of picture and frame operations are provided + in the base module `plain'. + + It is possible to insert verbatim `PostScript' commands in a + picture with one of the routines void postscript(picture pic=currentpicture, string s); + void postscript(picture pic=currentpicture, string s, pair min, + pair max) + Here `min' and `max' can be used to specify explicit bounds + associated with the resulting `PostScript' code. + + Verbatim TeX commands can be inserted in the intermediate `LaTeX' + output file with one of the functions void tex(picture pic=currentpicture, string s); + void tex(picture pic=currentpicture, string s, pair min, pair max) + Here `min' and `max' can be used to specify explicit bounds + associated with the resulting TeX code. + + To issue a global TeX command (such as a TeX macro definition) in + the TeX preamble (valid for the remainder of the top-level module) + use: void texpreamble(string s); + + The TeX environment can be reset to its initial state, clearing all + macro definitions, with the function void texreset(); + + The routine void usepackage(string s, string options=""); + provides a convenient abbreviation for texpreamble("\usepackage["+options+"]{"+s+"}"); + that can be used for importing `LaTeX' packages. + + + +File: asymptote.info, Node: Files, Next: Variable initializers, Prev: Frames and pictures, Up: Programming + +5.6 Files +========= + +`Asymptote' can read and write text files (including comma-separated +value) files and portable XDR (External Data Representation) binary +files. + + An input file must first be opened with `input(string name, bool +check=true, string comment="#")'; reading is then done by assignment: +file fin=input("test.txt"); +real a=fin; + + If the optional boolean argument `check' is `false', no check will +be made that the file exists. If the file does not exist or is not +readable, the function `bool error(file)' will return `true'. The +first character of the string `comment' specifies a comment character. +If this character is encountered in a data file, the remainder of the +line is ignored. When reading strings, a comment character followed +immediately by another comment character is treated as a single literal +comment character. + + If the `-globalwrite' (or `-nosafe') option is enabled, one can +change the current working directory to the contents of the string `s' +with the function `string cd(string s)', which returns the new working +directory. If `string s' is empty, the path is reset to the value it +had at program startup. + + When reading pairs, the enclosing parenthesis are optional. Strings +are also read by assignment, by reading characters up to but not +including a newline. In addition, `Asymptote' provides the function +`string getc(file)' to read the next character (treating the comment +character as an ordinary character) and return it as a string. + + A file named `name' can be open for output with +file output(string name, bool update=false); + If `update=false', any existing data in the file will be erased and +only write operations can be used on the file. If `update=true', any +existing data will be preserved, the position will be set to the +end-of-file, and both reading and writing operations will be enabled. +For security reasons, writing to files in directories other than the +current directory is allowed only if the `-globalwrite' (or `-nosafe') +command-line option is specified. + + There are two special files: `stdin', which reads from the keyboard, +and `stdout', which writes to the terminal. The implicit initializer +for files is `null'. + + Data of a built-in type `T' can be written to an output file by +calling one of the functions +write(string s="", T x, suffix suffix=endl ... T[]); +write(file file, string s="", T x, suffix suffix=none ... T[]); +write(file file=stdout, string s="", explicit T[] x ... T[][]); +write(file file=stdout, T[][]); +write(file file=stdout, T[][][]); +write(suffix suffix=endl); +write(file file, suffix suffix=none); + If `file' is not specified, `stdout' is used and terminated by default +with a newline. If specified, the optional identifying string `s' is +written before the data `x'. An arbitrary number of data values may be +listed when writing scalars or one-dimensional arrays. The `suffix' may +be one of the following: `none' (do nothing), `flush' (output buffered +data), `endl' (terminate with a newline and flush), `newl' (terminate +with a newline), `tab' (terminate with a tab), or `comma' (terminate +with a comma). Here are some simple examples of data output: +file fout=output("test.txt"); +write(fout,1); // Writes "1" +write(fout); // Writes a new line +write(fout,"List: ",1,2,3); // Writes "List: 1 2 3" + A file may also be opened with `xinput' or `xoutput', instead of +`input' or `output', to read or write double precision (64-bit) real +values and single precision (32-bit) integer values in Sun +Microsystem's XDR (External Data Representation) portable binary format +(available on all `UNIX' platforms). A file may also be opened with +`binput' or `boutput' to read or write double precision values in the +native (nonportable) machine binary format. The function `file +single(file,0.0)' may be used to set a file to read single precision +real XDR or binary values; calling `file single(file,0.0,false)' sets +it back to read doubles again. The functions `file single(file,0)' and +`file single(file,0,false)' can be used to change the default integer +precision (single). The functions `file single(file)' and `file +single(file,false)' may be used to set the precision for both real and +integer values. + + One can test a file for end-of-file with the boolean function +`eof(file)', end-of-line with `eol(file)', and for I/O errors with +`error(file)'. One can flush the output buffers with `flush(file)', +clear a previous I/O error with `clear(file)', and close the file with +`close(file)'. The function `int precision(file file=stdout, int +digits=0)' sets the number of digits of output precision for `file' to +`digits', provided `digits' is nonzero, and returns the previous +precision setting. The function `int tell(file)' returns the current +position in a file relative to the beginning. The routine `seek(file +file, int pos)' can be used to change this position, where a negative +value for the position `pos' is interpreted as relative to the +end-of-file. For example, one can rewind a file `file' with the command +`seek(file,0)' and position to the final character in the file with +`seek(file,-1)'. The command `seekeof(file)' sets the position to the +end of the file. + + Assigning `settings.scroll=n' for a positive integer `n' requests a +pause after every `n' output lines to `stdout'. One may then press +`Enter' to continue to the next `n' output lines, `s' followed by +`Enter' to scroll without further interruption, or `q' followed by +`Enter' to quit the current output operation. If `n' is negative, the +output scrolls a page at a time (i.e. by one less than the current +number of display lines). The default value, `settings.scroll=0', +specifies continuous scrolling. + + The routines +string getstring(string name="", string default="", string prompt="", + bool store=true); +int getint(string name="", int default=0, string prompt="", + bool store=true); +real getreal(string name="", real default=0, string prompt="", + bool store=true); +pair getpair(string name="", pair default=0, string prompt="", + bool store=true); +triple gettriple(string name="", triple default=(0,0,0), string prompt="", + bool store=true); + defined in the module `plain' may be used to prompt for a value from +`stdin' using the GNU `readline' library. If `store=true', the history +of values for `name' is stored in the file `".asy_history_"+name' +(*note history::). The most recent value in the history will be used to +provide a default value for subsequent runs. The default value +(initially `default') is displayed after `prompt'. These functions are +based on the internal routines +string readline(string prompt="", string name="", bool tabcompletion=false); +void saveline(string name, string value, bool store=true); + Here, `readline' prompts the user with the default value formatted +according to `prompt', while `saveline' is used to save the string +`value' in a local history named `name', optionally storing the local +history in a file `".asy_history_"+name'. + + The routine `history(string name, int n=1)' can be used to look up +the `n' most recent values (or all values up to `historylines' if +`n=0') entered for string `name'. The routine `history(int n=0)' +returns the interactive history. For example, +write(output("transcript.asy"),history()); + outputs the interactive history to the file `transcript.asy'. + + The function `int delete(string s)' deletes the file named by the +string `s'. Unless the `-globalwrite' (or `-nosafe') option is enabled, +the file must reside in the current directory. The function `int +rename(string from, string to)' may be used to rename file `from' to +file `to'. Unless the `-globalwrite' (or `-nosafe') option is enabled, +this operation is restricted to the current directory. The functions +int convert(string args="", string file="", string format=""); +int animate(string args="", string file="", string format=""); + call the `ImageMagick' commands `convert' and `animate', respectively, +with the arguments `args' and the file name constructed from the +strings `file' and `format'. If the setting `safe' is false, then the +function `int system(string s)' can be used to call the arbitrary system +command `s'. + + +File: asymptote.info, Node: Variable initializers, Next: Structures, Prev: Files, Up: Programming + +5.7 Variable initializers +========================= + +A variable can be assigned a value when it is declared, as in `int +x=3;' where the variable `x' is assigned the value `3'. As well as +literal constants such as `3', arbitary expressions can be used as +initializers, as in `real x=2*sin(pi/2);'. + + A variable is not added to the namespace until after the initializer +is evaluated, so for example, in + +int x=2; +int x=5*x; + the `x' in the initializer on the second line refers to the variable +`x' declared on the first line. The second line, then, declares a +variable `x' shadowing the original `x' and initializes it to the value +`10'. + + Variables of most types can be declared without an explicit +initializer and they will be initialized by the default initializer of +that type: + + * Variables of the numeric types `int', `real', and `pair' are all + initialized to zero; variables of type `triple' are initialized to + `O=(0,0,0)'. + + * `boolean' variables are initialized to `false'. + + * `string' variables are initialized to the empty string. + + * `transform' variables are initialized to the identity + transformation. + + * `path' and `guide' variables are initialized to `nullpath'. + + * `pen' variables are initialized to the default pen. + + * `frame' and `picture' variables are initialized to empty frames + and pictures, respectively. + + * `file' variables are initialized to `null'. + + The default initializers for user-defined array, structure, and +function types are explained in their respective sections. Some types, +such as `code', do not have default initializers. When a variable of +such a type is introduced, the user must initialize it by explicitly +giving it a value. + + The default initializer for any type `T' can be redeclared by +defining the function `T operator init()'. For instance, `int' +variables are usually initialized to zero, but in + +int operator init() { + return 3; +} +int y; + +the variable `y' is initialized to `3'. This example was given for +illustrative purposes; redeclaring the initializers of built-in types +is not recommended. Typically, `operator init' is used to define +sensible defaults for user-defined types. + + +File: asymptote.info, Node: Structures, Next: Operators, Prev: Variable initializers, Up: Programming + +5.8 Structures +============== + +Users may also define their own data types as structures, along with +user-defined operators, much as in C++. By default, structure members +are `public' (may be read and modified anywhere in the code), but may be +optionally declared `restricted' (readable anywhere but writeable only +inside the structure where they are defined) or `private' (readable and +writable only inside the structure). In a structure definition, the +keyword `this' can be used as an expression to refer to the enclosing +structure. Any code at the top-level scope within the structure is +executed on initialization. + + Variables hold references to structures. That is, in the example: + +struct T { + int x; +} + +T foo=new T; +T bar=foo; +bar.x=5; + + The variable `foo' holds a reference to an instance of the structure +`T'. When `bar' is assigned the value of `foo', it too now holds a +reference to the same instance as `foo' does. The assignment `bar.x=5' +changes the value of the field `x' in that instance, so that `foo.x' +will also be equal to `5'. + + The expression `new T' creates a new instance of the structure `T' +and returns a reference to that instance. In creating the new +instance, any code in the body of the record definition is executed. +For example: + +int Tcount=0; +struct T { + int x; + ++Tcount; +} + +T foo=new T; + + Here, the expression `new T' will produce a new instance of the +class, but will also cause `Tcount' to be incremented, so that it keeps +track of the number of instances produced. + + The expression `null' can be cast to any structure type to yield a +null reference, a reference that does not actually refer to any +instance of the structure. Trying to use a field of a null reference +will cause an error. + + The function `bool alias(T,T)' checks to see if two structure +references refer to the same instance of the structure (or both to +`null'). For example, in the example code at the start of the section, +`alias(foo,bar)' would return true, but `alias(foo,new T)' would return +false, as `new T' creates a new instance of the structure `T'. The +boolean operators `==' and `!=' are by default equivalent to `alias' and +`!alias' respectively, but may be overwritten for a particular type +(for example, to do a deep comparison). + + After the definition of a structure `T', a variable of type `T' is +initialized to a new instance (`new T') by default. During the +definition of the structure, however, variables of type `T' are +initialized to `null' by default. This special behaviour is to avoid +infinite recursion of creating new instances in code such as + +struct tree { + int value; + tree left; + tree right; +} + + Here is a simple example that illustrates the use of structures: + +struct S { + real a=1; + real f(real a) {return a+this.a;} +} + +S s; // Initializes s with new S; + +write(s.f(2)); // Outputs 3 + +S operator + (S s1, S s2) +{ + S result; + result.a=s1.a+s2.a; + return result; +} + +write((s+s).f(0)); // Outputs 2 + + It is often convenient to have functions that construct new +instances of a structure. Say we have a `Person' structure: + +struct Person { + string firstname; + string lastname; +} + +Person joe=new Person; +joe.firstname="Joe"; +joe.lastname="Jones"; + Creating a new Person is a chore; it takes three lines to create a new +instance and to initialize its fields (that's still considerably less +effort than creating a new person in real life, though). + + We can reduce the work by defining a constructor function +`Person(string,string)': + +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"); + + While it is now easier than before to create a new instance, we still +have to refer to the constructor by the qualified name `Person.Person'. +If we add the line + +from Person unravel Person; + immediately after the structure definition, then the constructor can +be used without qualification: `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. + + If, in the body of a structure, Asymptote encounters the definition +of a function of the form `void operator init(ARGS)', it implicitly +defines a constructor function of the arguments `ARGS' that uses the +`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 `Foo'): + + static Foo Foo(ARGS) { + Foo instance=new Foo; + instance.operator init(ARGS); + return instance; + } + + 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 `Person' +example can thus be implemented as: + +struct Person { + string firstname; + string lastname; + + void operator init(string firstname, string lastname) { + this.firstname=firstname; + this.lastname=lastname; + } +} + +Person joe=Person("Joe", "Jones"); + + The use of `operator init' to implicitly define constructors should +not be confused with its use to define default values for variables +(*note Variable initializers::). Indeed, in the first case, the return +type of the `operator init' must be `void' while in the second, it must +be the (non-`void') type of the variable. + + The function `cputime()' returns a structure `cputime' with +cumulative CPU times broken down into the fields `parent.user', +`parent.system', `child.user', and `child.system'. For convenience, the +incremental fields `change.user' and `change.system' indicate the +change in the corresponding total parent and child CPU times since the +last call to `cputime()'. The function +void write(file file=stdout, string s="", cputime c, + string format=cputimeformat, suffix suffix=none); + displays the incremental user cputime followed by "u", the incremental +system cputime followed by "s", the total user cputime followed by "U", +and the total system cputime followed by "S". + + Much like in C++, casting (*note Casts::) provides for an elegant +implementation of structure inheritance, including virtual functions: +struct parent { + real x=1; + void virtual(int) {write (0);} + void f() {virtual(1);} +} + +void write(parent p) {write(p.x);} + +struct child { + parent parent; + real y=2; + void virtual(int x) {write (x);} + parent.virtual=virtual; + void f()=parent.f; +} + +parent operator cast(child child) {return child.parent;} + +parent p; +child c; + +write(c); // Outputs 1; + +p.f(); // Outputs 0; +c.f(); // Outputs 1; + +write(c.parent.x); // Outputs 1; +write(c.y); // Outputs 2; + + For further examples of structures, see `Legend' and `picture' in +the `Asymptote' base module `plain'. + + +File: asymptote.info, Node: Operators, Next: Implicit scaling, Prev: Structures, Up: Programming + +5.9 Operators +============= + +* Menu: + +* Arithmetic & logical:: Basic mathematical operators +* Self & prefix operators:: Increment and decrement +* User-defined operators:: Overloading operators + + +File: asymptote.info, Node: Arithmetic & logical, Next: Self & prefix operators, Up: Operators + +5.9.1 Arithmetic & logical operators +------------------------------------ + +`Asymptote' uses the standard binary arithmetic operators. However, +when one integer is divided by another, both arguments are converted to +real values before dividing and a real quotient is returned (since this +is usually what is intended). The function `int quotient(int x, int y)' +returns the greatest integer less than or equal to `x/y'. In all other +cases both operands are promoted to the same type, which will also be +the type of the result: +`+' + addition + +`-' + subtraction + +`*' + multiplication + +`/' + division + +`%' + modulo; the result always has the same sign as the divisor. In + particular, this makes `q*quotient(p,q)+p%q == p' for all integers + `p' and nonzero integers `q'. + +`^' + power; if the exponent (second argument) is an int, recursive + multiplication is used; otherwise, logarithms and exponentials are + used (`**' is a synonym for `^'). + + + The usual boolean operators are also defined: +`==' + equals + +`!=' + not equals + +`<' + less than + +`<=' + less than or equals + +`>=' + greater than or equals + +`>' + greater than + +`&&' + and (with conditional evaluation of right-hand argument) + +`&' + and + +`||' + or (with conditional evaluation of right-hand argument) + +`|' + or + +`^' + xor + +`!' + not + + `Asymptote' also supports the C-like conditional syntax: +bool positive=(pi >= 0) ? true : false; + + The function `T interp(T a, T b, real t)' returns `(1-t)*a+t*b' for +nonintegral built-in arithmetic types `T'. If `a' and `b' are pens, +they are first promoted to the same color space. + + `Asymptote' also defines bitwise functions `int AND(int,int)', `int +OR(int,int)', `int XOR(int,int)', and `int NOT(int)'. + + +File: asymptote.info, Node: Self & prefix operators, Next: User-defined operators, Prev: Arithmetic & logical, Up: Operators + +5.9.2 Self & prefix operators +----------------------------- + +As in C, each of the arithmetic operators `+', `-', `*', `/', `%', and +`^' can be used as a self operator. The prefix operators `++' +(increment by one) and `--' (decrement by one) are also defined. For +example, +int i=1; +i += 2; +int j=++i; + +is equivalent to the code +int i=1; +i=i+2; +int j=i=i+1; + + However, postfix operators like `i++' and `i--' are not defined +(because of the inherent ambiguities that would arise with the `--' +path-joining operator). In the rare instances where `i++' and `i--' are +really needed, one can substitute the expressions `(++i-1)' and +`(--i+1)', respectively. + + +File: asymptote.info, Node: User-defined operators, Prev: Self & prefix operators, Up: Operators + +5.9.3 User-defined operators +---------------------------- + +The following symbols may be used with `operator' to define or redefine +operators on structures and built-in types: +- + * / % ^ ! < > == != <= >= & | ^^ .. :: -- --- ++ +<< >> $ $$ @ @@ + The operators on the second line have precedence one higher than the +boolean operators `<', `>', `<=', and `>='. + + Guide operators like `..' may be overloaded, say, to write a user +function that produces a new guide from a given guide: +guide dots(... guide[] g)=operator ..; + +guide operator ..(... guide[] g) { + guide G; + if(g.length > 0) { + write(g[0]); + G=g[0]; + } + for(int i=1; i < g.length; ++i) { + write(g[i]); + write(); + G=dots(G,g[i]); + } + return G; +} + +guide g=(0,0){up}..{SW}(100,100){NE}..{curl 3}(50,50)..(10,10); +write("g=",g); + + +File: asymptote.info, Node: Implicit scaling, Next: Functions, Prev: Operators, Up: Programming + +5.10 Implicit scaling +===================== + +If a numeric literal is in front of certain types of expressions, then +the two are multiplied: +int x=2; +real y=2.0; +real cm=72/2.540005; + +write(3x); +write(2.5x); +write(3y); +write(-1.602e-19 y); +write(0.5(x,y)); +write(2x^2); +write(3x+2y); +write(3(x+2y)); +write(3sin(x)); +write(3(sin(x))^2); +write(10cm); + + This produces the output +6 +5 +6 +-3.204e-19 +(1,1) +8 +10 +18 +2.72789228047704 +2.48046543129542 +283.464008929116 + + +File: asymptote.info, Node: Functions, Next: Arrays, Prev: Implicit scaling, Up: Programming + +5.11 Functions +============== + +`Asymptote' functions are treated as variables with a signature +(non-function variables have null signatures). Variables with the same +name are allowed, so long as they have distinct signatures. + + Functions arguments are passed by value. To pass an argument by +reference, simply enclose it in a structure (*note Structures::). + + Here are some significant features of `Asymptote' functions: + + 1. Variables with signatures (functions) and without signatures + (nonfunction variables) are distinct: int x, x(); + x=5; + x=new int() {return 17;}; + x=x(); // calls x() and puts the result, 17, in the scalar x + + 2. Traditional function definitions are allowed: int sqr(int x) + { + return x*x; + } + sqr=null; // but the function is still just a variable. + + 3. Casting can be used to resolve ambiguities: int a, a(), b, b(); // Valid: creates four variables. + a=b; // Invalid: assignment is ambiguous. + a=(int) b; // Valid: resolves ambiguity. + (int) (a=b); // Valid: resolves ambiguity. + (int) a=b; // Invalid: cast expressions cannot be L-values. + + int c(); + c=a; // Valid: only one possible assignment. + + 4. Anonymous (so-called "high-order") functions are also allowed: 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. + + 5. One may redefine a function `f', even for calls to `f' in + previously declared functions, by assigning another (anonymous or + named) function to it. However, if `f' is overloaded by a new + function definition, previous calls will still access the original + version of `f', as illustrated in this example: void f() { + write("hi"); + } + + void g() { + f(); + } + + g(); // writes "hi" + + f=new void() {write("bye");}; + + g(); // writes "bye" + + void f() {write("overloaded");}; + + f(); // writes "overloaded" + g(); // writes "bye" + + 6. Anonymous functions can be used to redefine a function variable + that has been declared (and implicitly initialized to the null + function) but not yet explicitly defined: void f(bool b); + + void g(bool b) { + if(b) f(b); + else write(b); + } + + f=new void(bool b) { + write(b); + g(false); + }; + + g(true); // Writes true, then writes false. + + + `Asymptote' is the only language we know of that treats functions as +variables, but allows overloading by distinguishing variables based on +their signatures. + + Functions are allowed to call themselves recursively. As in C++, +infinite nested recursion will generate a stack overflow (reported as a +segmentation fault, unless a fully working version of the GNU library +`libsigsegv' (e.g. 2.4 or later) is installed at configuration time). + +* 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 + + +File: asymptote.info, Node: Default arguments, Next: Named arguments, Up: Functions + +5.11.1 Default arguments +------------------------ + +`Asymptote' supports a more flexible mechanism for default function +arguments than C++: they may appear anywhere in the function prototype. +Because certain data types are implicitly cast to more sophisticated +types (*note Casts::) one can often avoid ambiguities by ordering +function arguments from the simplest to the most complicated. For +example, given +real f(int a=1, real b=0) {return a+b;} + then `f(1)' returns 1.0, but `f(1.0)' returns 2.0. + + The value of a default argument is determined by evaluating the +given `Asymptote' expression in the scope where the called function is +defined. + + +File: asymptote.info, Node: Named arguments, Next: Rest arguments, Prev: Default arguments, Up: Functions + +5.11.2 Named arguments +---------------------- + +It is sometimes difficult to remember the order in which arguments +appear in a function declaration. Named (keyword) arguments make calling +functions with multiple arguments easier. Unlike in the C and C++ +languages, an assignment in a function argument is interpreted as an +assignment to a parameter of the same name in the function signature, +_not within the local scope_. The command-line option `-d' may be used +to check `Asymptote' code for cases where a named argument may be +mistaken for a local assignment. + + When matching arguments to signatures, first all of the keywords are +matched, then the arguments without names are matched against the +unmatched formals as usual. For example, +int f(int x, int y) { + return 10x+y; +} +write(f(4,x=3)); + outputs 34, as `x' is already matched when we try to match the unnamed +argument `4', so it gets matched to the next item, `y'. + + For the rare occasions where it is desirable to assign a value to +local variable within a function argument (generally _not_ a good +programming practice), simply enclose the assignment in parentheses. +For example, given the definition of `f' in the previous example, +int x; +write(f(4,(x=3))); + is equivalent to the statements +int x; +x=3; +write(f(4,3)); + and outputs 43. + + As a technical detail, we point out that, since variables of the same +name but different signatures are allowed in the same scope, the code +int f(int x, int x()) { + return x+x(); +} +int seven() {return 7;} + is legal in `Asymptote', with `f(2,seven)' returning 9. A named +argument matches the first unmatched formal of the same name, so +`f(x=2,x=seven)' is an equivalent call, but `f(x=seven,2)' is not, as +the first argument is matched to the first formal, and `int ()' cannot +be implicitly cast to `int'. Default arguments do not affect which +formal a named argument is matched to, so if `f' were defined as +int f(int x=3, int x()) { + return x+x(); +} + then `f(x=seven)' would be illegal, even though `f(seven)' obviously +would be allowed. + + +File: asymptote.info, Node: Rest arguments, Next: Mathematical functions, Prev: Named arguments, Up: Functions + +5.11.3 Rest arguments +--------------------- + +Rest arguments allow one to write functions that take a variable number +of arguments: +// This function sums its arguments. +int sum(... int[] nums) { + int total=0; + for(int i=0; i < nums.length; ++i) + total += nums[i]; + return total; +} + +sum(1,2,3,4); // returns 10 +sum(); // returns 0 + +// This function subtracts subsequent arguments from the first. +int subtract(int start ... int[] subs) { + for(int i=0; i < subs.length; ++i) + start -= subs[i]; + return start; +} + +subtract(10,1,2); // returns 7 +subtract(10); // returns 10 +subtract(); // illegal + + Putting an argument into a rest array is called _packing_. One can +give an explicit list of arguments for the rest argument, so `subtract' +could alternatively be implemented as +int subtract(int start ... int[] subs) { + return start - sum(... subs); +} + + One can even combine normal arguments with rest arguments: +sum(1,2,3 ... new int[] {4,5,6}); // returns 21 + This builds a new six-element array that is passed to `sum' as `nums'. +The opposite operation, _unpacking_, is not allowed: +subtract(... new int[] {10, 1, 2}); + is illegal, as the start formal is not matched. + + If no arguments are packed, then a zero-length array (as opposed to +`null') is bound to the rest parameter. Note that default arguments are +ignored for rest formals and the rest argument is not bound to a +keyword. + + The overloading resolution in `Asymptote' is similar to the function +matching rules used in C++. Every argument match is given a score. +Exact matches score better than matches with casting, and matches with +formals (regardless of casting) score better than packing an argument +into the rest array. A candidate is maximal if all of the arguments +score as well in it as with any other candidate. If there is one +unique maximal candidate, it is chosen; otherwise, there is an +ambiguity error. + +int f(path g); +int f(guide g); +f((0,0)--(100,100)); // matches the second; the argument is a guide + +int g(int x, real y); +int g(real x, int x); + +g(3,4); // ambiguous; the first candidate is better for the first argument, + // but the second candidate is better for the second argument + +int h(... int[] rest); +int h(real x ... int[] rest); + +h(1,2); // the second definition matches, even though there is a cast, + // because casting is preferred over packing + +int i(int x ... int[] rest); +int i(real x, real y ... int[] rest); + +i(3,4); // ambiguous; the first candidate is better for the first argument, + // but the second candidate is better for the second one + + +File: asymptote.info, Node: Mathematical functions, Prev: Rest arguments, Up: Functions + +5.11.4 Mathematical functions +----------------------------- + +`Asymptote' has built-in versions of the standard `libm' mathematical +real(real) functions `sin', `cos', `tan', `asin', `acos', `atan', +`exp', `log', `pow10', `log10', `sinh', `cosh', `tanh', `asinh', +`acosh', `atanh', `sqrt', `cbrt', `fabs', `expm1', `log1p', as well as +the identity function `identity'. `Asymptote' also defines the order +`n' Bessel functions of the first kind `J(int n, real)' and second kind +`Y(int n, real)', as well as the gamma function `gamma', the error +function `erf', and the complementary error function `erfc'. The +standard real(real, real) functions `atan2', `hypot', `fmod', +`remainder' are also included. + + The functions `degrees(real radians)' and `radians(real degrees)' +can be used to convert between radians and degrees. The function +`Degrees(real radians)' returns the angle in degrees in the interval +[0,360). For convenience, `Asymptote' defines variants `Sin', `Cos', +`Tan', `aSin', `aCos', and `aTan' of the standard trigonometric +functions that use degrees rather than radians. We also define complex +versions of the `sqrt', `sin', `cos', `exp', `log', and `gamma' +functions. + + The functions `floor', `ceil', and `round' differ from their usual +definitions in that they all return an int value rather than a real +(since that is normally what one wants). The functions `Floor', +`Ceil', and `Round' are respectively similar, except that if the result +cannot be converted to a valid int, they return `intMax' for positive +arguments and `intMin' for negative arguments, rather than generating +an integer overflow. We also define a function `sgn', which returns +the sign of its real argument as an integer (-1, 0, or 1). + + There is an `abs(int)' function, as well as an `abs(real)' function +(equivalent to `fabs(real)'), an `abs(pair)' function (equivalent to +`length(pair)'). + + Random numbers can be seeded with `srand(int)' and generated with +the `int rand()' function, which returns a random integer between 0 and +the integer `randMax'. The `unitrand()' function returns a random +number uniformly distributed in the interval [0,1]. A Gaussian random +number generator `Gaussrand' and a collection of statistics routines, +including `histogram', are provided in the base file `stats.asy'. The +functions `factorial(int n)', which returns n!, and `choose(int n, int +k)', which returns n!/(k!(n-k)!), are also defined. + + When configured with the GNU Scientific Library (GSL), available from +`http://www.gnu.org/software/gsl/', `Asymptote' contains an internal +module `gsl' that defines the airy functions `Ai(real)', `Bi(real)', +`Ai_deriv(real)', `Bi_deriv(real)', `zero_Ai(int)', `zero_Bi(int)', +`zero_Ai_deriv(int)', `zero_Bi_deriv(int)', the Bessel functions +`I(int, real)', `K(int, real)', `j(int, real)', `y(int, real)', +`i_scaled(int, real)', `k_scaled(int, real)', `J(real, real)', `Y(real, +real)', `I(real, real)', `K(real, real)', `zero_J(real, int)', the +elliptic functions `F(real, real)', `E(real, real)', and `P(real, +real)', the exponential/trigonometric integrals `Ei', `Si', and `Ci', +the Legendre polynomials `Pl(int, real)', and the Riemann zeta function +`zeta(real)'. For example, to compute the sine integral `Si' of 1.0: +import gsl; +write(Si(1.0)); + + `Asymptote' also provides a few general purpose numerical routines: + +``real newton(int iterations=100, real f(real), real fprime(real), real x, bool verbose=false);'' + Use Newton-Raphson iteration to solve for a root of a real-valued + differentiable function `f', given its derivative `fprime' and an + initial guess `x'. Diagnostics for each iteration are printed if + `verbose=true'. If the iteration fails after the maximum allowed + number of loops (`iterations'), `realMax' is returned. + +``real newton(int iterations=100, real f(real), real fprime(real), real x1, real x2, bool verbose=false);'' + Use bracketed Newton-Raphson bisection to solve for a root of a + real-valued differentiable function `f' within an interval + [`x1',`x2'] (on which the endpoint values of `f' have opposite + signs), given its derivative `fprime'. Diagnostics for each + iteration are printed if `verbose=true'. If the iteration fails + after the maximum allowed number of loops (`iterations'), + `realMax' is returned. + +``real simpson(real f(real), real a, real b, real acc=realEpsilon, real dxmax=b-a)'' + returns the integral of `f' from `a' to `b' using adaptive Simpson + integration. + + + +File: asymptote.info, Node: Arrays, Next: Casts, Prev: Functions, Up: Programming + +5.12 Arrays +=========== + +* Menu: + +* Slices:: Python-style array slices + + Appending `[]' to a built-in or user-defined type yields an array. +The array element `i' of an array `A' can be accessed as `A[i]'. By +default, attempts to access or assign to an array element using a +negative index generates an error. Reading an array element with an +index beyond the length of the array also generates an error; however, +assignment to an element beyond the length of the array causes the +array to be resized to accommodate the new element. One can also index +an array `A' with an integer array `B': the array `A[B]' is formed by +indexing array `A' with successive elements of array `B'. A convenient +Java-style shorthand exists for iterating over all elements of an +array; see *note array iteration::. + + The declaration +real[] A; + +initializes `A' to be an empty (zero-length) array. Empty arrays should +be distinguished from null arrays. If we say +real[] A=null; + +then `A' cannot be dereferenced at all (null arrays have no length and +cannot be read from or assigned to). + + Arrays can be explicitly initialized like this: +real[] A={0,1,2}; + + Array assignment in `Asymptote' does a shallow copy: only the +pointer is copied (if one copy if modified, the other will be too). +The `copy' function listed below provides a deep copy of an array. + + Every array `A' of type `T[]' has the virtual members + * `int length', + + * `void cyclic(bool b)', + + * `bool cyclicflag', + + * `int[] keys', + + * `T push(T x)', + + * `void append(T[] a)', + + * `T pop()', + + * `void insert(int i ... T[] x)', + + * `void delete(int i, int j=i)', + + * `void delete()', and + + * `bool initialized(int n)'. + + The member `A.length' evaluates to the length of the array. Setting +`A.cyclic(true)' signifies that array indices should be reduced modulo +the current array length. Reading from or writing to a nonempty cyclic +array never leads to out-of-bounds errors or array resizing. The member +`A.cyclicflag' returns the current setting of the `cyclic' flag. + + The member `A.keys' evaluates to an array of integers containing the +indices of initialized entries in the array in ascending order. Hence, +for an array of length `n' with all entries initialized, `A.keys' +evaluates to the array of integers from `0' to `n-1' inclusive. A new +keys array is produced each time `A.keys' is evaluated. + + The functions `A.push' and `A.append' append their arguments onto +the end of the array, while `A.insert(int i ... T[] x)' inserts `x' +into the array at index `i'. For convenience `A.push' returns the +pushed item. The function `A.pop()' pops and returns the last element, +while `A.delete(int i, int j=i)' deletes elements with indices in the +range [`i',`j'], shifting the position of all higher-indexed elements +down. If no arguments are given, `A.delete()' provides a convenient way +of deleting all elements of `A'. The routine `A.initialized(int n)' can +be used to examine whether the element at index `n' is initialized. +Like all `Asymptote' functions, `cyclic', `push', `append', `pop', +`insert', `delete', and `initialized' can be "pulled off" of the array +and used on their own. For example, +int[] A={1}; +A.push(2); // A now contains {1,2}. +A.append(A); // A now contains {1,2,1,2}. +int f(int)=A.push; +f(3); // A now contains {1,2,1,2,3}. +int g()=A.pop; +write(g()); // Outputs 3. +A.delete(0); // A now contains {2,1,2}. +A.delete(0,1); // A now contains {2}. +A.insert(1,3); // A now contains {2,3}. +A.insert(1 ... A); // A now contains {2,2,3,3} +A.insert(2,4,5); // A now contains {2,2,4,5,3,3}. + + The `[]' suffix can also appear after the variable name; this is +sometimes convenient for declaring a list of variables and arrays of +the same type: +real a,A[]; + This declares `a' to be `real' and implicitly declares `A' to be of +type `real[]'. + + In the following list of built-in array functions, `T' represents a +generic type. Note that the internal functions `alias', `array', +`copy', `concat', `sequence', `map', and `transpose', which depend on +type `T[]', are defined only after the first declaration of a variable +of type `T[]'. + +`new T[]' + returns a new empty array of type `T[]'; + +`new T[] {list}' + returns a new array of type `T[]' initialized with `list' (a comma + delimited list of elements). + +`new T[n]' + returns a new array of `n' elements of type `T[]'. These `n' + array elements are not initialized unless they are arrays + themselves (in which case they are each initialized to empty + arrays). + +`T[] array(int n, T value, int depth=intMax)' + returns an array consisting of `n' copies of value. By default, if + `value' is itself an array, a deep copy of that array is made for + each entry in the new array. If `depth' is specified, this deep + copying only recurses to the number of levels specified. + +`int[] sequence(int n)' + if `n >= 1' returns the array `{0,1,...,n-1}' (otherwise returns a + null array); + +`int[] sequence(int n, int m)' + if `m >= n' returns an array `{n,n+1,...,m}' (otherwise returns a + null array); + +`T[] sequence(T f(int), int n)' + if `n >= 1' returns the sequence `{f_i :i=0,1,...n-1}' given a + function `T f(int)' and integer `int n' (otherwise returns a null + array); + +`T[] map(T f(T), T[] a)' + returns the array obtained by applying the function `f' to each + element of the array `a'. This is equivalent to `sequence(new + T(int i) {return f(a[i]);},a.length)'. + +`int[] reverse(int n)' + if `n >= 1' returns the array `{n-1,n-2,...,0}' (otherwise returns + a null array); + +`int[] complement(int[] a, int n)' + returns the complement of the integer array `a' in + `{0,1,2,...,n-1}', so that `b[complement(a,b.length)]' yields the + complement of `b[a]'. + +`real[] uniform(real a, real b, int n)' + if `n >= 1' returns a uniform partition of `[a,b]' into `n' + subintervals (otherwise returns a null array); + +`int find(bool[], int n=1)' + returns the index of the `n'th `true' value or -1 if not found. + If `n' is negative, search backwards from the end of the array for + the `-n'th value; + +`int search(T[] a, T key)' + For built-in ordered types `T', searches a sorted ordered array + `a' of `n' elements to find an interval containing `key', + returning `-1' if `key' is less than the first element, `n-1' if + `key' is greater than or equal to the last element, and otherwise + the index corresponding to the left-hand (smaller) endpoint. + +`T[] copy(T[] a)' + returns a deep copy of the array `a'; + +`T[][] copy(T[][] a)' + returns a deep copy of the array `a'; + +`T[][][] copy(T[][][] a)' + returns a deep copy of the array `a'; + +`T[] concat(... T[][] a)' + returns a new array formed by concatenating the arrays given as + arguments; + +`bool alias(T[] a, T[] b)' + returns `true' if the arrays `a' and `b' are identical; + +`T[] sort(T[] a)' + For built-in ordered types `T', returns a copy of `a' sorted in + ascending order; + +`T[][] sort(T[][] a)' + For built-in ordered types `T', returns a copy of `a' with the rows + sorted by the first column, breaking ties with successively higher + columns. For example: string[][] a={{"bob","9"},{"alice","5"},{"pete","7"}, + {"alice","4"}}; + // Row sort (by column 0, using column 1 to break ties): + write(sort(a)); + + produces alice 4 + alice 5 + bob 9 + pete 7 + +`T[] sort(T[] a, bool compare(T i, T j))' + returns a copy of `a' stably sorted in ascending order such that + element `i' precedes element `j' if `compare(i,j)' is true. + +`T[][] transpose(T[][] a)' + returns the transpose of `a'. + +`T[][][] transpose(T[][][] a, int[] perm)' + returns the 3D transpose of `a' obtained by applying the + permutation `perm' of `new int[]{0,1,2}' to the indices of each + entry. + +`T sum(T[] a)' + For arithmetic types `T', returns the sum of `a'. In the case + where `T' is `bool', the number of true elements in `a' is + returned. + +`T min(T[] a)' + +`T min(T[][] a)' + +`T min(T[][][] a)' + For built-in ordered types `T', returns the minimum element of `a'. + +`T max(T[] a)' + +`T max(T[][] a)' + +`T max(T[][][] a)' + For built-in ordered types `T', returns the maximum element of `a'. + +`T[] min(T[] a, T[] b)' + For built-in ordered types `T', and arrays `a' and `b' of the same + length, returns an array composed of the minimum of the + corresponding elements of `a' and `b'. + +`T[] max(T[] a, T[] b)' + For built-in ordered types `T', and arrays `a' and `b' of the same + length, returns an array composed of the maximum of the + corresponding elements of `a' and `b'. + +`pair[] pairs(real[] x, real[] y);' + For arrays `x' and `y' of the same length, returns the pair array + `sequence(new pair(int i) {return (x[i],y[i]);},x.length)'. + +`pair[] fft(pair[] a, int sign=1)' + returns the Fast Fourier Transform of `a' (if the optional `FFTW' + package is installed), using the given `sign'. Here is a simple + example: int n=4; + pair[] f=sequence(n); + write(f); + pair[] g=fft(f,-1); + write(); + write(g); + f=fft(g,1); + write(); + write(f/n); + +`real dot(real[] a, real[] b)' + returns the dot product of the vectors `a' and `b'. + +`real[] tridiagonal(real[] a, real[] b, real[] c, real[] f);' + Solve the periodic tridiagonal problem L`x'=`f' and return the + solution `x', where `f' is an n vector and L is the n \times n + matrix [ b[0] c[0] a[0] ] + [ a[1] b[1] c[1] ] + [ a[2] b[2] c[2] ] + [ ... ] + [ c[n-1] a[n-1] b[n-1] ] + For Dirichlet boundary conditions (denoted here by `u[-1]' and + `u[n]'), replace `f[0]' by `f[0]-a[0]u[-1]' and + `f[n-1]-c[n-1]u[n]'; then set `a[0]=c[n-1]=0'. + +`real[] solve(real[][] a, real[] b, bool warn=true)' + Solve the linear equation `a'x=`b' by LU decomposition and return + the solution x, where `a' is an n \times n matrix and `b' is an + array of length n. For example: import math; + real[][] a={{1,-2,3,0},{4,-5,6,2},{-7,-8,10,5},{1,50,1,-2}}; + real[] b={7,19,33,3}; + real[] x=solve(a,b); + write(a); write(); + write(b); write(); + write(x); write(); + write(a*x); + If `a' is a singular matrix and `warn' is `false', return an + empty array. If the matrix `a' is tridiagonal, the routine + `tridiagonal' provides a more efficient algorithm (*note + tridiagonal::). + +`real[][] solve(real[][] a, real[][] b, bool warn=true)' + Solve the linear equation `a'x=`b' and return the solution x, + where `a' is an n \times n matrix and `b' is an n \times m matrix. + If `a' is a singular matrix and `warn' is `false', return an empty + matrix. + +`real[][] identity(int n);' + returns the n \times n identity matrix. + +`real[][] diagonal(... real[] a)' + returns the diagonal matrix with diagonal entries given by a. + +`real[][] inverse(real[][] a)' + returns the inverse of a square matrix `a'. + +``real[] quadraticroots(real a, real b, real c);'' + This numerically robust solver returns the real roots of the + quadratic equation ax^2+bx+c=0, in ascending order. Multiple roots + are listed separately. + +``pair[] quadraticroots(explicit pair a, explicit pair b, explicit pair c);'' + This numerically robust solver returns the two complex roots of the + quadratic equation ax^2+bx+c=0. + +``real[] cubicroots(real a, real b, real c, real d);'' + This numerically robust solver returns the real roots of the cubic + equation ax^3+bx^2+cx+d=0. Multiple roots are listed separately. + + + `Asymptote' includes a full set of vectorized array instructions for +arithmetic (including self) and logical operations. These +element-by-element instructions are implemented in C++ code for speed. +Given +real[] a={1,2}; +real[] b={3,2}; + then `a == b' and `a >= 2' both evaluate to the vector `{false, true}'. To +test whether all components of `a' and `b' agree, use the boolean +function `all(a == b)'. One can also use conditionals like `(a >= 2) ? +a : b', which returns the array `{3,2}', or `write((a >= 2) ? a : +null', which returns the array `{2}'. + + All of the standard built-in `libm' functions of signature +`real(real)' also take a real array as an argument, effectively like an +implicit call to `map'. + + As with other built-in types, arrays of the basic data types can be +read in by assignment. In this example, the code +file fin=input("test.txt"); +real[] A=fin; + +reads real values into `A' until the end-of-file is reached (or an I/O +error occurs). If line mode is set with `line(file)', then reading will +stop once the end of the line is reached instead (line mode may be +cleared with `line(file,false)'): +file fin=input("test.txt"); +real[] A=line(fin); + + 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, set with +`word(file)' and cleared with `word(file,false)', which causes string +reads to respect white-space delimiters, instead of the default +end-of-line delimiter: +file fin=word(line(input("test.txt"))); +real[] A=fin; + + Another useful mode is comma-separated-value mode, set with +`csv(file)' and cleared with `csv(file,false)', which causes reads to +respect comma delimiters: +file fin=csv(input("test.txt")); +real[] A=fin; + + To restrict the number of values read, use the `dimension(file,int)' +function: +file fin=input("test.txt"); +real[] A=dimension(fin,10); + + This reads 10 values into A, unless end-of-file (or end-of-line in +line mode) occurs first. Attempting to read beyond the end of the file +will produce a runtime error message. Specifying a value of 0 for the +integer limit is equivalent to the previous example of reading until +end-of-file (or end-of-line in line mode) is encountered. + + Two- and three-dimensional arrays of the basic data types can be read +in like this: +file fin=input("test.txt"); +real[][] A=dimension(fin,2,3); +real[][][] B=dimension(fin,2,3,4); + Again, an integer limit of zero means no restriction. + + Sometimes the array dimensions are stored with the data as integer +fields at the beginning of an array. Such arrays can be read in with the +functions `read1', `read2', and `read3', respectively: +file fin=input("test.txt"); +real[] A=read1(fin); +real[][] B=read2(fin); +real[][][] C=read3(fin); + + One, two, and three-dimensional arrays of the basic data types can be +output with the functions `write(file,T[])', `write(file,T[][])', +`write(file,T[][][])', respectively. + + +File: asymptote.info, Node: Slices, Up: Arrays + +5.12.1 Slices +------------- + +Asymptote allows a section of an array to be addressed as a slice using +a Python-like syntax. If `A' is an array, the expression `A[m:n]' +returns a new array consisting of the elements of `A' with indices from +`m' up to but not including `n'. For example, + +int[] x={0,1,2,3,4,5,6,7,8,9}; +int[] y=x[2:6]; // y={2,3,4,5}; +int[] z=x[5:10]; // z={5,6,7,8,9}; + + If the left index is omitted, it is taken be `0'. If the right +index is omitted it is taken to be the length of the array. If both +are omitted, the slice then goes from the start of the array to the +end, producing a non-cyclic deep copy of the array. For example: + +int[] x={0,1,2,3,4,5,6,7,8,9}; +int[] y=x[:4]; // y={0,1,2,3} +int[] z=x[5:]; // z={5,6,7,8,9} +int[] w=x[:]; // w={0,1,2,3,4,5,6,7,8,9}, distinct from array x. + + If A is a non-cyclic array, it is illegal to use negative values for +either of the indices. If the indices exceed the length of the array, +however, they are politely truncated to that length. + + For cyclic arrays, the slice `A[m:n]' still consists of the cells +with indices in the set [`m',`n'), but now negative values and values +beyond the length of the array are allowed. The indices simply wrap +around. For example: + +int[] x={0,1,2,3,4,5,6,7,8,9}; +x.cyclic(true); +int[] y=x[8:15]; // y={8,9,0,1,2,3,4}. +int[] z=x[-5:5]; // z={5,6,7,8,9,0,1,2,3,4} +int[] w=x[-3:17]; // w={7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6} + + Notice that with cyclic arrays, it is possible to include the same +element of the original array multiple times within a slice. +Regardless of the original array, arrays produced by slices are always +non-cyclic. + + If the left and right indices of a slice are the same, the result is +an empty array. If the array being sliced is empty, the result is an +empty array. Any slice with a left index greater than its right index +will yield an error. + + Slices can also be assigned to, changing the value of the original +array. If the array being assigned to the slice has a different length +than the slice itself, elements will be inserted or removed from the +array to accommodate it. For instance: + +string[] toppings={"mayo", "salt", "ham", "lettuce"}; +toppings[0:2]=new string[] {"mustard", "pepper"}; + // Now toppings={"mustard", "pepper", "ham", "lettuce"} +toppings[2:3]=new string[] {"turkey", "bacon" }; + // Now toppings={"mustard", "pepper", "turkey", "bacon", "lettuce"} +toppings[0:3]=new string[] {"tomato"}; + // Now toppings={"tomato", "bacon", "lettuce"} + + If an array is assigned to a slice of itself, a copy of the original +array is assigned to the slice. That is, code such as `x[m:n]=x' is +equivalent to `x[m:n]=copy(x)'. One can use the shorthand `x[m:m]=y' +to insert the contents of the array `y' into the array `x' starting at +the location just before `x[m]'. + + For a cyclic array, a slice is bridging if it addresses cells up to +the end of the array and then continues on to address cells at the +start of the array. For instance, if `A' is a cyclic array of length +10, `A[8:12]', `A[-3:1]', and `A[5:25]' are bridging slices whereas +`A[3:7]', `A[7:10]', `A[-3:0]' and `A[103:107]' are not. Bridging +slices can only be assigned to if the number of elements in the slice +is exactly equal to the number of elements we are assigning to it. +Otherwise, there is no clear way to decide which of the new entries +should be `A[0]' and an error is reported. Non-bridging slices may be +assigned an array of any length. + + For a cyclic array `A' an expression of the form +`A[A.length:A.length]' is equivalent to the expression `A[0:0]' and so +assigning to this slice will insert values at the start of the array. +`A.append()' can be used to insert values at the end of the array. + + It is illegal to assign to a slice of a cyclic array that repeats +any of the cells. + + +File: asymptote.info, Node: Casts, Next: Import, Prev: Arrays, Up: Programming + +5.13 Casts +========== + +`Asymptote' implicitly casts `int' to `real', `int' to `pair', `real' +to `pair', `pair' to `path', `pair' to `guide', `path' to `guide', +`guide' to `path', `real' to `pen', `pair[]' to `guide[]', `pair[]' to +`path[]', `path' to `path[]', and `guide' to `path[]', along with +various three-dimensional casts defined in `three.asy'. Implicit casts +are automatically attempted on assignment and when trying to match +function calls with possible function signatures. Implicit casting can +be inhibited by declaring individual arguments `explicit' in the +function signature, say to avoid an ambiguous function call in the +following example, which outputs 0: + +int f(pair a) {return 0;} +int f(explicit real x) {return 1;} + +write(f(0)); + + Other conversions, say `real' to `int' or `real' to `string', +require an explicit cast: +int i=(int) 2.5; +string s=(string) 2.5; + +real[] a={2.5,-3.5}; +int[] b=(int []) a; +write(stdout,b); // Outputs 2,-3 + + Casting to user-defined types is also possible using `operator cast': +struct rpair { + real radius; + real angle; +} + +pair operator cast(rpair x) { + return (x.radius*cos(x.angle),x.radius*sin(x.angle)); +} + +rpair x; +x.radius=1; +x.angle=pi/6; + +write(x); // Outputs (0.866025403784439,0.5) + + One must use care when defining new cast operators. Suppose that in +some code one wants all integers to represent multiples of 100. To +convert them to reals, one would first want to multiply them by 100. +However, the straightforward implementation +real operator cast(int x) {return x*100;} + is equivalent to an infinite recursion, since the result `x*100' needs +itself to be cast from an integer to a real. Instead, we want to use +the standard conversion of int to real: +real convert(int x) {return x*100;} +real operator cast(int x)=convert; + + Explicit casts are implemented similarly, with `operator ecast'. + + +File: asymptote.info, Node: Import, Next: Static, Prev: Casts, Up: Programming + +5.14 Import +=========== + +While `Asymptote' provides many features by default, some applications +require specialized features contained in external `Asymptote' modules. +For instance, the lines +access graph; +graph.axes(); + draw x and y axes on a two-dimensional graph. Here, the command looks +up the module under the name `graph' in a global dictionary of modules +and puts it in a new variable named `graph'. The module is a +structure, and we can refer to its fields as we usually would with a +structure. + + Often, one wants to use module functions without having to specify +the module name. The code +from graph access axes; + adds the `axes' field of `graph' into the local name space, so that +subsequently, one can just write `axes()'. If the given name is +overloaded, all types and variables of that name are added. To add +more than one name, just use a comma-separated list: + +from graph access axes, xaxis, yaxis; + Wild card notation can be used to add all non-private fields and types +of a module to the local name space: + +from graph access *; + + Similarly, one can add the non-private fields and types of a +structure to the local environment with the `unravel' keyword: + +struct matrix { + real a,b,c,d; +} + +real det(matrix m) { + unravel m; + return a*d-b*c; +} + Alternatively, one can unravel selective fields: +real det(matrix m) { + from m unravel a,b,c as C,d; + return a*d-b*C; +} + + The command +import graph; + is a convenient abbreviation for the commands +access graph; +unravel graph; + That is, `import graph' first loads a module into a structure called +`graph' and then adds its non-private fields and types to the local +environment. This way, if a member variable (or function) is +overwritten with a local variable (or function of the same signature), +the original one can still be accessed by qualifying it with the module +name. + + Wild card importing will work fine in most cases, but one does not +usually know all of the internal types and variables of a module, which +can also change as the module writer adds or changes features of the +module. As such, it is prudent to add `import' commands at the start +of an `Asymptote' file, so that imported names won't shadow locally +defined functions. Still, imported names may shadow other imported +names, depending on the order in which they were imported, and imported +functions may cause overloading resolution problems if they have the +same name as local functions defined later. + + To rename modules or fields when adding them to the local +environment, use `as': + +access graph as graph2d; +from graph access xaxis as xline, yaxis as yline; + + The command +import graph as graph2d; + is a convenient abbreviation for the commands +access graph as graph2d; +unravel graph2d; + + Except for a few built-in modules, such as `settings', all modules +are implemented as `Asymptote' files. When looking up a module that +has not yet been loaded, `Asymptote' searches the standard search paths +(*note Search paths::) for the matching file. The file corresponding +to that name is read and the code within it is interpreted as the body +of a structure defining the module. + + If the file name contains nonalphanumeric characters, enclose it +with quotation marks: + +`access "/usr/local/share/asymptote/graph.asy" as graph;' + +`from "/usr/local/share/asymptote/graph.asy" access axes;' + +`import "/usr/local/share/asymptote/graph.asy" as graph;' + + It is an error if modules import themselves (or each other in a +cycle). The module name to be imported must be known at compile time. + + However, you can import an `Asymptote' module determined by the +string `s' at runtime like this: +eval("import "+s,true); + + To conditionally execute an array of asy files, use +void asy(string format, bool overwrite ... string[] s); + The file will only be processed, using output format `format', if +overwrite is `true' or the output file is missing. + + One can evaluate an `Asymptote' expression (without any return +value, however) contained in the string `s' with: +void eval(string s, bool embedded=false); + It is not necessary to terminate the string `s' with a semicolon. If +`embedded' is `true', the string will be evaluated at the top level of +the current environment. If `embedded' is `false' (the default), the +string will be evaluated in an independent environment, sharing the same +`settings' module (*note settings::). + + One can evaluate arbitrary `Asymptote' code (which may contain +unescaped quotation marks) with the command +void eval(code s, bool embedded=false); + Here `code' is a special type used with `quote {}' to enclose +`Asymptote code' like this: +real a=1; +code s=quote { + write(a); +}; +eval(s,true); // Outputs 1 + + To include the contents of a file `graph' verbatim (as if the +contents of the file were inserted at that point), use one of the forms: +include graph; + +`include "/usr/local/share/asymptote/graph.asy";' + + To list all global functions and variables defined in a module named +by the contents of the string `s', use the function +void list(string s, bool imports=false); + Imported global functions and variables are also listed if `imports' +is `true'. + + +File: asymptote.info, Node: Static, Prev: Import, Up: Programming + +5.15 Static +=========== + +Static qualifiers allocate the memory address of a variable in a higher +enclosing level. + + For a function body, the variable is allocated in the block where the +function is defined; so in the code +struct s { + int count() { + static int c=0; + ++c; + return c; + } +} + +there is one instance of the variable `c' for each object `s' (as +opposed to each call of `count'). + + Similarly, in +int factorial(int n) { + int helper(int k) { + static int x=1; + x *= k; + return k == 1 ? x : helper(k-1); + } + return helper(n); +} + +there is one instance of `x' for every call to `factorial' (and not for +every call to `helper'), so this is a correct, but ugly, implementation +of factorial. + + Similarly, a static variable declared within a structure is +allocated in the block where the structure is defined. Thus, +struct A { + struct B { + static pair z; + } +} + +creates one object `z' for each object of type `A' created. + + In this example, +int pow(int n, int k) { + struct A { + static int x=1; + void helper() { + x *= n; + } + } + for(int i=0; i < k; ++i) { + A a; + a.helper(); + } + return A.x; +} + +there is one instance of `x' for each call to `pow', so this is an ugly +implementation of exponentiation. + + Loop constructs allocate a new frame in every iteration. This is so +that higher-order functions can refer to variables of a specific +iteration of a loop: + +void f(); +for(int i=0; i < 10; ++i) { + int x=i; + if(x==5) { + f=new void () { write(x); } + } +} +f(); + + Here, every iteration of the loop has its own variable `x', so `f()' +will write `5'. If a variable in a loop is declared static, it will be +allocated where the enclosing function or structure was defined (just +as if it were declared static outside of the loop). For instance, in: + +void f() { + static int x; + for(int i=0; i < 10; ++i) { + static int y; + } +} + both `x' and `y' will be allocated in the same place, which is also +where `f' is also allocated. + + Statements may also be declared static, in which case they are run +at the place where the enclosing function or structure is defined. +Declarations or statements not enclosed in a function or structure +definition are already at the top level, so static modifiers are +meaningless. A warning is given in such a case. + + Since structures can have static fields, it is not always clear for +a qualified name whether the qualifier is a variable or a type. For +instance, in: + +struct A { + static int x; +} +pair A; + +int y=A.x; + does the `A' in `A.x' refer to the structure or to the pair variable. +It is the convention in Asymptote that, if there is a non-function +variable with the same name as the qualifier, the qualifier refers to +that variable, and not to the type. This is regardless of what fields +the variable actually possesses. + + +File: asymptote.info, Node: LaTeX usage, Next: Base modules, Prev: Programming, Up: Top + +6 `LaTeX' usage +*************** + +`Asymptote' comes with a convenient `LaTeX' style file `asymptote.sty' +that makes `LaTeX' `Asymptote'-aware. Entering `Asymptote' code +directly into the `LaTeX' source file, at the point where it is needed, +keeps figures organized and avoids the need to invent new file names +for each figure. Simply add the line `\usepackage{asymptote}' at the +beginning of your file and enclose your `Asymptote' code within a +`\begin{asy}...\end{asy}' environment. As with the `LaTeX' `comment' +environment, the `\end{asy}' command must appear on a line by itself, +with no leading spaces or trailing commands/comments. + + The sample `LaTeX' file below, named `latexusage.tex', can be run as +follows: +latex latexusage +asy latexusage +latex latexusage + +or + +pdflatex latexusage +asy latexusage +pdflatex latexusage + + To switch between using `latex' and `pdflatex' you may first need to +remove the files `latexusage-*', `latexusage_.pre', and +`latexusage.aux'. + + One can specify `width', `height', `viewportwidth', +`viewportheight', and `attach' `keyval'-style options to the `asy' +environment. The current version (1.06) of `asymptote.sty' supports the +embedding of 3D PRC files, either inline or, using the `attach' option +with the `attachfile2' (or older `attachfile') `LaTeX' package, as +annotated (but printable) attachments. For many applications, the +annotated attachment method tends to be more convenient. The default +value of `viewportwidth' is `\the\linewidth' for inline 3D figures and +`0' for attachments. + + If the `inline' option is given to the `asymptote.sty' package, +inline `LaTeX' code is generated instead of EPS or PDF files. This +makes LaTeX symbols visible to the `\begin{asy}...\end{asy}' +environment. In this mode, Asymptote correctly aligns LaTeX symbols +defined outside of `\begin{asy}...\end{asy}', but treats their size as +zero; an optional second string can be given to `Label' to provide an +estimate of the unknown label size. + + Note that if `latex' is used with the `inline' option, the labels +might not show up in DVI viewers that cannot handle raw `PostScript' +code. One can use `dvips'/`dvipdf' to produce `PostScript'/PDF output +(we recommend using the modified version of `dvipdf' in the `Asymptote' +patches directory, which accepts the `dvips -z' hyperdvi option). + + An excellent tutorial by Dario Teixeira on integrating `Asymptote' +and `LaTeX' is available at `http://dario.dse.nl/projects/asylatex/'. + + Here now is `latexusage.tex': + +\documentclass[12pt]{article} + +% Use this form to include EPS (latex) or PDF (pdflatex) files: +\usepackage{asymptote} + +% Use this form with latex or pdflatex to include inline LaTeX code: +%\usepackage[inline]{asymptote} + +% Enable this line to support PDF hyperlinks: +%\usepackage[setpagesize=false]{hyperref} + +% Enable this line for PDF attachments with asy environment option attach=true: +%\usepackage[dvips]{attachfile2} + +\begin{document} + +\begin{asydef} +// Global Asymptote definitions can be put here. +usepackage("bm"); +texpreamble("\def\V#1{\bm{#1}}"); +// One can globally override the default toolbar settings here: +// settings.toolbar=true; +\end{asydef} + +Here is a venn diagram produced with Asymptote, drawn to width 4cm: + +\def\A{A} +\def\B{\V{B}} + +%\begin{figure} +\begin{center} +\begin{asy} +size(4cm,0); +pen colour1=red; +pen colour2=green; + +pair z0=(0,0); +pair z1=(-1,0); +pair z2=(1,0); +real r=1.5; +path c1=circle(z1,r); +path c2=circle(z2,r); +fill(c1,colour1); +fill(c2,colour2); + +picture intersection=new picture; +fill(intersection,c1,colour1+colour2); +clip(intersection,c2); + +add(intersection); + +draw(c1); +draw(c2); + +//draw("$\A$",box,z1); // Requires [inline] package option. +//draw(Label("$\B$","$B$"),box,z2); // Requires [inline] package option. +draw("$A$",box,z1); +draw("$\V{B}$",box,z2); + +pair z=(0,-2); +real m=3; +margin BigMargin=Margin(0,m*dot(unit(z1-z),unit(z0-z))); + +draw(Label("$A\cap B$",0),conj(z)--z0,Arrow,BigMargin); +draw(Label("$A\cup B$",0),z--z0,Arrow,BigMargin); +draw(z--z1,Arrow,Margin(0,m)); +draw(z--z2,Arrow,Margin(0,m)); + +shipout(bbox(0.25cm)); +\end{asy} +%\caption{Venn diagram}\label{venn} +\end{center} +%\end{figure} + +Each graph is drawn in its own environment. One can specify the width +and height to \LaTeX\ explicitly. This 3D example can be viewed +interactively either with Adobe Reader or Asymptote's fast OpenGL-based +renderer. It is often desirable to embed such files as annotated attachments; +this requires the optional \verb+\usepackage{attachfile2}+ package and +the \verb+{attach=true}+ option: +\begin{center} +\begin{asy}[height=4cm,attach=false] +import three; +currentprojection=orthographic(5,4,2); +draw(unitcube,blue); +label("$V-E+F=2$",(0,1,0.5),3Y,blue+fontsize(17)); +\end{asy} +\end{center} + +One can also scale the figure to the full line width: +\begin{center} +\begin{asy}[width=\the\linewidth] +pair z0=(0,0); +pair z1=(2,0); +pair z2=(5,0); +pair zf=z1+0.75*(z2-z1); + +draw(z1--z2); +dot(z1,red+0.15cm); +dot(z2,darkgreen+0.3cm); +label("$m$",z1,1.2N,red); +label("$M$",z2,1.5N,darkgreen); +label("$\hat{\ }$",zf,0.2*S,fontsize(24)+blue); + +pair s=-0.2*I; +draw("$x$",z0+s--z1+s,N,red,Arrows,Bars,PenMargins); +s=-0.5*I; +draw("$\bar{x}$",z0+s--zf+s,blue,Arrows,Bars,PenMargins); +s=-0.95*I; +draw("$X$",z0+s--z2+s,darkgreen,Arrows,Bars,PenMargins); +\end{asy} +\end{center} +\end{document} + + + +File: asymptote.info, Node: Base modules, Next: Options, Prev: LaTeX usage, Up: Top + +7 Base modules +************** + +`Asymptote' currently ships with the following base modules: + +* Menu: + +* plain:: Default `Asymptote' base file +* simplex:: Linear programming: simplex method +* math:: Extend `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 +* 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 PDF and MPEG movies +* embed:: Embedding movies, sounds, and 3D objects +* slide:: Making presentations with `Asymptote' +* MetaPost:: `MetaPost' compatibility routines +* unicode:: Accept `unicode' (UTF-8) characters +* latin1:: Accept `ISO 8859-1' characters +* babel:: Interface to `LaTeX' `babel' package +* labelpath:: Drawing curved labels +* labelpath3:: Drawing curved labels in 3D +* annotate:: Annotate your 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 +* slopefield:: Slope fields + + +File: asymptote.info, Node: plain, Next: simplex, Up: Base modules + +7.1 `plain' +=========== + +This is the default `Asymptote' base file, which defines key parts of +the drawing language (such as the `picture' structure). + + By default, an implicit `private import plain;' occurs before +translating a file and before the first command given in interactive +mode. This also applies when translating files for module definitions +(except when translating `plain', of course). This means that the +types and functions defined in `plain' are accessible in almost all +`Asymptote' code. Use the `-noautoplain' command-line option to disable +this feature. + + +File: asymptote.info, Node: simplex, Next: math, Prev: plain, Up: Base modules + +7.2 `simplex' +============= + +This package solves the two-variable linear programming problem using +the simplex method. It is used by the module `plain' for automatic +sizing of pictures. + + +File: asymptote.info, Node: math, Next: interpolate, Prev: simplex, Up: Base modules + +7.3 `math' +========== + +This package extends `Asymptote''s mathematical capabilities with +intersection algorithms and matrix arithmetic: + +`void drawline(picture pic=currentpicture, pair P, pair Q, pen p=currentpen);' + draw the visible portion of the (infinite) line going through `P' + and `Q', without altering the size of picture `pic', using pen `p'. + +`real intersect(triple P, triple Q, triple n, triple Z);' + returns the intersection time of the extension of the line segment + `PQ' with the plane perpendicular to `n' and passing through `Z'. + +`triple intersectionpoint(triple n0, triple P0, triple n1, triple P1);' + Return any point on the intersection of the two planes with normals + `n0' and `n1' passing through points `P0' and `P1', respectively. + If the planes are parallel, return `(infinity,infinity,infinity)'. + +``pair[] quarticroots(real a, real b, real c, real d, real e);'' + returns the four complex roots of the quartic equation + ax^4+bx^3+cx^2+dx+e=0. + + + +File: asymptote.info, Node: interpolate, Next: geometry, Prev: math, Up: Base modules + +7.4 `interpolate' +================= + +This module implements Lagrange, Hermite, and standard cubic spline +interpolation in `Asymptote', as illustrated in the example +`interpolate1.asy'. + + +File: asymptote.info, Node: geometry, Next: trembling, Prev: interpolate, Up: Base modules + +7.5 `geometry' +============== + +This module, written by Philippe Ivaldi, provides an extensive set of +geometry routines, including `perpendicular' symbols and a `triangle' +structure. It is documented here: +`http://asymptote.sourceforge.net/geometry_en.pdf', including an +extensive set of examples, +`http://piprim.tuxfamily.org/asymptote/geometry/index.html', and an +index: + + `http://piprim.tuxfamily.org/asymptote/geometry/modules/geometry.asy.index.type.html' + + +File: asymptote.info, Node: trembling, Next: stats, Prev: geometry, Up: Base modules + +7.6 `trembling' +=============== + +This module, written by Philippe Ivaldi and illustrated in the example +`floatingdisk.asy', allows one to draw wavy lines, as if drawn by hand. +Further examples are posted at +`http://piprim.tuxfamily.org/asymptote/trembling/index.html' + + +File: asymptote.info, Node: stats, Next: patterns, Prev: trembling, Up: Base modules + +7.7 `stats' +=========== + +This package implements a Gaussian random number generator and a +collection of statistics routines, including `histogram' and +`leastsquares'. + + +File: asymptote.info, Node: patterns, Next: markers, Prev: stats, Up: Base modules + +7.8 `patterns' +============== + +This package implements `Postscript' tiling patterns and includes +several convenient pattern generation routines. + + +File: asymptote.info, Node: markers, Next: tree, Prev: patterns, Up: Base modules + +7.9 `markers' +============= + +This package implements specialized routines for marking paths and +angles. The principal mark routine provided by this package is +markroutine markinterval(int n=1, frame f, bool rotated=false); + which centers `n' copies of frame `f' within uniformly space intervals +in arclength along the path, optionally rotated by the angle of the +local tangent. + + The `marker' (*note marker::) routine can be used to construct new +markers from these predefined frames: + +frame stickframe(int n=1, real size=0, pair space=0, real angle=0, + pair offset=0, pen p=currentpen); + +frame circlebarframe(int n=1, real barsize=0, + real radius=0,real angle=0, + pair offset=0, pen p=currentpen, + filltype filltype=NoFill, bool above=false); + +frame crossframe(int n=3, real size=0, pair space=0, + real angle=0, pair offset=0, pen p=currentpen); + +frame tildeframe(int n=1, real size=0, pair space=0, + real angle=0, pair offset=0, pen p=currentpen); + + For convenience, this module also constructs the markers +`StickIntervalMarker', `CrossIntervalMarker', +`CircleBarIntervalMarker', and `TildeIntervalMarker' from the above +frames. The example `markers1.asy' illustrates the use of these markers: + + + + +This package also provides a routine for marking an angle AOB: +void markangle(picture pic=currentpicture, Label L="", + int n=1, real radius=0, real space=0, + pair A, pair O, pair B, arrowbar arrow=None, + pen p=currentpen, margin margin=NoMargin, + marker marker=nomarker); + as illustrated in the example `markers2.asy'. + + + + + +File: asymptote.info, Node: tree, Next: binarytree, Prev: markers, Up: Base modules + +7.10 `tree' +=========== + +This package implements an example of a dynamic binary search tree. + + +File: asymptote.info, Node: binarytree, Next: drawtree, Prev: tree, Up: Base modules + +7.11 `binarytree' +================= + +This module can be used to draw an arbitrary binary tree and includes an +input routine for the special case of a binary search tree, as +illustrated in the example `binarytreetest.asy': + +import binarytree; + +picture pic,pic2; + +binarytree bt=binarytree(1,2,4,nil,5,nil,nil,0,nil,nil,3,6,nil,nil,7); +draw(pic,bt); + +binarytree st=searchtree(10,5,2,1,3,4,7,6,8,9,15,13,12,11,14,17,16,18,19); +draw(pic2,st,blue); + +add(pic.fit(),(0,0),10N); +add(pic2.fit(),(0,0),10S); + + + + +File: asymptote.info, Node: drawtree, Next: syzygy, Prev: binarytree, Up: Base modules + +7.12 `drawtree' +=============== + +This is a simple tree drawing module used by the example `treetest.asy'. + + +File: asymptote.info, Node: syzygy, Next: feynman, Prev: drawtree, Up: Base modules + +7.13 `syzygy' +============= + +This module automates the drawing of braids, relations, and syzygies, +along with the corresponding equations, as illustrated in the example +`knots.asy'. + + +File: asymptote.info, Node: feynman, Next: roundedpath, Prev: syzygy, Up: Base modules + +7.14 `feynman' +============== + +This package, contributed by Martin Wiebusch, is useful for drawing +Feynman diagrams, as illustrated by the examples `eetomumu.asy' and +`fermi.asy'. + + +File: asymptote.info, Node: roundedpath, Next: animation, Prev: feynman, Up: Base modules + +7.15 `roundedpath' +================== + +This package, contributed by Stefan Knorr, is useful for rounding the +sharp corners of paths, as illustrated in the example file +`roundpath.asy'. + + +File: asymptote.info, Node: animation, Next: embed, Prev: roundedpath, Up: Base modules + +7.16 `animation' +================ + +This module allows one to generate animations, as illustrated by the +files `wheel.asy', `wavelet.asy', and `cube.asy' in the `animations' +subdirectory of the examples directory. These animations use the +`ImageMagick' `convert' program to `merge' multiple images into a GIF +or MPEG movie. + + The related `animate' module, derived from the `animation' module, +generates higher-quality portable clickable PDF movies, with optional +controls. This requires installing the package + + `http://www.ctan.org/tex-archive/macros/latex/contrib/animate/animate.sty' + (version 2007/11/30 or later) in a new directory `animate' in the +local `LaTeX' directory (for example, in +`/usr/local/share/texmf/tex/latex/animate'). On `UNIX' systems, one +must then execute the command `texhash'. + + The example `pdfmovie.asy' in the `animations' directory, along with +the slide presentations `slidemovies.asy' and `intro.asy', illustrate +the use of embedded PDF movies. The examples `inlinemovie.tex' and +`inlinemovie3.tex' show how to generate and embed PDF movies directly +within a `LaTeX' file (*note LaTeX usage::). The member function +string pdf(fit fit=NoBox, real delay=animationdelay, string options="", + bool keep=settings.keep, bool multipage=true); + of the `animate' structure accepts any of the `animate.sty' options, +as described here: + + `http://www.ctan.org/tex-archive/macros/latex/contrib/animate/doc/animate.pdf' + + +File: asymptote.info, Node: embed, Next: slide, Prev: animation, Up: Base modules + +7.17 `embed' +============ + +This module provides an interface to the `LaTeX' package (included with +`MikTeX') + + `http://www.ctan.org/tex-archive/macros/latex/contrib/movie15' + for embedding movies, sounds, and 3D objects into a PDF document. However, +`XeLaTeX' users need to rename the modified version +`movie15_dvipdfmx.sty' from + + `http://asymptote.svn.sourceforge.net/viewvc/asymptote/trunk/asymptote/patches/' + to `movie15.sty' and place it in their `LaTeX' path. + + The latest version (2008/10/08) of the `movie15' package requires +both `pdflatex' version 1.20 or later and the file + + `http://www.ctan.org/tex-archive/macros/latex/contrib/oberdiek/ifdraft.dtx' + which can be installed by placing it in a directory `ifdraft' in the +local `LaTeX' directory (e.g. +`/usr/local/share/texmf/tex/latex/ifdraft') and executing in that +directory the commands: +tex ifdraft.dtx +texhash + + An example of embedding `U3D' code is provided in the file +`embeddedu3d.asy'. As of version 7.0.8, `Adobe Reader' supports the +`U3D' format under Linux. + + Unfortunately, Adobe has not yet made available an embedded movie +plugin for the Linux version of `Adobe Reader'. A portable method for +embedding movie files, which should work on any platform and does not +require the `movie15' or `ifdraft' packages, is provided by using the +`external' module instead of `embed'. An example of these interfaces is +provided in the file `embeddedmovie.asy' and `externalmovie.asy' in the +`animations' subdirectory of the examples directory. For a higher +quality movie generated directly by `Asymptote', use the `animate' +module along with the `animate.sty' package to embed a portable PDF +animation (*note animate::). + + +File: asymptote.info, Node: slide, Next: MetaPost, Prev: embed, Up: Base modules + +7.18 `slide' +============ + +This package provides a simple yet high-quality facility for making +presentation slides, including portable embedded PDF animations (see +the file `slidemovies.asy'). A simple example is provided in the file +`slidedemo.asy'. + + +File: asymptote.info, Node: MetaPost, Next: unicode, Prev: slide, Up: Base modules + +7.19 `MetaPost' +=============== + +This package provides some useful routines to help `MetaPost' users +migrate old `MetaPost' code to `Asymptote'. Further contributions here +are welcome. + + Unlike `MetaPost', `Asymptote' does not implicitly solve linear +equations and therefore does not have the notion of a `whatever' +unknown. The routine `extension' (*note extension::) provides a useful +replacement for a common use of `whatever': finding the intersection +point of the lines through `P', `Q' and `p', `q'. For less common +occurrences of `whatever', one can use the built-in explicit linear +equation solver `solve' instead. + + +File: asymptote.info, Node: unicode, Next: latin1, Prev: MetaPost, Up: Base modules + +7.20 `unicode' +============== + +Import this package at the beginning of the file to instruct `LaTeX' to +accept `unicode' (UTF-8) standardized international characters. To use +Cyrillic fonts, you will need to change the font encoding: +import unicode; +texpreamble("\usepackage{mathtext}\usepackage[russian]{babel}"); +defaultpen(font("T2A","cmr","m","n")); + Support for Chinese, Japanese, and Korean fonts is provided by the CJK +package: + + `http://www.ctan.org/tex-archive/languages/chinese/CJK/' + The following commands enable the CJK song family (within a label, +you can also temporarily switch to another family, say kai, by +prepending `"\CJKfamily{kai}"' to the label string): +texpreamble("\usepackage{CJK} +\AtBeginDocument{\begin{CJK*}{GBK}{song}} +\AtEndDocument{\clearpage\end{CJK*}}"); + + +File: asymptote.info, Node: latin1, Next: babel, Prev: unicode, Up: Base modules + +7.21 `latin1' +============= + +If you don't have `LaTeX' support for `unicode' installed, you can +enable support for Western European languages (ISO 8859-1) by importing +the module `latin1'. This module can be used as a template for +providing support for other ISO 8859 alphabets. + + +File: asymptote.info, Node: babel, Next: labelpath, Prev: latin1, Up: Base modules + +7.22 `babel' +============ + +This module implements the `LaTeX' `babel' package in `Asymptote'. For +example: +import babel; +babel("german"); + + +File: asymptote.info, Node: labelpath, Next: labelpath3, Prev: babel, Up: Base modules + +7.23 `labelpath' +================ + +This module uses the `PSTricks' `pstextpath' macro to fit labels along +a path (properly kerned, as illustrated in the example file +`curvedlabel.asy'), using the command +void labelpath(picture pic=currentpicture, Label L, path g, + string justify=Centered, pen p=currentpen); + Here `justify' is one of `LeftJustified', `Centered', or +`RightJustified'. The x component of a shift transform applied to the +Label is interpreted as a shift along the curve, whereas the y +component is interpreted as a shift away from the curve. All other +Label transforms are ignored. This package requires the `latex' tex +engine and inherits the limitations of the `PSTricks' `\pstextpath' +macro. + + +File: asymptote.info, Node: labelpath3, Next: annotate, Prev: labelpath, Up: Base modules + +7.24 `labelpath3' +================= + +This module, contributed by Jens Schwaiger, implements a 3D version of +`labelpath' that does not require the `PSTricks' package. An example +is provided in `curvedlabel3.asy'. + + +File: asymptote.info, Node: annotate, Next: CAD, Prev: labelpath3, Up: Base modules + +7.25 `annotate' +=============== + +This module supports PDF annotations for viewing with `Adobe Reader', +via the function +void annotate(picture pic=currentpicture, string title, string text, + pair position); + Annotations are illustrated in the example file `annotation.asy'. +Currently, annotations are only implemented for the `latex' (default) +and `tex' TeX engines. + + +File: asymptote.info, Node: CAD, Next: graph, Prev: annotate, Up: Base modules + +7.26 `CAD' +========== + +This package, contributed by Mark Henning, provides basic pen +definitions and measurement functions for simple 2D CAD drawings +according to DIN 15. It is documented separately, in the file `CAD.pdf'. + + +File: asymptote.info, Node: graph, Next: palette, Prev: CAD, Up: Base modules + +7.27 `graph' +============ + +This package implements two-dimensional linear and logarithmic graphs, +including automatic scale and tick selection (with the ability to +override manually). A graph is a `guide' (that can be drawn with the +draw command, with an optional legend) constructed with one of the +following routines: + + * guide graph(picture pic=currentpicture, real f(real), real a, real b, + int n=ngraph, real T(real)=identity, + interpolate join=operator --); + guide[] graph(picture pic=currentpicture, real f(real), real a, real b, + int n=ngraph, real T(real)=identity, bool3 cond(real), + interpolate join=operator --); + + Returns a graph using the scaling information for picture `pic' + (*note automatic scaling::) of the function `f' on the interval + [`T'(`a'),`T'(`b')], sampling at `n' points evenly spaced in + [`a',`b'], optionally restricted by the bool3 function `cond' on + [`a',`b']. If `cond' is: + * `true', the point is added to the existing guide; + + * `default', the point is added to a new guide; + + * `false', the point is omitted and a new guide is begun. + Th points are connected using the interpolation specified by + `join': + * `operator --' (linear interpolation; the abbreviation + `Straight' is also accepted); + + * `operator ..' (piecewise Bezier cubic spline interpolation; + the abbreviation `Spline' is also accepted); + + * `Hermite' (standard cubic spline interpolation using boundary + condition `notaknot', `natural', `periodic', `clamped(real + slopea, real slopeb)'), or `monotonic'. The abbreviation + `Hermite' is equivalent to `Hermite(notaknot)' for + nonperiodic data and `Hermite(periodic)' for periodic data). + + + * guide graph(picture pic=currentpicture, real x(real), real y(real), + real a, real b, int n=ngraph, real T(real)=identity, + interpolate join=operator --); + guide[] graph(picture pic=currentpicture, real x(real), real y(real), + real a, real b, int n=ngraph, real T(real)=identity, + bool3 cond(real), interpolate join=operator --); + + Returns a graph using the scaling information for picture `pic' of + the parametrized function (`x'(t),`y'(t)) for t in the interval + [`T'(`a'),`T'(`b')], sampling at `n' points evenly spaced in + [`a',`b'], optionally restricted by the bool3 function `cond' on + [`a',`b'], using the given interpolation type. + + * guide graph(picture pic=currentpicture, pair z(real), real a, real b, + int n=ngraph, real T(real)=identity, + interpolate join=operator --); + guide[] graph(picture pic=currentpicture, pair z(real), real a, real b, + int n=ngraph, real T(real)=identity, bool3 cond(real), + interpolate join=operator --); + + Returns a graph using the scaling information for picture `pic' of + the parametrized function `z'(t) for t in the interval + [`T'(`a'),`T'(`b')], sampling at `n' points evenly spaced in + [`a',`b'], optionally restricted by the bool3 function `cond' on + [`a',`b'], using the given interpolation type. + + * guide graph(picture pic=currentpicture, pair[] z, + interpolate join=operator --); + guide[] graph(picture pic=currentpicture, pair[] z, bool3[] cond, + interpolate join=operator --); + + Returns a graph using the scaling information for picture `pic' of + the elements of the array `z', optionally restricted to those + indices for which the elements of the boolean array `cond' are + `true', using the given interpolation type. + + * guide graph(picture pic=currentpicture, real[] x, real[] y, + interpolate join=operator --); + guide[] graph(picture pic=currentpicture, real[] x, real[] y, + bool3[] cond, interpolate join=operator --); + + Returns a graph using the scaling information for picture `pic' of + the elements of the arrays (`x',`y'), optionally restricted to + those indices for which the elements of the boolean array `cond' + are `true', using the given interpolation type. + + * guide polargraph(picture pic=currentpicture, real f(real), real a, + real b, int n=ngraph, interpolate join=operator --); + + Returns a polar-coordinate graph using the scaling information for + picture `pic' of the function `f' on the interval [`a',`b'], + sampling at `n' evenly spaced points, with the given interpolation + type. + + + + + An axis can be drawn on a picture with one of the following commands: + + * void xaxis(picture pic=currentpicture, Label L="", axis axis=YZero, + real xmin=-infinity, real xmax=infinity, pen p=currentpen, + ticks ticks=NoTicks, arrowbar arrow=None, bool above=false); + + Draw an x axis on picture `pic' from x=`xmin' to x=`xmax' using + pen `p', optionally labelling it with Label `L'. The relative + label location along the axis (a real number from [0,1]) defaults + to 1 (*note Label::), so that the label is drawn at the end of the + axis. An infinite value of `xmin' or `xmax' specifies that the + corresponding axis limit will be automatically determined from the + picture limits. The optional `arrow' argument takes the same + values as in the `draw' command (*note arrows::). The axis is + drawn before any existing objects in the current picture unless + `above=true'. The axis placement is determined by one of the + following `axis' types: + + `YZero(bool extend=true)' + Request an x axis at y=0 (or y=1 on a logarithmic axis) + extending to the full dimensions of the picture, unless + `extend'=false. + + `YEquals(real Y, bool extend=true)' + Request an x axis at y=`Y' extending to the full dimensions + of the picture, unless `extend'=false. + + `Bottom(bool extend=false)' + Request a bottom axis. + + `Top(bool extend=false)' + Request a top axis. + + `BottomTop(bool extend=false)' + Request a bottom and top axis. + + + Custom axis types can be created by following the examples in + `graph.asy'. One can easily override the default values for the + standard axis types: import graph; + + YZero=new axis(bool extend=true) { + return new void(picture pic, axisT axis) { + real y=pic.scale.x.scale.logarithmic ? 1 : 0; + axis.value=I*pic.scale.y.T(y); + axis.position=1; + axis.side=right; + axis.align=2.5E; + axis.value2=Infinity; + axis.extend=extend; + }; + }; + YZero=YZero(); + + The default tick option is `NoTicks'. The options `LeftTicks', + `RightTicks', or `Ticks' can be used to draw ticks on the left, + right, or both sides of the path, relative to the direction in + which the path is drawn. These tick routines accept a number of + optional arguments: ticks LeftTicks(Label format="", ticklabel ticklabel=null, + bool beginlabel=true, bool endlabel=true, + int N=0, int n=0, real Step=0, real step=0, + bool begin=true, bool end=true, tickmodifier modify=None, + real Size=0, real size=0, bool extend=false, + pen pTick=nullpen, pen ptick=nullpen); + + If any of these parameters are omitted, reasonable defaults will + be chosen: + `Label format' + override the default tick label format (`defaultformat', + initially "$%.4g$"), rotation, pen, and alignment (for + example, `LeftSide', `Center', or `RightSide') relative to + the axis. To enable `LaTeX' math mode fonts, the format + string should begin and end with `$' *note format::. If the + format string is `trailingzero', trailing zeros will be added + to the tick labels; if the format string is `"%"', the tick + label will be suppressed; + + `ticklabel' + is a function `string(real x)' returning the label (by + default, format(format.s,x)) for each major tick value `x'; + + `bool beginlabel' + include the first label; + + `bool endlabel' + include the last label; + + `int N' + when automatic scaling is enabled (the default; *note + automatic scaling::), divide a linear axis evenly into this + many intervals, separated by major ticks; for a logarithmic + axis, this is the number of decades between labelled ticks; + + `int n' + divide each interval into this many subintervals, separated + by minor ticks; + + `real Step' + the tick value spacing between major ticks (if `N'=`0'); + + `real step' + the tick value spacing between minor ticks (if `n'=`0'); + + `bool begin' + include the first major tick; + + `bool end' + include the last major tick; + + `tickmodifier modify;' + an optional function that takes and returns a `tickvalue' + structure having real[] members `major' and `minor' + consisting of the tick values (to allow modification of the + automatically generated tick values); + + `real Size' + the size of the major ticks (in `PostScript' coordinates); + + `real size' + the size of the minor ticks (in `PostScript' coordinates); + + `bool extend;' + extend the ticks between two axes (useful for drawing a grid + on the graph); + + `pen pTick' + an optional pen used to draw the major ticks; + + `pen ptick' + an optional pen used to draw the minor ticks. + + + For convenience, the predefined tickmodifier `OmitTick(... real[] + x)' tickmodifier can be used to remove specific auto-generated + ticks and their labels. The `OmitFormat(string s=defaultformat ... + real[] x)' ticklabel can be used to remove specific tick labels + but not the corresponding ticks. The tickmodifier `NoZero' is an + abbreviation for `OmitTick(0)' and the ticklabel `NoZeroFormat' is + an abbrevation for `OmitFormat(0)'. + + It is also possible to specify custom tick locations with + `LeftTicks', `RightTicks', and `Ticks' by passing explicit real + arrays `Ticks' and (optionally) `ticks' containing the locations + of the major and minor ticks, respectively: ticks LeftTicks(Label format="", ticklabel ticklabel=null, + bool beginlabel=true, bool endlabel=true, + real[] Ticks, real[] ticks=new real[], + real Size=0, real size=0, bool extend=false, + pen pTick=nullpen, pen ptick=nullpen) + + * void yaxis(picture pic=currentpicture, Label L="", axis axis=XZero, + real ymin=-infinity, real ymax=infinity, pen p=currentpen, + ticks ticks=NoTicks, arrowbar arrow=None, bool above=false); + + Draw a y axis on picture `pic' from y=`ymin' to y=`ymax' using pen + `p', optionally labelling it with Label `L'. The relative location + of the label (a real number from [0,1]) defaults to 1 (*note + Label::). An infinite value of `ymin' or `ymax' specifies that the + corresponding axis limit will be automatically determined from the + picture limits. The optional `arrow' argument takes the same + values as in the `draw' command (*note arrows::). The axis is + drawn before any existing objects in the current picture unless + `above=true'. The tick type is specified by `ticks' and the axis + placement is determined by one of the following `axis' types: + + `XZero(bool extend=true)' + Request a y axis at x=0 (or x=1 on a logarithmic axis) + extending to the full dimensions of the picture, unless + `extend'=false. + + `XEquals(real X, bool extend=true)' + Request a y axis at x=`X' extending to the full dimensions of + the picture, unless `extend'=false. + + `Left(bool extend=false)' + Request a left axis. + + `Right(bool extend=false)' + Request a right axis. + + `LeftRight(bool extend=false)' + Request a left and right axis. + + + * For convenience, the functions void xequals(picture pic=currentpicture, Label L="", real x, + bool extend=false, real ymin=-infinity, real ymax=infinity, + pen p=currentpen, ticks ticks=NoTicks, bool above=true, + arrowbar arrow=None); + and void yequals(picture pic=currentpicture, Label L="", real y, + bool extend=false, real xmin=-infinity, real xmax=infinity, + pen p=currentpen, ticks ticks=NoTicks, bool above=true, + arrowbar arrow=None); + can be respectively used to call `yaxis' and `xaxis' with the + appropriate axis types `XEquals(x,extend)' and + `YEquals(y,extend)'. This is the recommended way of drawing + vertical or horizontal lines and axes at arbitrary locations. + + * void axes(picture pic=currentpicture, Label xlabel="", Label ylabel="", + pair min=(-infinity,-infinity), pair max=(infinity,infinity), + pen p=currentpen, arrowbar arrow=None, bool above=false); + This convenience routine draws both x and y axes on picture `pic' + from `min' to `max', with optional labels `xlabel' and `ylabel' + and any arrows specified by `arrow'. The axes are drawn on top of + existing objects in the current picture only if `above=true'. + + * void axis(picture pic=currentpicture, Label L="", path g, + pen p=currentpen, ticks ticks, ticklocate locate, + arrowbar arrow=None, int[] divisor=new int[], + bool above=false, bool opposite=false); + + This routine can be used to draw on picture `pic' a general axis + based on an arbitrary path `g', using pen `p'. One can optionally + label the axis with Label `L' and add an arrow `arrow'. The tick + type is given by `ticks'. The optional integer array `divisor' + specifies what tick divisors to try in the attempt to produce + uncrowded tick labels. A `true' value for the flag `opposite' + identifies an unlabelled secondary axis (typically drawn opposite + a primary axis). The axis is drawn before any existing objects in + the current picture unless `above=true'. The tick locator + `ticklocate' is constructed by the routine ticklocate ticklocate(real a, real b, autoscaleT S=defaultS, + real tickmin=-infinity, real tickmax=infinity, + real time(real)=null, pair dir(real)=zero); + where `a' and `b' specify the respective tick values at + `point(g,0)' and `point(g,length(g))', `S' specifies the + autoscaling transformation, the function `real time(real v)' + returns the time corresponding to the value `v', and `pair + dir(real t)' returns the absolute tick direction as a function of + `t' (zero means draw the tick perpendicular to the axis). + + * These routines are useful for manually putting ticks and labels on + axes (if the variable `Label' is given as the `Label' argument, + the `format' argument will be used to format a string based on the + tick location): void xtick(picture pic=currentpicture, Label L="", explicit pair z, + pair dir=N, string format="", + real size=Ticksize, pen p=currentpen); + void xtick(picture pic=currentpicture, Label L="", real x, + pair dir=N, string format="", + real size=Ticksize, pen p=currentpen); + void ytick(picture pic=currentpicture, Label L="", explicit pair z, + pair dir=E, string format="", + real size=Ticksize, pen p=currentpen); + void ytick(picture pic=currentpicture, Label L="", real y, + pair dir=E, string format="", + real size=Ticksize, pen p=currentpen); + void tick(picture pic=currentpicture, pair z, + pair dir, real size=Ticksize, pen p=currentpen); + void labelx(picture pic=currentpicture, Label L="", explicit pair z, + align align=S, string format="", pen p=nullpen); + void labelx(picture pic=currentpicture, Label L="", real x, + align align=S, string format="", pen p=nullpen); + 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=nullpen); + void labely(picture pic=currentpicture, Label L="", real y, + align align=W, string format="", pen p=nullpen); + void labely(picture pic=currentpicture, Label L, + string format="", explicit pen p=nullpen); + + Here are some simple examples of two-dimensional graphs: + + 1. This example draws a textbook-style graph of y= exp(x), with the y + axis starting at y=0: import graph; + size(150,0); + + real f(real x) {return exp(x);} + pair F(real x) {return (x,f(x));} + + xaxis("$x$"); + yaxis("$y$",0); + + draw(graph(f,-4,2,operator ..),red); + + labely(1,E); + label("$e^x$",F(1),SE); + + + + 2. The next example draws a scientific-style graph with a legend. + The position of the legend can be adjusted either explicitly or by + using the graphical user interface `xasy' (*note GUI::). If an + `UnFill(real xmargin=0, real ymargin=xmargin)' or `Fill(pen)' + option is specified to `add', the legend will obscure any + underlying objects. Here we illustrate how to clip the portion of + the picture covered by a label: + + import graph; + + size(400,200,IgnoreAspect); + + real Sin(real t) {return sin(2pi*t);} + real Cos(real t) {return cos(2pi*t);} + + draw(graph(Sin,0,1),red,"$\sin(2\pi x)$"); + draw(graph(Cos,0,1),blue,"$\cos(2\pi x)$"); + + xaxis("$x$",BottomTop,LeftTicks); + yaxis("$y$",LeftRight,RightTicks(trailingzero)); + + label("LABEL",point(0),UnFill(1mm)); + + add(legend(),point(E),20E,UnFill); + + + + To specify a fixed size for the graph proper, use `attach': import graph; + + size(250,200,IgnoreAspect); + + real Sin(real t) {return sin(2pi*t);} + real Cos(real t) {return cos(2pi*t);} + + draw(graph(Sin,0,1),red,"$\sin(2\pi x)$"); + draw(graph(Cos,0,1),blue,"$\cos(2\pi x)$"); + + xaxis("$x$",BottomTop,LeftTicks); + yaxis("$y$",LeftRight,RightTicks(trailingzero)); + + label("LABEL",point(0),UnFill(1mm)); + + attach(legend(),truepoint(E),20E,UnFill); + A legend can have multiple entries per line: import graph; + size(8cm,6cm,IgnoreAspect); + + typedef real realfcn(real); + realfcn F(real p) { + return new real(real x) {return sin(p*x);}; + }; + + for(int i=1; i < 5; ++i) + draw(graph(F(i*pi),0,1),Pen(i), + "$\sin("+(i == 1 ? "" : (string) i)+"\pi x)$"); + xaxis("$x$",BottomTop,LeftTicks); + yaxis("$y$",LeftRight,RightTicks(trailingzero)); + + attach(legend(2),(point(S).x,truepoint(S).y),10S,UnFill); + + + + 3. This example draws a graph of one array versus another (both of + the same size) using custom tick locations and a smaller font size + for the tick labels on the y axis. import graph; + + size(200,150,IgnoreAspect); + + real[] x={0,1,2,3}; + real[] y=x^2; + + draw(graph(x,y),red); + + xaxis("$x$",BottomTop,LeftTicks); + yaxis("$y$",LeftRight, + RightTicks(Label(fontsize(8)),new real[]{0,4,9})); + + + + 4. This example shows how to graph columns of data read from a file. import graph; + + size(200,150,IgnoreAspect); + + file in=line(input("filegraph.dat")); + real[][] a=dimension(in,0,0); + a=transpose(a); + + real[] x=a[0]; + real[] y=a[1]; + + draw(graph(x,y),red); + + xaxis("$x$",BottomTop,LeftTicks); + yaxis("$y$",LeftRight,RightTicks); + + + + 5. The next example draws two graphs of an array of coordinate pairs, + using frame alignment and data markers. In the left-hand graph, the + markers, constructed with marker marker(path g, markroutine markroutine=marknodes, + pen p=currentpen, filltype filltype=NoFill, + bool above=true); + using the path `unitcircle' (*note filltype::), are drawn below + each node. Any frame can be converted to a marker, using marker marker(frame f, markroutine markroutine=marknodes, + bool above=true); + In the right-hand graph, the unit n-sided regular polygon + `polygon(int n)' and the unit n-point cyclic cross `cross(int n, + bool round=true, real r=0)' (where `r' is an optional "inner" + radius) are used to build a custom marker frame. Here + `markuniform(bool centered=false, int n, bool rotated=false)' adds + this frame at `n' uniformly spaced points along the arclength of + the path, optionally rotated by the angle of the local tangent to + the path (if centered is true, the frames will be centered within + `n' evenly spaced arclength intervals). Alternatively, one can use + markroutine `marknodes' to request that the marks be placed at each + Bezier node of the path, or markroutine `markuniform(pair z(real + t), real a, real b, int n)' to place marks at points `z(t)' for n + evenly spaced values of `t' in `[a,b]'. + + These markers are predefined: marker[] Mark={ + marker(scale(circlescale)*unitcircle), + marker(polygon(3)),marker(polygon(4)), + marker(polygon(5)),marker(invert*polygon(3)), + marker(cross(4)),marker(cross(6)) + }; + + marker[] MarkFill={ + marker(scale(circlescale)*unitcircle,Fill),marker(polygon(3),Fill), + marker(polygon(4),Fill),marker(polygon(5),Fill), + marker(invert*polygon(3),Fill) + }; + + The example also illustrates the `errorbar' routines: + + void errorbars(picture pic=currentpicture, pair[] z, pair[] dp, + pair[] dm={}, bool[] cond={}, pen p=currentpen, + real size=0); + + void errorbars(picture pic=currentpicture, real[] x, real[] y, + real[] dpx, real[] dpy, real[] dmx={}, real[] dmy={}, + bool[] cond={}, pen p=currentpen, real size=0); + + Here, the positive and negative extents of the error are given by + the absolute values of the elements of the pair array `dp' and the + optional pair array `dm'. If `dm' is not specified, the positive + and negative extents of the error are assumed to be equal. import graph; + + picture pic; + real xsize=200, ysize=140; + size(pic,xsize,ysize,IgnoreAspect); + + pair[] f={(5,5),(50,20),(90,90)}; + pair[] df={(0,0),(5,7),(0,5)}; + + errorbars(pic,f,df,red); + draw(pic,graph(pic,f),"legend", + marker(scale(0.8mm)*unitcircle,red,FillDraw(blue),above=false)); + + scale(pic,true); + + xaxis(pic,"$x$",BottomTop,LeftTicks); + yaxis(pic,"$y$",LeftRight,RightTicks); + add(pic,legend(pic),point(pic,NW),20SE,UnFill); + + picture pic2; + size(pic2,xsize,ysize,IgnoreAspect); + + frame mark; + filldraw(mark,scale(0.8mm)*polygon(6),green,green); + draw(mark,scale(0.8mm)*cross(6),blue); + + draw(pic2,graph(pic2,f),marker(mark,markuniform(5))); + + scale(pic2,true); + + xaxis(pic2,"$x$",BottomTop,LeftTicks); + yaxis(pic2,"$y$",LeftRight,RightTicks); + + yequals(pic2,55.0,red+Dotted); + xequals(pic2,70.0,red+Dotted); + + // Fit pic to W of origin: + add(pic.fit(),(0,0),W); + + // Fit pic2 to E of (5mm,0): + add(pic2.fit(),(5mm,0),E); + + + + 6. A custom mark routine can be also be specified: import graph; + + size(200,100,IgnoreAspect); + + markroutine marks() { + return new void(picture pic=currentpicture, frame f, path g) { + path p=scale(1mm)*unitcircle; + for(int i=0; i <= length(g); ++i) { + pair z=point(g,i); + frame f; + if(i % 4 == 0) { + fill(f,p); + add(pic,f,z); + } else { + if(z.y > 50) { + pic.add(new void(frame F, transform t) { + path q=shift(t*z)*p; + unfill(F,q); + draw(F,q); + }); + } else { + draw(f,p); + add(pic,f,z); + } + } + } + }; + } + + pair[] f={(5,5),(40,20),(55,51),(90,30)}; + + draw(graph(f),marker(marks())); + + scale(true); + + xaxis("$x$",BottomTop,LeftTicks); + yaxis("$y$",LeftRight,RightTicks); + + + + 7. This example shows how to label an axis with arbitrary strings. import graph; + + size(400,150,IgnoreAspect); + + real[] x=sequence(12); + real[] y=sin(2pi*x/12); + + scale(false); + + string[] month={"Jan","Feb","Mar","Apr","May","Jun", + "Jul","Aug","Sep","Oct","Nov","Dec"}; + + draw(graph(x,y),red,MarkFill[0]); + + xaxis(BottomTop,LeftTicks(new string(real x) { + return month[round(x % 12)];})); + yaxis("$y$",LeftRight,RightTicks(4)); + + + + 8. The next example draws a graph of a parametrized curve. The calls + to xlimits(picture pic=currentpicture, real min=-infinity, + real max=infinity, bool crop=NoCrop); + and the analogous function `ylimits' can be uncommented to set + the respective axes limits for picture `pic' to the specified + `min' and `max' values. Alternatively, the function void limits(picture pic=currentpicture, pair min, pair max, bool crop=NoCrop); + can be used to limit the axes to the box having opposite vertices + at the given pairs). Existing objects in picture `pic' will be + cropped to lie within the given limits if `crop'=`Crop'. The + function `crop(picture pic)' can be used to crop a graph to the + current graph limits. import graph; + + size(0,200); + + real x(real t) {return cos(2pi*t);} + real y(real t) {return sin(2pi*t);} + + draw(graph(x,y,0,1)); + + //xlimits(0,1,Crop); + //ylimits(-1,0,Crop); + + xaxis("$x$",BottomTop,LeftTicks); + yaxis("$y$",LeftRight,RightTicks(trailingzero)); + + + + The next example illustrates how one can extract a common axis + scaling factor. import graph; + + axiscoverage=0.9; + size(200,IgnoreAspect); + + real[] x={-1e-11,1e-11}; + real[] y={0,1e6}; + + real xscale=round(log10(max(x))); + real yscale=round(log10(max(y)))-1; + + draw(graph(x*10^(-xscale),y*10^(-yscale)),red); + + xaxis("$x/10^{"+(string) xscale+"}$",BottomTop,LeftTicks); + yaxis("$y/10^{"+(string) yscale+"}$",LeftRight,RightTicks(trailingzero)); + + + + Axis scaling can be requested and/or automatic selection of the + axis limits can be inhibited with one of these `scale' routines: void scale(picture pic=currentpicture, scaleT x, scaleT y); + + void scale(picture pic=currentpicture, bool xautoscale=true, + bool yautoscale=xautoscale, bool zautoscale=yautoscale); + + This sets the scalings for picture `pic'. The `graph' routines + accept an optional `picture' argument for determining the + appropriate scalings to use; if none is given, it uses those set + for `currentpicture'. + + Two frequently used scaling routines `Linear' and `Log' are + predefined in `graph'. + + All picture coordinates (including those in paths and those given + to the `label' and `limits' functions) are always treated as linear + (post-scaled) coordinates. Use pair Scale(picture pic=currentpicture, pair z); + to convert a graph coordinate into a scaled picture coordinate. + + The x and y components can be individually scaled using the + analogous routines real ScaleX(picture pic=currentpicture, real x); + real ScaleY(picture pic=currentpicture, real y); + + The predefined scaling routines can be given two optional boolean + arguments: `automin=false' and `automax=automin'. These default to + `false' but can be respectively set to `true' to enable automatic + selection of "nice" axis minimum and maximum values. The `Linear' + scaling can also take as optional final arguments a multiplicative + scaling factor and intercept (e.g. for a depth axis, `Linear(-1)' + requests axis reversal). + + For example, to draw a log/log graph of a function, use + `scale(Log,Log)': import graph; + + size(200,200,IgnoreAspect); + + real f(real t) {return 1/t;} + + scale(Log,Log); + + draw(graph(f,0.1,10)); + + //xlimits(1,10); + //ylimits(0.1,1); + + dot(Label("(3,5)",align=S),Scale((3,5))); + + xaxis("$x$",BottomTop,LeftTicks); + yaxis("$y$",LeftRight,RightTicks); + + + + By extending the ticks, one can easily produce a logarithmic grid: import graph; + size(200,200,IgnoreAspect); + + real f(real t) {return 1/t;} + + scale(Log,Log); + draw(graph(f,0.1,10),red); + pen thin=linewidth(0.5*linewidth()); + xaxis("$x$",BottomTop,LeftTicks(begin=false,end=false,extend=true, + ptick=thin)); + yaxis("$y$",LeftRight,RightTicks(begin=false,end=false,extend=true, + ptick=thin)); + + + + One can also specify custom tick locations and formats for + logarithmic axes: import graph; + + size(300,175,IgnoreAspect); + scale(Log,Log); + draw(graph(identity,5,20)); + xlimits(5,20); + ylimits(1,100); + xaxis("$M/M_\odot$",BottomTop,LeftTicks(DefaultFormat, + new real[] {6,10,12,14,16,18})); + yaxis("$\nu_{\rm upp}$ [Hz]",LeftRight,RightTicks(DefaultFormat)); + + + + It is easy to draw logarithmic graphs with respect to other bases: import graph; + size(200,IgnoreAspect); + + // Base-2 logarithmic scale on y-axis: + + real log2(real x) {static real log2=log(2); return log(x)/log2;} + real pow2(real x) {return 2^x;} + + scaleT yscale=scaleT(log2,pow2,logarithmic=true); + scale(Linear,yscale); + + real f(real x) {return 1+x^2;} + + draw(graph(f,-4,4)); + + yaxis("$y$",ymin=1,ymax=f(5),RightTicks(Label(Fill(white))),EndArrow); + xaxis("$x$",xmin=-5,xmax=5,LeftTicks,EndArrow); + + + + Here is an example of "broken" linear x and logarithmic y axes + that omit the segments [3,8] and [100,1000], respectively. In the + case of a logarithmic axis, the break endpoints are automatically + rounded to the nearest integral power of the base. import graph; + + size(200,150,IgnoreAspect); + + // Break the x axis at 3; restart at 8: + real a=3, b=8; + + // Break the y axis at 100; restart at 1000: + real c=100, d=1000; + + scale(Broken(a,b),BrokenLog(c,d)); + + real[] x={1,2,4,6,10}; + real[] y=x^4; + + draw(graph(x,y),red,MarkFill[0]); + + xaxis("$x$",BottomTop,LeftTicks(Break(a,b))); + yaxis("$y$",LeftRight,RightTicks(Break(c,d))); + + label(rotate(90)*Break,(a,point(S).y)); + label(rotate(90)*Break,(a,point(N).y)); + label(Break,(point(W).x,ScaleY(c))); + label(Break,(point(E).x,ScaleY(c))); + + + + 9. `Asymptote' can draw secondary axes with the routines picture secondaryX(picture primary=currentpicture, void f(picture)); + picture secondaryY(picture primary=currentpicture, void f(picture)); + + In this example, `secondaryY' is used to draw a secondary linear y + axis against a primary logarithmic y axis: import graph; + texpreamble("\def\Arg{\mathop {\rm Arg}\nolimits}"); + + size(10cm,5cm,IgnoreAspect); + + real ampl(real x) {return 2.5/(1+x^2);} + real phas(real x) {return -atan(x)/pi;} + + scale(Log,Log); + draw(graph(ampl,0.01,10)); + ylimits(0.001,100); + + xaxis("$\omega\tau_0$",BottomTop,LeftTicks); + yaxis("$|G(\omega\tau_0)|$",Left,RightTicks); + + picture q=secondaryY(new void(picture pic) { + scale(pic,Log,Linear); + draw(pic,graph(pic,phas,0.01,10),red); + ylimits(pic,-1.0,1.5); + yaxis(pic,"$\Arg G/\pi$",Right,red, + LeftTicks("$% #.1f$", + begin=false,end=false)); + yequals(pic,1,Dotted); + }); + label(q,"(1,0)",Scale(q,(1,0)),red); + add(q); + + + + A secondary logarithmic y axis can be drawn like this: import graph; + + size(9cm,6cm,IgnoreAspect); + string data="secondaryaxis.csv"; + + file in=line(csv(input(data))); + + string[] titlelabel=in; + string[] columnlabel=in; + + real[][] a=dimension(in,0,0); + a=transpose(a); + real[] t=a[0], susceptible=a[1], infectious=a[2], dead=a[3], larvae=a[4]; + real[] susceptibleM=a[5], exposed=a[6],infectiousM=a[7]; + + scale(true); + + draw(graph(t,susceptible,t >= 10 & t <= 15)); + draw(graph(t,dead,t >= 10 & t <= 15),dashed); + + xaxis("Time ($\tau$)",BottomTop,LeftTicks); + yaxis(Left,RightTicks); + + picture secondary=secondaryY(new void(picture pic) { + scale(pic,Linear(true),Log(true)); + draw(pic,graph(pic,t,infectious,t >= 10 & t <= 15),red); + yaxis(pic,Right,red,LeftTicks(begin=false,end=false)); + }); + + add(secondary); + label(shift(5mm*N)*"Proportion of crows",point(NW),E); + + + + 10. Here is a histogram example, which uses the `stats' module. import graph; + import stats; + + size(400,200,IgnoreAspect); + + int n=10000; + real[] a=new real[n]; + for(int i=0; i < n; ++i) a[i]=Gaussrand(); + + draw(graph(Gaussian,min(a),max(a)),blue); + + // Optionally calculate "optimal" number of bins a la Shimazaki and Shinomoto. + int N=bins(a); + + histogram(a,min(a),max(a),N,normalize=true,low=0,lightred,black,bars=false); + + xaxis("$x$",BottomTop,LeftTicks); + yaxis("$dP/dx$",LeftRight,RightTicks(trailingzero)); + + + + 11. Here is an example of reading column data in from a file and a + least-squares fit, using the `stats' module. size(400,200,IgnoreAspect); + + import graph; + import stats; + + file fin=line(input("leastsquares.dat")); + + real[][] a=dimension(fin,0,0); + a=transpose(a); + + real[] t=a[0], rho=a[1]; + + // Read in parameters from the keyboard: + //real first=getreal("first"); + //real step=getreal("step"); + //real last=getreal("last"); + + real first=100; + real step=50; + real last=700; + + // Remove negative or zero values of rho: + t=rho > 0 ? t : null; + rho=rho > 0 ? rho : null; + + scale(Log(true),Linear(true)); + + int n=step > 0 ? ceil((last-first)/step) : 0; + + real[] T,xi,dxi; + + for(int i=0; i <= n; ++i) { + real first=first+i*step; + real[] logrho=(t >= first & t <= last) ? log(rho) : null; + real[] logt=(t >= first & t <= last) ? -log(t) : null; + + if(logt.length < 2) break; + + // Fit to the line logt=L.m*logrho+L.b: + linefit L=leastsquares(logt,logrho); + + T.push(first); + xi.push(L.m); + dxi.push(L.dm); + } + + draw(graph(T,xi),blue); + errorbars(T,xi,dxi,red); + + crop(); + + ylimits(0); + + xaxis("$T$",BottomTop,LeftTicks); + yaxis("$\xi$",LeftRight,RightTicks); + + + + 12. Here is an example that illustrates the general `axis' routine. import graph; + size(0,100); + + path g=ellipse((0,0),1,2); + + scale(true); + + axis(Label("C",align=10W),g,LeftTicks(endlabel=false,8,end=false), + ticklocate(0,360,new real(real v) { + path h=(0,0)--max(abs(max(g)),abs(min(g)))*dir(v); + return intersect(g,h)[0];})); + + + + 13. To draw a vector field of `n' arrows evenly spaced along the + arclength of a path, use the routine picture vectorfield(path vector(real), path g, int n, bool truesize=false, + pen p=currentpen, arrowbar arrow=Arrow); + as illustrated in this simple example of a flow field: import graph; + defaultpen(1.0); + + size(0,150,IgnoreAspect); + + real arrowsize=4mm; + real arrowlength=2arrowsize; + + typedef path vector(real); + + // Return a vector interpolated linearly between a and b. + vector vector(pair a, pair b) { + return new path(real x) { + return (0,0)--arrowlength*interp(a,b,x); + }; + } + + real f(real x) {return 1/x;} + + real epsilon=0.5; + path g=graph(f,epsilon,1/epsilon); + + int n=3; + draw(g); + xaxis("$x$"); + yaxis("$y$"); + + add(vectorfield(vector(W,W),g,n,true)); + add(vectorfield(vector(NE,NW),(0,0)--(point(E).x,0),n,true)); + add(vectorfield(vector(NE,NE),(0,0)--(0,point(N).y),n,true)); + + + + 14. To draw a vector field of `nx'\times`ny' arrows in `box(a,b)', use + the routine picture vectorfield(path vector(pair), pair a, pair b, + int nx=nmesh, int ny=nx, bool truesize=false, + real maxlength=truesize ? 0 : maxlength(a,b,nx,ny), + bool cond(pair z)=null, pen p=currentpen, + arrowbar arrow=Arrow, margin margin=PenMargin) + as illustrated in this example: import graph; + size(100); + + pair a=(0,0); + pair b=(2pi,2pi); + + path vector(pair z) {return (0,0)--(sin(z.x),cos(z.y));} + + add(vectorfield(vector,a,b)); + + + + 15. The following scientific graphs, which illustrate many features of + `Asymptote''s graphics routines, were generated from the examples + `diatom.asy' and `westnile.asy', using the comma-separated data in + `diatom.csv' and `westnile.csv'. + + + + + + + +File: asymptote.info, Node: palette, Next: three, Prev: graph, Up: Base modules + +7.28 `palette' +============== + +`Asymptote' can also generate color density images and palettes. The +following palettes are predefined in `palette.asy': + +`pen[] Grayscale(int NColors=256)' + a grayscale palette; + +`pen[] Rainbow(int NColors=32766)' + a rainbow spectrum; + +`pen[] BWRainbow(int NColors=32761)' + a rainbow spectrum tapering off to black/white at the ends; + +`pen[] BWRainbow2(int NColors=32761)' + a double rainbow palette tapering off to black/white at the ends, + with a linearly scaled intensity. + +`pen[] Wheel(int NColors=32766)' + a full color wheel palette; + +`pen[] Gradient(int NColors=256 ... pen[] p)' + a palette varying linearly over the specified array of pens, using + NColors in each interpolation interval; + + + The function `cmyk(pen[] Palette)' may be used to convert any of +these palettes to the CMYK colorspace. + + A color density plot using palette `palette' can be generated from a +function `f'(x,y) and added to a picture `pic': +bounds image(picture pic=currentpicture, real f(real,real), + range range=Full, pair initial, pair final, + int nx=ngraph, int ny=nx, pen[] palette, bool antialias=false) + The function `f' will be sampled at `nx' and `ny' evenly spaced points +over a rectangle defined by the points `initial' and `final', +respecting the current graphical scaling of `pic'. The color space is +scaled according to the z axis scaling (*note automatic scaling::). A +bounds structure for the function values is returned: +struct bounds { + real min; + real max; + // Possible tick intervals: + int[] divisor; +} + This information can be used for generating an optional palette bar. +The palette color space corresponds to a range of values specified by +the argument `range', which can be `Full', `Automatic', or an explicit +range `Range(real min, real max)'. Here `Full' specifies a range +varying from the minimum to maximum values of the function over the +sampling interval, while `Automatic' selects "nice" limits. The +example `imagecontour.asy' illustrates how level sets (contour lines) +can be drawn on a color density plot (*note contour::). + + A color density plot can also be generated from an explicit real[][] +array `data': +bounds image(picture pic=currentpicture, real[][] f, range range=Full, + pair initial, pair final, pen[] palette, + bool transpose=(initial.x < final.x && initial.y < final.y), + bool copy=true, bool antialias=false); + If the initial point is to the left and below the final point, by +default the array indices are interpreted according to the Cartesian +convention (first index: x, second index: y) rather than the usual +matrix convention (first index: -y, second index: x). + + To construct an image from an array of irregularly spaced points and +an array of values `f' at these points, use one of the routines +bounds image(picture pic=currentpicture, pair[] z, real[] f, + range range=Full, pen[] palette) +bounds image(picture pic=currentpicture, real[] x, real[] y, real[] f, + range range=Full, pen[] palette) + + An optionally labelled palette bar may be generated with the routine +void palette(picture pic=currentpicture, Label L="", bounds bounds, + pair initial, pair final, axis axis=Right, pen[] palette, + pen p=currentpen, paletteticks ticks=PaletteTicks, + bool copy=true, bool antialias=false); + The color space of `palette' is taken to be over bounds `bounds' with +scaling given by the z scaling of `pic'. The palette orientation is +specified by `axis', which may be one of `Right', `Left', `Top', or +`Bottom'. The bar is drawn over the rectangle from `initial' to +`final'. The argument `paletteticks' is a special tick type (*note +ticks::) that takes the following arguments: +paletteticks PaletteTicks(Label format="", ticklabel ticklabel=null, + bool beginlabel=true, bool endlabel=true, + int N=0, int n=0, real Step=0, real step=0, + pen pTick=nullpen, pen ptick=nullpen); + + The image and palette bar can be fit to a frame and added and +optionally aligned to a picture at the desired location: + +size(12cm,12cm); + +import graph; +import palette; + +int n=256; +real ninv=2pi/n; +real[][] v=new real[n][n]; + +for(int i=0; i < n; ++i) + for(int j=0; j < n; ++j) + v[i][j]=sin(i*ninv)*cos(j*ninv); + +pen[] Palette=BWRainbow(); + +picture bar; + +bounds range=image(v,(0,0),(1,1),Palette); +palette(bar,"$A$",range,(0,0),(0.5cm,8cm),Right,Palette, + PaletteTicks("$%+#.1f$")); +add(bar.fit(),point(E),30E); + + + +Here is an example that uses logarithmic scaling of the function values: + +import graph; +import palette; + +size(10cm,10cm,IgnoreAspect); + +real f(real x, real y) { + return 0.9*pow10(2*sin(x/5+2*y^0.25)) + 0.1*(1+cos(10*log(y))); +} + +scale(Linear,Log,Log); + +pen[] Palette=BWRainbow(); + +bounds range=image(f,Automatic,(0,1),(100,100),nx=200,Palette); + +xaxis("$x$",BottomTop,LeftTicks,above=true); +yaxis("$y$",LeftRight,RightTicks,above=true); + +palette("$f(x,y)$",range,(0,200),(100,250),Top,Palette, + PaletteTicks(ptick=linewidth(0.5*linewidth()))); + + + +One can also draw an image directly from a two-dimensional pen array: +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); + as illustrated in the following example: + +size(200); + +import palette; + +int n=256; +real ninv=2pi/n; +pen[][] v=new pen[n][n]; + +for(int i=0; i < n; ++i) + for(int j=0; j < n; ++j) + v[i][j]=rgb(0.5*(1+sin(i*ninv)),0.5*(1+cos(j*ninv)),0); + +image(v,(0,0),(1,1)); + + + +For convenience, the module `palette' also defines functions that may +be used to construct a pen array from a given function and palette: +pen[] palette(real[] f, pen[] palette); +pen[][] palette(real[][] f, pen[] palette); + + +File: asymptote.info, Node: three, Next: obj, Prev: palette, Up: Base modules + +7.29 `three' +============ + +This module fully extends the notion of guides and paths in `Asymptote' +to three dimensions. It introduces the new types guide3, path3, and +surface. Guides in three dimensions are specified with the same syntax +as in two dimensions except that triples `(x,y,z)' are used in place of +pairs `(x,y)' for the nodes and direction specifiers. This +generalization of John Hobby's spline algorithm is shape-invariant under +three-dimensional rotation, scaling, and shifting, and reduces in the +planar case to the two-dimensional algorithm used in `Asymptote', +`MetaPost', and `MetaFont' [cf. J. C. Bowman, Proceedings in Applied +Mathematics and Mechanics, 7:1, 2010021-2010022 (2007)]. + + For example, a unit circle in the XY plane may be filled and drawn +like this: + +import three; + +size(100); + +path3 g=(1,0,0)..(0,1,0)..(-1,0,0)..(0,-1,0)..cycle; +draw(g); +draw(O--Z,red+dashed,Arrow3); +draw(((-1,-1,0)--(1,-1,0)--(1,1,0)--(-1,1,0)--cycle)); +dot(g,red); + + +and then distorted into a saddle: + +import three; + +size(100,0); +path3 g=(1,0,0)..(0,1,1)..(-1,0,0)..(0,-1,1)..cycle; +draw(g); +draw(((-1,-1,0)--(1,-1,0)--(1,1,0)--(-1,1,0)--cycle)); +dot(g,red); + + +Module `three' provides constructors for converting two-dimensional +paths to three-dimensional ones, and vice-versa: +path3 path3(path p, triple plane(pair)=XYplane); +path path(path3 p, pair P(triple)=xypart); + + A Bezier surface, the natural two-dimensional generalization of +Bezier curves, is defined in `three_surface.asy' as a structure +containing an array of Bezier patches. Surfaces may drawn with one of +the routines +void draw(picture pic=currentpicture, surface s, int nu=1, int nv=1, + material surfacepen=currentpen, pen meshpen=nullpen, + light light=currentlight, light meshlight=light); +void draw(picture pic=currentpicture, surface s, int nu=1, int nv=1, + material[] surfacepen, pen meshpen, + light light=currentlight, light meshlight=light); +void draw(picture pic=currentpicture, surface s, int nu=1, int nv=1, + material[] surfacepen, pen[] meshpen=nullpens, + light light=currentlight, light meshlight=light); + The parameters `nu' and `nv' specify the number of subdivisions for +drawing optional mesh lines for each Bezier patch. Here material is a +structure defined in `three_light.asy': +struct material { + pen[] p; // diffusepen,ambientpen,emissivepen,specularpen + real opacity; + real shininess; + real granularity; +... +} + These material properties are used to implement `OpenGL'-style +lighting, based on the Phong-Blinn specular model. Sample Bezier +surfaces are contained in the example files `BezierSurface.asy', +`teapot.asy', and `parametricsurface.asy'. + + The examples `elevation.asy' and `sphericalharmonic.asy' illustrate +how to draw a surface with patch-dependent colors. The examples +`vertexshading' and `smoothelevation' illustrate vertex-dependent +colors, which is supported for both `Asymptote''s native `OpenGL' +renderer and two-dimensional projections. Since the PRC output format +does not currently support vertex shading of Bezier surfaces, PRC +patches are shaded with the mean of the four vertex colors. + + A surface can be constructed from a cyclic `path3' with the +constructor +surface surface(path3 external, triple[] internal=new triple[], + triple[] normals=new triple[], pen[] colors=new pen[], + bool3 planar=default); + and then filled: +draw(surface(path3(polygon(5))),red); +draw(surface(unitcircle3),red); +draw(surface(unitcircle3,new pen[] {red,green,blue,black})); + The last example constructs a patch with vertex-specific colors. A +three-dimensional planar surface in the plane `plane' can be +constructed from a two-dimensional cyclic path `g' with the constructor +surface surface(path p, triple plane(pair)=XYplane); + and then filled: +draw(surface((0,0)--E+2N--2E--E+N..0.2E..cycle),red); + Planar Bezier surfaces patches are constructed using Orest Shardt's +`bezulate' routine, which decomposes (possibly nonsimply connected) +regions bounded by nonselfintersecting cyclic paths (according to the +`zerowinding' fill rule) into subregions bounded by cyclic paths of +length `4' or less. + + Arbitrary thick three-dimensional curves and line caps (which the +`OpenGL' standard does not require implementations to provide) are +constructed with the routine +surface tube(path3 g, real width); + which returns a tube of diameter `width' centered on `g'. This can +make files slow to render, especially with the `Adobe Reader' renderer. +The setting `thick=false' can be used to disable this feature and force +all lines to be drawn with `linewidth(0)' (one pixel wide, regardless +of the resolution). By default mesh and contour lines in +three-dimensions are always drawn thin, unless an explicit line width +is given in the pen parameter or the setting `thin' is set to `false'. +The pens `thin()' and `thick()' defined in plain_pens.asy can also be +used to override these defaults for specific draw commands. + +There are four choices for viewing 3D `Asymptote' output: + 1. Use the native `Asymptote' adaptive `OpenGL'-based renderer (with + the command-line option `-V' and the default settings + `outformat=""' and `render=-1'). If you encounter warnings from + your graphics card driver, try specifying `-glOptions=-indirect' + on the command line. On `UNIX' systems with graphics support for + multisampling, we recommend installing the latest SVN (antialiased) + version of the `freeglut' library (*note multisampling::); the + sample width can be controlled with the setting `multisample'. An + initial screen position can be specified with the pair setting + `position', where negative values are interpreted as relative to + the corresponding maximum screen dimension. The mouse bindings are: + * Left: rotate + + * shift Left: zoom + + * ctrl Left: shift + + * Middle: menu + + * Wheel: zoom + + * Right: zoom + + * Right double click: menu + + * shift Right: rotate about the X axis + + * ctrl Right: rotate about the Y axis + + * alt Right: rotate about the Z axis + The keyboard shortcuts are: + * h: home + + * f: toggle fitscreen + + * x: spin about the X axis + + * y: spin about the Y axis + + * z: spin about the Z axis + + * s: stop spinning + + * m: rendering mode (solid/mesh/patch) + + * e: export + + * c: show camera parameters + + * +: expand + + * =: expand + + * -: shrink + + * _: shrink + + * q: exit + + * Ctrl-q: exit + + 2. Render the scene to a specified rasterized format `outformat' at + the resolution of `n' pixels per `bp', as specified by the setting + `render=n'. A negative value of `n' is interpreted as `|2n|' for + EPS and PDF formats and `|n|' for other formats. The default value + of `render' is -1. By default, the scene is internally rendered + at twice the specified resolution; this can be disabled by setting + `antialias=1'. High resolution rendering is done by tiling the + image. If your graphics card allows it, the rendering can be made + more efficient by increasing the maximum tile size `maxtile' + beyond the screen dimensions (indicated by `maxtile=(0,0)'. The + tile size is also limited by the setting `maxviewport', which + restricts the maximum width and height of the viewport. On `UNIX' + systems some graphics drivers support batch mode (`-noV') + rendering in an iconified window; this can be enabled with the + setting `iconify=true'. Other `UNIX' graphics drivers may require + the command line setting `-glOptions=-indirect'. + + 3. Embed the 3D PRC format in a PDF file and view the resulting PDF + file with version `8.0' or later of `Adobe Reader'. In addition + to the default `settings.prc=true', this requires + `settings.outformat="pdf"', which can be specified by the command + line option `-f pdf', put in the `Asymptote' configuration file + (*note configuration file::), or specified in the script before + `three.asy' (or `graph3.asy') is imported. Version 2008/10/08 or + later of the `movie15' package is also required (*note embed::). + The example `pdb.asy' illustrates how one can generate a list of + predefined views (see `100d.views'). A stationary preview image + with a resolution of `n' pixels per `bp' can be embedded with the + setting `render=n'; this allows the file to be viewed with other + `PDF' viewers. Alternatively, the file `externalprc.tex' + illustrates how the resulting PRC and rendered image files can be + extracted and processed in a separate `LaTeX' file. However, see + *note LaTeX usage:: for an easier way to embed three-dimensional + `Asymptote' pictures within `LaTeX'. The open-source PRC + specification is available from + `http://livedocs.adobe.com/acrobat_sdk/9/Acrobat9_HTMLHelp/API_References/PRCReference/PRC_Format_Specification/'. + + 4. Project the scene to a two-dimensional vector (EPS or PDF) format + with `render=0'. Only limited hidden surface removal facilities + are currently available with this approach (*note PostScript3D::). + + + Automatic picture sizing in three dimensions is accomplished with +double deferred drawing. The maximal desired dimensions of the scene in +each of the three dimensions can optionally be specified with the +routine +void size3(picture pic=currentpicture, real x, real y=x, real z=y, + bool keepAspect=pic.keepAspect); + The resulting simplex linear programming problem is then solved to +produce a 3D version of a frame (actually implemented as a 3D picture). +The result is then fit with another application of deferred drawing to +the viewport dimensions corresponding to the usual two-dimensional +picture `size' parameters. The global pair `viewportmargin' may be used +to add horizontal and vertical margins to the viewport dimensions. +Alternatively, a minimum `viewportsize' may be specified. + + For convenience, the `three' module defines `O=(0,0,0)', +`X=(1,0,0)', `Y=(0,1,0)', and `Z=(0,0,1)', along with a unitcircle in +the XY plane: +path3 unitcircle3=X..Y..-X..-Y..cycle; + + A general (approximate) circle can be drawn perpendicular to the +direction `normal' with the routine +path3 circle(triple c, real r, triple normal=Z); + + A circular arc centered at `c' with radius `r' from +`c+r*dir(theta1,phi1)' to `c+r*dir(theta2,phi2)', drawing +counterclockwise relative to the normal vector +`cross(dir(theta1,phi1),dir(theta2,phi2))' if `theta2 > theta1' or if +`theta2 == theta1' and `phi2 >= phi1', can be constructed with +path3 arc(triple c, real r, real theta1, real phi1, real theta2, real phi2, + triple normal=O); + The normal must be explicitly specified if `c' and the endpoints are +colinear. If `r' < 0, the complementary arc of radius `|r|' is +constructed. For convenience, an arc centered at `c' from triple `v1' +to `v2' (assuming `|v2-c|=|v1-c|') in the direction CCW +(counter-clockwise) or CW (clockwise) may also be constructed with +path3 arc(triple c, triple v1, triple v2, triple normal=O, + bool direction=CCW); + When high accuracy is needed, the routines `Circle' and `Arc' defined +in `graph3' may be used instead. See *note GaussianSurface:: for an +example of a three-dimensional circular arc. + + The representation `O--O+u--O+u+v--O+v--cycle' of the plane passing +through point `O' with normal `cross(u,v)' is returned by +path3 plane(triple u, triple v, triple O=O); + A three-dimensional box with opposite vertices at triples `v1' and +`v2' may be drawn with the function +path3[] box(triple v1, triple v2); + For example, a unit box is predefined as +path3[] unitbox=box(O,(1,1,1)); + `Asymptote' also provides optimized definitions for the +three-dimensional paths `unitsquare3' and `unitcircle3', along with the +surfaces `unitdisk', `unitplane', `unitcube', `unitcylinder', +`unitcone', `unitsolidcone', `unitfrustum(real t1, real t2)', +`unitsphere', and `unithemisphere'. + +These projections to two dimensions are predefined: +`oblique' + +`oblique(real angle);' + The point `(x,y,z)' is projected to `(x-0.5z,y-0.5z)'. If an + optional real argument is given, the negative z axis is drawn at + this angle in degrees. The projection `obliqueZ' is a synonym for + `oblique'. + +`obliqueX' + +`obliqueX(real angle)' + The point `(x,y,z)' is projected to `(y-0.5x,z-0.5x)'. If an + optional real argument is given, the negative x axis is drawn at + this angle in degrees. + +`obliqueY' + +`obliqueY(real angle)' + The point `(x,y,z)' is projected to `(x+0.5y,z+0.5y)'. If an + optional real argument is given, the positive y axis is drawn at + this angle in degrees. + +`orthographic(triple camera, triple up=Z, triple target=O, + 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 `unit(camera)', + orienting the camera so that, if possible, the vector `up' points + upwards. Parallel lines are projected to parallel lines. The + bounding volume is expanded to include `target' if + `showtarget=true'. If `center=true', the target will be adjusted + to the center of the bounding volume. + +`orthographic(real x, real y, real z, triple up=Z, triple target=O, + bool showtarget=true, bool center=false)' + This is equivalent to + `orthographic((x,y,z),up,target,showtarget,center)'. + +`perspective(triple camera, triple up=Z, triple target=O, + bool showtarget=true, bool autoadjust=true, + bool center=autoadjust)' + This projects from three to two dimensions, taking account of + perspective, as seen from the location `camera' looking at + `target', orienting the camera so that, if possible, the vector + `up' points upwards. If `render=0', projection of + three-dimensional cubic Bezier splines is implemented by + approximating a two-dimensional nonuniform rational B-spline + (NURBS) with a two-dimensional Bezier curve containing additional + nodes and control points. If `autoadjust=true', the camera will + automatically be adjusted to lie outside the bounding volume for + all possible interactive rotations about `target'. If + `center=true', the target will be adjusted to the center of the + bounding volume. + +`perspective(real x, real y, real z, triple up=Z, triple target=O, + bool showtarget=true, bool autoadjust=true, + bool center=autoadjust)' + This is equivalent to `perspective((x,y,z),up,target,showtarget, + autoadjust,center)'. + +The default projection, `currentprojection', is initially set to +`perspective(5,4,2)'. + + We also define standard orthographic views used in technical drawing: +projection LeftView=orthographic(-X,showtarget=true); +projection RightView=orthographic(X,showtarget=true); +projection FrontView=orthographic(-Y,showtarget=true); +projection BackView=orthographic(Y,showtarget=true); +projection BottomView=orthographic(-Z,showtarget=true); +projection TopView=orthographic(Z,showtarget=true); + The function +void addViews(picture dest=currentpicture, picture src, bool group=true, + filltype filltype=NoFill, bool above=true); + adds picture `pic' as seen with `FrontView' aligned above the +projection `TopView' and to the right of the projection `RightView'. + + A triple or path3 can be projected to a pair or path, with +`project(triple, projection P=currentprojection)' or `project(path3, +projection P=currentprojection)'. + + It is occasionally useful to be able to invert a projection, sending +a pair `z' onto the plane perpendicular to `normal' and passing through +`point': +triple invert(pair z, triple normal, triple point, + projection P=currentprojection); + A pair `z' on the projection plane can be inverted to a triple with +the routine +triple invert(pair z, projection P=currentprojection); + A pair direction `dir' on the projection plane can be inverted to a +triple direction relative to a point `v' with the routine +triple invert(pair dir, triple v, projection P=currentprojection). + + Three-dimensional objects may be transformed with one of the +following built-in transform3 types: + +`shift(triple v)' + translates by the triple `v'; + +`xscale3(real x)' + scales by `x' in the x direction; + +`yscale3(real y)' + scales by `y' in the y direction; + +`zscale3(real z)' + scales by `z' in the z direction; + +`scale3(real s)' + scales by `s' in the x, y, and z directions; + +`scale(real x, real y, real z)' + scales by `x' in the x direction, by `y' in the y direction, and + by `z' in the z direction; + +`rotate(real angle, triple v)' + rotates by `angle' in degrees about an axis `v' through the origin; + +`rotate(real angle, triple u, triple v)' + rotates by `angle' in degrees about the axis `u--v'; + +`reflect(triple u, triple v, triple w)' + reflects about the plane through `u', `v', and `w'. + + Three-dimensional TeX Labels, which are by default drawn as Bezier +surfaces directly on the projection plane, can be transformed from the +`XY' plane by any of the above transforms or mapped to a specified +two-dimensional plane with the transform3 types `XY', `YZ', `ZX', `YX', +`ZY', `ZX'. There are also modified versions of these transforms that +take an optional argument `projection P=currentprojection' that rotate +and/or flip the label so that it is more readable from the initial +viewpoint. + + A transform3 that projects in the direction `dir' onto the plane +with normal `n' through point `O' is returned by +transform3 planeproject(triple n, triple O=O, triple dir=n); + One can use +triple normal(path3 p); + to find the unit normal vector to a planar three-dimensional path `p'. +As illustrated in the example `planeproject.asy', a transform3 that +projects in the direction `dir' onto the plane defined by a planar path +`p' is returned by +transform3 planeproject(path3 p, triple dir=normal(p)); + + The functions +surface extrude(path p, triple axis=Z); +surface extrude(Label L, triple axis=Z); + return the surface obtained by extruding path `p' or Label `L' along +`axis'. + + Three-dimensional versions of the path functions `length', `size', +`point', `dir', `accel', `radius', `precontrol', `postcontrol', +`arclength', `arctime', `reverse', `subpath', `intersect', +`intersections', `intersectionpoint', `intersectionpoints', `min', +`max', `cyclic', and `straight' are also defined. + + The routine +real[][] intersections(path3 p, surface s, real fuzz=-1); + returns the intersection times of a path `p' with a surface `s' as a +sorted array of real arrays of length 2, and +triple[] intersectionpoints(path3 p, surface s, real fuzz=-1); + returns the corresponding intersection points. Here, the computations +are performed to the absolute error specified by `fuzz', or if `fuzz < +0', to machine precision. + + Here is an example showing all five guide3 connectors: + +import graph3; + +size(200); + +currentprojection=orthographic(500,-500,500); + +triple[] z=new triple[10]; + +z[0]=(0,100,0); z[1]=(50,0,0); z[2]=(180,0,0); + +for(int n=3; n <= 9; ++n) + z[n]=z[n-3]+(200,0,0); + +path3 p=z[0]..z[1]---z[2]::{Y}z[3] +&z[3]..z[4]--z[5]::{Y}z[6] +&z[6]::z[7]---z[8]..{Y}z[9]; + +draw(p,grey+linewidth(4mm)+opacity(0.5)); + +xaxis3(Label(XY()*"$x$",align=-3Y),red,above=true); +yaxis3(Label(XY()*"$y$",align=-3X),red,above=true); + +dot(z); + + + +Three-dimensional versions of bars or arrows can be drawn with one of +the specifiers `None', `Blank', `BeginBar3', `EndBar3' (or equivalently +`Bar3'), `Bars3', `BeginArrow3', `MidArrow3', `EndArrow3' (or +equivalently `Arrow3'), `Arrows3', `BeginArcArrow3', `EndArcArrow3' (or +equivalently `ArcArrow3'), `MidArcArrow3', and `ArcArrows3'. +Three-dimensional bars accept the optional arguments `(real size=0, +triple dir=O)'. If `size=O', the default bar length is used; if +`dir=O', the bar is drawn perpendicular to the path and the initial +viewing direction. The predefined three-dimensional arrowhead styles +are `DefaultHead3', `HookHead3', `TeXHead3'. Versions of the +two-dimensional arrowheads lifted to three-dimensional space and +aligned according to the initial viewpoint (or an optionally specified +`normal' vector) are also defined: `DefaultHead2(triple normal=O)', +`HookHead2(triple normal=O)', `TeXHead2(triple normal=O)'. These are +illustrated in the example `arrows3.asy'. + + Module `three' also defines the three-dimensional margins +`NoMargin3', `BeginMargin3', `EndMargin3', `Margin3', `Margins3', +`BeginPenMargin2', `EndPenMargin2', `PenMargin2', `PenMargins2', +`BeginPenMargin3', `EndPenMargin3', `PenMargin3', `PenMargins3', +`BeginDotMargin3', `EndDotMargin3', `DotMargin3', `DotMargins3', +`Margin3', and `TrueMargin3'. + + Further three-dimensional examples are provided in the files +`near_earth.asy', `conicurv.asy', and (in the `animations' +subdirectory) `cube.asy'. + + Limited support for projected vector graphics (effectively +three-dimensional nonrendered `PostScript') is available with the +setting `render=0'. This currently only works for piecewise planar +surfaces, such as those produced by the parametric `surface' routines +in the `graph3' module. Surfaces produced by the `solids' package will +also be properly rendered if the parameter `nslices' is sufficiently +large. + + In the module `bsp', hidden surface removal of planar pictures is +implemented using a binary space partition and picture clipping. A +planar path is first converted to a structure `face' derived from +`picture'. A `face' may be given to a two-dimensional drawing routine +in place of any `picture' argument. An array of such faces may then be +drawn, removing hidden surfaces: +void add(picture pic=currentpicture, face[] faces, + projection P=currentprojection); + Labels may be projected to two dimensions, using projection `P', onto +the plane passing through point `O' with normal `cross(u,v)' by +multiplying it on the left by the transform +transform transform(triple u, triple v, triple O=O, + projection P=currentprojection); + + Here is an example that shows how a binary space partition may be +used to draw a two-dimensional vector graphics projection of three +orthogonal intersecting planes: + +size(6cm,0); +import bsp; + +real u=2.5; +real v=1; + +currentprojection=oblique; + +path3 y=plane((2u,0,0),(0,2v,0),(-u,-v,0)); +path3 l=rotate(90,Z)*rotate(90,Y)*y; +path3 g=rotate(90,X)*rotate(90,Y)*y; + +face[] faces; +filldraw(faces.push(y),project(y),yellow); +filldraw(faces.push(l),project(l),lightgrey); +filldraw(faces.push(g),project(g),green); + +add(faces); + + + + +File: asymptote.info, Node: obj, Next: graph3, Prev: three, Up: Base modules + +7.30 `obj' +========== + +This module allows one to construct surfaces from simple obj files, as +illustrated in the example files `galleon.asy' and `triceratops.asy'. + + +File: asymptote.info, Node: graph3, Next: grid3, Prev: obj, Up: Base modules + +7.31 `graph3' +============= + +This module implements three-dimensional versions of the functions in +`graph.asy'. To draw an x axis in three dimensions, use the routine +void xaxis3(picture pic=currentpicture, Label L="", axis axis=YZZero, + real xmin=-infinity, real xmax=infinity, pen p=currentpen, + ticks3 ticks=NoTicks3, arrowbar3 arrow=None, bool above=false); + Analogous routines `yaxis' and `zaxis' can be used to draw y and z +axes in three dimensions. There is also a routine for drawing all +three axis: +void axes3(picture pic=currentpicture, + Label xlabel="", Label ylabel="", Label zlabel="", + triple min=(-infinity,-infinity,-infinity), + triple max=(infinity,infinity,infinity), + pen p=currentpen, arrowbar3 arrow=None); + +The predefined three-dimensional axis types are +axis YZEquals(real y, real z, triple align=O, bool extend=false); +axis XZEquals(real x, real z, triple align=O, bool extend=false); +axis XYEquals(real x, real y, triple align=O, bool extend=false); +axis YZZero(triple align=O, bool extend=false); +axis XZZero(triple align=O, bool extend=false); +axis XYZero(triple align=O, bool extend=false); +axis Bounds(int type=Both, int type2=Both, triple align=O, bool extend=false); + The optional `align' parameter to these routines can be used to +specify the default axis and tick label alignments. The `Bounds' axis +accepts two type parameters, each of which must be one of `Min', `Max', +or `Both'. These parameters specify which of the four possible +three-dimensional bounding box edges should be drawn. + + The three-dimensional tick options are `NoTicks3', `InTicks', +`OutTicks', and `InOutTicks'. These specify the tick directions for the +`Bounds' axis type; other axis types inherit the direction that would +be used for the `Bounds(Min,Min)' axis. + + Here is an example of a helix and bounding box axes with ticks and +axis labels, using orthographic projection: + +import graph3; + +size(0,200); +size3(200,IgnoreAspect); + +currentprojection=orthographic(4,6,3); + +real x(real t) {return cos(2pi*t);} +real y(real t) {return sin(2pi*t);} +real z(real t) {return t;} + +path3 p=graph(x,y,z,0,2.7,operator ..); + +draw(p,Arrow3); + +scale(true); + +xaxis3(XZ()*"$x$",Bounds(),red,InTicks(Label,2,2)); +yaxis3(YZ()*"$y$",Bounds(),red,InTicks(beginlabel=false,Label,2,2)); +zaxis3(XZ()*"$z$",Bounds(),red,InTicks); + + + +The next example illustrates three-dimensional x, y, and z axes, +without autoscaling of the axis limits: + +import graph3; + +size(0,200); +size3(200,IgnoreAspect); + +currentprojection=perspective(5,2,2); + +scale(Linear,Linear,Log); + +xaxis3("$x$",0,1,red,OutTicks(2,2)); +yaxis3("$y$",0,1,red,OutTicks(2,2)); +zaxis3("$z$",1,30,red,OutTicks(beginlabel=false)); + + + +One can also place ticks along a general three-dimensional axis: + +import graph3; + +size(0,100); + +path3 g=yscale3(2)*unitcircle3; +currentprojection=perspective(10,10,10); + +axis(Label("C",position=0,align=15X),g,InTicks(endlabel=false,8,end=false), + ticklocate(0,360,new real(real v) { + path3 h=O--max(abs(max(g)),abs(min(g)))*dir(90,v); + return intersect(g,h)[0];}, + new triple(real t) {return cross(dir(g,t),Z);})); + + + +Surface plots of matrices and functions over the region `box(a,b)' in +the XY plane are also implemented: +surface surface(real[][] f, pair a, pair b, bool[][] cond={}); +surface surface(real[][] f, pair a, pair b, splinetype splinetype, + bool[][] cond={}); +surface surface(real[][] f, real[] x, real[] y, + splinetype splinetype=null, 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 splinetype, 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); + The final version draws a parametric surface for a function f(u,v) +over the parameter space `box(a,b)', as illustrated in the example +`parametricsurface.asy'. The boolean array or function `cond' can be +used to control which surface mesh cells are actually drawn (by default +all mesh cells over `box(a,b)' are drawn). Surface lighting is +illustrated in the example files `parametricsurface.asy' and `sinc.asy'. +Lighting can be disabled by setting `light=nolight', as in this example +of a Gaussian surface: + +import graph3; + +size(200,0); + +currentprojection=perspective(10,8,4); + +real f(pair z) {return 0.5+exp(-abs(z)^2);} + +draw((-1,-1,0)--(1,-1,0)--(1,1,0)--(-1,1,0)--cycle); + +draw(arc(0.12Z,0.2,90,60,90,25),ArcArrow3); + +surface s=surface(f,(-1,-1),(1,1),nx=5,Spline); + +xaxis3(Label("$x$"),red,Arrow3); +yaxis3(Label("$y$"),red,Arrow3); +zaxis3(XYZero(extend=true),red,Arrow3); + +draw(s,lightgray,meshpen=black+thick(),nolight); + +label("$O$",O,-Z+Y,red); + + +A mesh can be drawn without surface filling by specifying `nullpen' for +the surfacepen. + + A vector field of `nu'\times`nv' arrows on a parametric surface `f' +over `box(a,b)' can be drawn with the routine +picture vectorfield(path3 vector(pair v), triple f(pair z), pair a, pair b, + int nu=nmesh, int nv=nu, bool truesize=false, + real maxlength=truesize ? 0 : maxlength(f,a,b,nu,nv), + bool cond(pair z)=null, pen p=currentpen, + arrowbar3 arrow=Arrow3, margin3 margin=PenMargin3) + as illustrated in the examples `vectorfield3.asy' and +`vectorfieldsphere.asy'. + + +File: asymptote.info, Node: grid3, Next: solids, Prev: graph3, Up: Base modules + +7.32 `grid3' +============ + +This module, contributed by Philippe Ivaldi, can be used for drawing 3D +grids. Here is an example (further examples can be found in `grid3.asy' +and at `http://piprim.tuxfamily.org/asymptote/grid3/'): + +import grid3; + +size(8cm,0,IgnoreAspect); +currentprojection=orthographic(0.5,1,0.5); + +scale(Linear, Linear, Log); + +limits((-2,-2,1),(0,2,100)); + +grid3(XYZgrid); + +xaxis3(Label("$x$",position=EndPoint,align=S),Bounds(Min,Min), + OutTicks()); +yaxis3(Label("$y$",position=EndPoint,align=S),Bounds(Min,Min),OutTicks()); +zaxis3(Label("$z$",position=EndPoint,align=(-1,0.5)),Bounds(Min,Min), + OutTicks(beginlabel=false)); + + + + +File: asymptote.info, Node: solids, Next: tube, Prev: grid3, Up: Base modules + +7.33 `solids' +============= + +This solid geometry package defines a structure `revolution' that can +be used to fill and draw surfaces of revolution. The following example +uses it to display the outline of a circular cylinder of radius 1 with +axis `O--1.5unit(Y+Z)' with perspective projection: + +import solids; + +size(0,100); + +revolution r=cylinder(O,1,1.5,Y+Z); +draw(r,heavygreen); + + + +Further illustrations are provided in the example files `cylinder.asy', +`cones.asy', `hyperboloid.asy', and `torus.asy'. + + The structure `skeleton' contains the three-dimensional wireframe +used to visualize a volume of revolution: +struct skeleton { + struct curve { + path3[] front; + path3[] back; + } + // transverse skeleton (perpendicular to axis of revolution) + curve transverse; + // longitudinal skeleton (parallel to axis of revolution) + curve longitudinal; +} + + +File: asymptote.info, Node: tube, Next: flowchart, Prev: solids, Up: Base modules + +7.34 `tube' +=========== + +This package extends the routine `tube' defined in `three_arrows.asy' +to arbitrary cross sections, colors, and spine transformations. The +routine +surface tube(path3 g, coloredpath section, + transform T(real)=new transform(real t) {return identity();}, + real corner=1, real relstep=0); + draws a tube along `g' with cross section `section', after applying +the transformation `T(t)' at `relpoint(g,t)'. The parameter `corner' +controls the number of elementary tubes at the angular points of `g'. A +nonzero value of `relstep' specifies a fixed relative time step (in the +sense of `relpoint(g,t)') to use in constructing elementary tubes along +`g'. The type `coloredpath' is a generalization of `path' to which a +`path' can be cast: +struct coloredpath +{ + path p; + pen[] pens(real); + int colortype=coloredSegments; +} + Here `p' defines the cross section and the method `pens(real t)' +returns an array of pens (interpreted as a cyclic array) used for +shading the tube patches at `relpoint(g,t)'. If +`colortype=coloredSegments', the tube patches are filled as if each +segment of the section was colored with the pen returned by `pens(t)', +whereas if `colortype=coloredNodes', the tube components are vertex +shaded as if the nodes of the section were colored. + + A `coloredpath' can be constructed with one of the routines: +coloredpath coloredpath(path p, pen[] pens(real), + int colortype=coloredSegments); +coloredpath coloredpath(path p, pen[] pens=new pen[] {currentpen}, + int colortype=coloredSegments); +coloredpath coloredpath(path p, pen pen(real)); + In the second case, the pens are independent of the relative time. In +the third case, the array of pens contains only one pen, which depends +of the relative time. + + The casting of `path' to `coloredpath' allows the use of a `path' +instead of a `coloredpath'; in this case the shading behaviour is the +default shading behavior for a surface. + + An example of `tube' is provided in the file `trefoilknot.asy'. +Further examples can be found at +`http://piprim.tuxfamily.org/asymptote/tube/'. + + +File: asymptote.info, Node: flowchart, Next: contour, Prev: tube, Up: Base modules + +7.35 `flowchart' +================ + +This package provides routines for drawing flowcharts. The primary +structure is a `block', which represents a single block on the +flowchart. The following eight functions return a position on the +appropriate edge of the block, given picture transform `t': + +pair block.top(transform t=identity()); +pair block.left(transform t=identity()); +pair block.right(transform t=identity()); +pair block.bottom(transform t=identity()); +pair block.topleft(transform t=identity()); +pair block.topright(transform t=identity()); +pair block.bottomleft(transform t=identity()); +pair block.bottomright(transform t=identity()); + + +To obtain an arbitrary position along the boundary of the block in user +coordinates, use: +pair block.position(real x, transform t=identity()); + + +The center of the block in user coordinates is stored in `block.center' +and the block size in `PostScript' coordinates is given by `block.size'. + +A frame containing the block is returned by +frame block.draw(pen p=currentpen); + + + The following block generation routines accept a Label, string, or +frame for their object argument: + +"rectangular block with an optional header (and padding `dx' around header and body):" + block rectangle(object header, object body, pair center=(0,0), + pen headerpen=mediumgray, pen bodypen=invisible, + pen drawpen=currentpen, + real dx=3, real minheaderwidth=minblockwidth, + real minheaderheight=minblockwidth, + real minbodywidth=minblockheight, + real minbodyheight=minblockheight); + block rectangle(object body, pair center=(0,0), + pen fillpen=invisible, pen drawpen=currentpen, + real dx=3, real minwidth=minblockwidth, + real minheight=minblockheight); + +"diamond-shaped flowchart block:" + block diamond(object body, pair center=(0,0), + pen fillpen=invisible, pen drawpen=currentpen, + real ds=5, real dw=1, + real height=20, real minwidth=minblockwidth, + real minheight=minblockheight); + +"circular flowchart block:" + block circle(object body, pair center=(0,0), pen fillpen=invisible, + pen drawpen=currentpen, real dr=3, + real mindiameter=mincirclediameter); + +"rectangular flowchart block with rounded corners:" + block roundrectangle(object body, pair center=(0,0), + pen fillpen=invisible, pen drawpen=currentpen, + real ds=5, real dw=0, real minwidth=minblockwidth, + real minheight=minblockheight); + +"rectangular flowchart block with beveled edges:" + block bevel(object body, pair center=(0,0), pen fillpen=invisible, + pen drawpen=currentpen, real dh=5, real dw=5, + real minwidth=minblockwidth, real minheight=minblockheight); + + + To draw paths joining the pairs in `point' with right-angled lines, +use the routine: +path path(pair point[] ... flowdir dir[]); + The entries in `dir' identify whether successive segments between the +pairs specified by `point' should be drawn in the `Horizontal' or +`Vertical' direction. + + Here is a simple flowchart example: + +size(0,300); + +import flowchart; + +block block1=rectangle(Label("Example",magenta), + pack(Label("Start:",heavygreen),"",Label("$A:=0$",blue), + "$B:=1$"),(-0.5,3),palegreen,paleblue,red); +block block2=diamond(Label("Choice?",blue),(0,2),palegreen,red); +block block3=roundrectangle("Do something",(-1,1)); +block block4=bevel("Don't do something",(1,1)); +block block5=circle("End",(0,0)); + +draw(block1); +draw(block2); +draw(block3); +draw(block4); +draw(block5); + +add(new void(picture pic, transform t) { + draw(pic,path(new pair[]{block1.right(t),block2.top(t)},Horizontal), + Arrow,PenMargin); + draw(pic,Label("Yes",0.5,NW),path(new pair[]{block2.left(t),block3.top(t)}, + Horizontal),Arrow,PenMargin); + draw(pic,Label("No",0.5,NE),path(new pair[]{block2.right(t),block4.top(t)}, + Horizontal),Arrow,PenMargin); + draw(pic,path(new pair[]{block3.bottom(t),block5.left(t)},Vertical), + Arrow,PenMargin); + draw(pic,path(new pair[]{block4.bottom(t),block5.right(t)},Vertical), + Arrow,PenMargin); + }); + + + + +File: asymptote.info, Node: contour, Next: contour3, Prev: flowchart, Up: Base modules + +7.36 `contour' +============== + +This package draws contour lines. To construct contours corresponding +to the values in an array `c' for a function `f' on `box(a,b)', use +guide[][] contour(real f(real, real), pair a, pair b, + real[] c, int nx=ngraph, int ny=nx, + interpolate join=operator --); + The integers `nx' and `ny' define the resolution. The default +resolution, `ngraph x ngraph' (here `ngraph' defaults to `100'), can be +increased for greater accuracy. The default interpolation operator is +`operator --' (linear). Spline interpolation (`operator ..') may +produce smoother contours but it can also lead to overshooting. + + To construct contours for an array of data values on a uniform +two-dimensional lattice on `box(a,b)', use +guide[][] contour(real[][] f, real[][] midpoint=new real[][], + pair a, pair b, real[] c, + interpolate join=operator --); + + To construct contours for an array of data values on a nonoverlapping +regular mesh specified by the two-dimensional array `z', optionally +specifying the values of `f' at the mesh midpoints, use +guide[][] contour(pair[][] z, real[][] f, + real[][] midpoint=new real[][], real[] c, + interpolate join=operator --); + + To construct contours for an array of values `f' specified at +irregularly positioned points `z', use the routine +guide[][] contour(pair[] z, real[] f, real[] c, + interpolate join=operator --); + The contours themselves can be drawn with one of the routines +void draw(picture pic=currentpicture, Label[] L=new Label[], + guide[][] g, pen p=currentpen) + +void draw(picture pic=currentpicture, Label[] L=new Label[], + guide[][] g, pen[] p) + + The following simple example draws the contour at value `1' for the +function z=x^2+y^2, which is a unit circle: + +import contour; +size(75); + +real f(real a, real b) {return a^2+b^2;} +draw(contour(f,(-1,-1),(1,1),new real[] {1})); + + + +The next example draws and labels multiple contours for the function +z=x^2-y^2 with the resolution `100 x 100', using a dashed pen for +negative contours and a solid pen for positive (and zero) contours: + +import contour; + +size(200); + +real f(real x, real y) {return x^2-y^2;} +int n=10; +real[] c=new real[n]; +for(int i=0; i < n; ++i) c[i]=(i-n/2)/n; + +pen[] p=sequence(new pen(int i) { + return (c[i] >= 0 ? solid : dashed)+fontsize(6); + },c.length); + +Label[] Labels=sequence(new Label(int i) { + return Label(c[i] != 0 ? (string) c[i] : "",Relative(unitrand()),(0,0), + UnFill(1bp)); + },c.length); + +draw(Labels,contour(f,(-1,-1),(1,1),c),p); + + + +The next example illustrates how contour lines can be drawn on color +density images: + +import graph; +import palette; +import contour; + +size(10cm,10cm,IgnoreAspect); + +pair a=(0,0); +pair b=(2pi,2pi); + +real f(real x, real y) {return cos(x)*sin(y);} + +int N=200; +int Divs=10; +int divs=2; + +defaultpen(1bp); +pen Tickpen=black; +pen tickpen=gray+0.5*linewidth(currentpen); +pen[] Palette=BWRainbow(); + +scale(false); + +bounds range=image(f,Automatic,a,b,N,Palette); + +// Major contours + +real[] Cvals=uniform(range.min,range.max,Divs); +draw(contour(f,a,b,Cvals,N,operator --),Tickpen); + +// Minor contours +real[] cvals; +for(int i=0; i < Cvals.length-1; ++i) + cvals.append(uniform(Cvals[i],Cvals[i+1],divs)[1:divs]); +draw(contour(f,a,b,cvals,N,operator --),tickpen); + +xaxis("$x$",BottomTop,LeftTicks,above=true); +yaxis("$y$",LeftRight,RightTicks,above=true); + +palette("$f(x,y)$",range,point(NW)+(0,0.5),point(NE)+(0,1),Top,Palette, + PaletteTicks(N=Divs,n=divs,Tickpen,tickpen)); + + + +Finally, here is an example that illustrates the construction of +contours from irregularly spaced data: + +import contour; + +size(200); + +int n=100; + +pair[] points=new pair[n]; +real[] values=new real[n]; + +real f(real a, real b) {return a^2+b^2;} + +real r() {return 1.1*(rand()/randMax*2-1);} + +for(int i=0; i < n; ++i) { + points[i]=(r(),r()); + values[i]=f(points[i].x,points[i].y); +} + +draw(contour(points,values,new real[]{0.25,0.5,1},operator ..),blue); + + + +In the above example, the contours of irregularly spaced data are +constructed by first creating a triangular mesh from an array `z' of +pairs: + +int[][] triangulate(pair[] z); + +size(200); +int np=100; +pair[] points; + +real r() {return 1.2*(rand()/randMax*2-1);} + +for(int i=0; i < np; ++i) + points.push((r(),r())); + +int[][] trn=triangulate(points); + +for(int i=0; i < trn.length; ++i) { + draw(points[trn[i][0]]--points[trn[i][1]]); + draw(points[trn[i][1]]--points[trn[i][2]]); + draw(points[trn[i][2]]--points[trn[i][0]]); +} + +for(int i=0; i < np; ++i) + dot(points[i],red); + + + +The example `Gouraudcontour' illustrates how to produce color density +images over such irregular triangular meshes. `Asymptote' uses a +robust version of Paul Bourke's Delaunay triangulation algorithm based +on the public-domain exact arithmetic predicates written by Jonathan +Shewchuk. + + +File: asymptote.info, Node: contour3, Next: slopefield, Prev: contour, Up: Base modules + +7.37 `contour3' +=============== + +This package draws surfaces described as the null space of real-valued +functions of (x,y,z) or real[][][] matrices. Its usage is illustrated +in the example file `magnetic.asy'. + + +File: asymptote.info, Node: slopefield, Prev: contour3, Up: Base modules + +7.38 `slopefield' +================= + +To draw a slope field for the differential equation dy/dx=f(x,y) (or +dy/dx=f(x)), use: +picture slopefield(real f(real,real), pair a, pair b, + int nx=nmesh, int ny=nx, + real tickfactor=0.5, pen p=currentpen, + arrowbar arrow=None); + Here, the points `a' and `b' are the lower left and upper right +corners of the rectangle in which the slope field is to be drawn, `nx' +and `ny' are the respective number of ticks in the x and y directions, +`tickfactor' is the fraction of the minimum cell dimension to use for +drawing ticks, and `p' is the pen to use for drawing the slope fields. +The return value is a picture that can be added to `currentpicture' via +the `add(picture)' command. + + The function +path curve(pair c, real f(real,real), pair a, pair b); + takes a point (`c') and a slope field-defining function `f' and +returns, as a path, the curve passing through that point. The points +`a' and `b' represent the rectangular boundaries over which the curve +is interpolated. + + Both `slopefield' and `curve' alternatively accept a function `real +f(real)' that depends on x only, as seen in this example: + +import slopefield; + +size(200); + +real func(real x) {return 2x;} +add(slopefield(func,(-3,-3),(3,3),20,Arrow)); +draw(curve((0,0),func,(-3,-3),(3,3)),red); + + |