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ePiX 1.2

Table of Contents

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ePiX manual

This is the manual for ePiX, Version 1.2 Copyright © 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2017 Andrew D. Hwang


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1 Introduction

ePiX, a collection of batch utilities, creates mathematically accurate figures, plots, and animations containing LaTeX typography. The input syntax is easy to learn, and the user interface resembles that of LaTeX itself: You prepare a scene description in a text editor, then “compile” the input file into a picture. LaTeX- and web-compatible output types include a LaTeX picture-like environment written with PSTricks, tikz, or eepic macros; vector images (eps, ps, and pdf); and bitmapped images and movies (png, mng, and gif).

ePiX’s strengths include:

ePiX facilitates logical structuring of mathematical figures, as opposed to visual structuring (or WYSIWYG), analogous to the relationship between LaTeX and a word processor. Stylistic defaults streamline the creation of simple figures, but there are few internal restrictions on the contents or appearance of a figure; aesthetic and practical decisions are left to you.

If you are a:

This manual is relatively conversational, and occasionally redundant, especially between portions meant for readers at different levels of familiarity. Throughout, you are assumed to be familiar with LaTeX and basic linear algebra: the description of points, vectors, lines, and planes in three-dimensional space. Other material, such as C++ syntax, is introduced as needed.


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1.1 Software Dependencies

If you run GNU/Linux, a BSD, or Solaris, you probably have (and can surely install) all the external software needed to use ePiX. On Mac OS X, you will need the Apple developer tools and an X server (such as XQuartz), and the free fink package manager to build a GNU environment. For Windows, you’ll need to install Cygwin and several packages. Detailed instructions are given below.

“Under the hood”, an input file is successively converted to a LaTeX picture; dvi; PostScript, pdf or eps; and if desired, to a bitmapped image or movie. Four shell scripts—epix, laps, elaps, and flix—automate the various file format conversions.

ePiX consists of a C++ library, header, and shell scripts, and requires GNU bash and a compiler for normal use. For complete functionality, you need g++ (Version 3.2 or later), bash, a text editor (ePiX works particularly well with emacs), a LaTeX distribution, Ghostscript, gv (or your favorite PS/PDF previewer), and GraphicsMagick. GNU grep and sed are good to have. You may need additional “developer packages” (binutils, make) in order to build ePiX. The more up to date your software is, the better your experience is likely to be, but bleeding edge versions are not necessary, or even always desirable.

Aside from their reliance on specific programs, ePiX’s shell scripts are written using Unix-style pathnames. Thus, the most straightforward way to use ePiX is to install a GNU environment.

Jay Belanger’s emacs mode allows you to write, compile, and view ePiX figures without leaving emacs. If you use another editor, you’ll want to create template source files so you don’t have to type boilerplate code each time you write a new figure.


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1.1.1 Setting up an Environment Under Windows

Cygwin can be used to run ePiX under Windows. Download setup.exe from http://www.cygwin.com, then install the packages you need. The following are recommended, and sufficient for the actions described in this manual.

(Archive)    bzip2, tar
(Devel)      binutils, coreutils, gcc, gcc-g++, make, sed
(Editors)    emacs, emacs-X11, vim
(Graphics)   GraphicsMagick, ghostscript-base, ghostscript-x11, gv
(Publishing) tetex (all)
(Shells)     bash, bash-completion
(X11)        X-start-menu-icons, X-startup-scripts,
             XFree86-lib-compat, xorg-x11-fscl, xorg-x11-fsrv

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1.2 Installation

ePiX is distributed over the World-Wide Web as source code. Packages may be found at http://mathcs.holycross.edu/~ahwang/epix/ePiX.html The latest stable release is also on the CTAN mirrors, in the graphics directory. (Some users of Red Hat have reported file permission problems when unpacking the CTAN tarballs. If you encounter this difficulty, please try downloading the sources from the project main page.) Unpack the compressed tar file with the appropriate command:

  tar -zxvf epix-x.y.z.tar.gz
  tar -jxvf epix-x.y.z.tar.bz2

(x.y.z is the version number) or, if your tar doesn’t do decompression,

  gunzip -c epix-x.y.z.tar.gz | tar -xvf -
  bzcat epix-x.y.z.tar.bz2 | tar -xvf -

cd to the source directory, epix-x.y.z. The INSTALL file contains detailed installation instructions. If you’re impatient, the short of it is ./configure [--options]; make; make install. Run ./configure --help for a list of options.

By default, ePiX installs in subdirectories of /usr/local; if you want to install elsewhere, supply ./configure with the appropriate --prefix. You may also want to consult POST-INSTALL for information on setting your PATH variable so your shell can find ePiX. The manual and sample files are in /usr/local/share/doc/epix.


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1.2.1 Development

There are two mailing lists, one for user questions, one for development discussion. Please visit http://savannah.nongnu.org/mail/?group=epix to subscribe.


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2 Getting Started

This chapter describes the basics of creating figures in ePiX for readers familiar with LaTeX but completely new to C++. No detailed knowledge of C++ is needed to use ePiX, only a bit of grammar that is easily absorbed by example.

Section See section-running, describes the commands (shell scripts) comprising ePiX, and explains how to set up a graphical environment using standard *nix programs. Section See section-overview, briefly describes figure creation. Section See section-tutorial, presents a few files side-by-side with their output, and should be read at a computer so you can run the exercises.


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2.1 Running ePiX

An “input file” is a human-written figure specification containing ePiX commands; an “output file” is either a LaTeX picture-like environment or a stand-alone vector graphic. Conversion (“compiling” a figure) is accomplished with four shell scripts, laps, epix, elaps, and flix. Each script has a preferred extension for its input files, and is invoked with a command of the form

  <script> [options] <input file(s)>

Often, no options are necessary. <script> --help describes <script>’s options.

By default, output file names are constructed by replacing the input extension with the (preferred) output extension. For brevity, extensions may be omitted. If the script has doubts about your intent, it proceeds with default behavior and prints a warning message.

The author is a great fan of TAB completion, under which a shell, based on what has been typed so far, automatically fills in a command when the TAB key is pressed. ePiX comes with code snippets that complete intelligently when the first part of a command is one of the shell scripts. For example, if command completion is active, typing epix TAB prints only names of epix input files. To use this feature, you must install Ian MacDonald’s bash completion package. The INSTALL and POST_INSTALL files contain details.

Shell Scripts

laps

laps performs LaTeX to PostScript/PDF conversion, and is independent of the rest of ePiX. By default, laps invokes LaTeX and dvips. The option --pdf creates a PDF file by post-processing the PostScript with ps2pdf. Other TeX-family processors (pslatex, pdftex, etc.) may be used instead of LaTeX by invoking laps with an appropriate option.

epix

epix compiles an input file into a LaTeX picture. The input file should have extension xp (for eXtended Picture). Jay Belanger’s emacs mode is Highly Recommended; the installation procedure can be found in the POST_INSTALL file. When this mode is active, emacs automatically inserts a preamble template when a new xp file is created. In addition, you can peruse ePiX’s info file, or format, compile, and preview files, all from within emacs.

Output files from the script epix have extension eepic, after Conrad Kwok’s enhancements to the LaTeX picture environment. In Version 1.2, however, an output file may contain PSTricks or tikz macros instead of eepic macros. A command such as [Image] incorporates an output file into a LaTeX document. The preamble must contain usepackage lines appropriate to the output format.

elaps

elaps creates stand-alone vector images (eps or pdf) from epix input files or eepic figures, even those not produced with epix. elaps automatically loads the LaTeX packages needed for features described in this manual. Additional LaTeX packages and dvips options may be specified on the command line. An elaps output file is placed in a LaTeX document with an \includegraphics command; the preamble must contain \usepackage{graphicx} [sic].

flix

flix creates bitmapped images and movies, Section See section-animation, . Input files should have extension flx, and must contain a valid epix header as well as additional code. Jay Belanger’s emacs mode facilitates creation of flix files.


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2.2 The Drawing Model

To draw a figure manually, you select a sheet of paper of appropriate size and add paths, markers, and labels. These scene elements have attributes (line color, line width, fill color, font size, etc.)\ affecting their appearance.

ePiX’s drawing model behaves similarly. A picture command sets the canvas (or logical drawing area, a Cartesian rectangle) and the true size of the final figure. A begin command initializes the “virtual paper” for drawing. Subsequent drawing commands add objects to the scene: lines, curves, function plots, labels, and the like. The printed appearance of scene elements is determined by the current “attribute state” and controlled by style declaration which remain in effect until superseded. Finally, an end command closes the figure and prints it to the output file.

In this manual, command descriptions follow a few conventions. Optional arguments are enclosed in square brackets. A len argument is either a number (representing a length in pt) or a double-quoted string containing a number and a two-letter LaTeX length unit, such as "1.5pt" or "6cm". A color argument is a named primary (Red(), Cyan(), White(), etc.), or a Color specified by densities (RGB(r,g,b), CMY(c,m,y), etc.)

\noindent A complete list of style commands is found in Section See section-quickref, , starting on page section-quickref.

To handle three-dimensional scenes, a camera performs point projection from a selected spatial location to the canvas. As a user, you’ll need to control relatively few of the camera’s parameters. Keep in mind, however, that elements are added to a scene in the same order their commands appear in the output file, and that later elements generally cover earlier ones. Some three-dimensional scenes require manual ordering of the input file; such ordering is dependent on the camera’s location.

By default, ePiX draws thin, solid, black lines, performs no filling of regions, prints text in a 12 pt Roman font, and looks down the z axis from a large distance, giving orthogonal projection on the (x,y) plane. When the camera is moved, the z axis points vertically up on the page.


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2.3 Tutorial

This section presents sample input files side-by-side with their output so you can compare what you write with what you’ll see on the screen or page. ePiX provides standard drawing capabilities, but like all software has its own idioms and personality. The basic syntax, which comes from C/C++, should be mostly self-explanatory. One-line comments begin with the string “//”.

To use the sample files interactively, you’ll need working software: bash, emacs, ePiX, g++, gv, and a running X server. To complete your “GUI”, start gv and select “Watch file” from the “State” menu. The loaded file will update automatically when it changes.

Create a “scratch” directory, cd into it, and run the command

  tar -zxf /usr/local/share/doc/epix/sample_src.tar.gz

(Change the path as appropriate for your installation.) This unpacks copies of the sample files into your scratch directory, including all the files mentioned in this manual. The README file serves as a table of contents.

Open a sample file in emacs, compile it from the drop-down menu (or with the keyboard command), then open the EPS file in gv. Now you’re ready to follow the tutorial interactively. A few suggested exercises are included with each file. Naturally, as you study more files, you’ll be able to make more interesting changes on your own.

Basic Drawing

The first sample, hello.xp, contains code needed to specify the figure’s size, followed by the classic greeting. The border command draws a box around the figure in the specified color and width, and serves here merely to delimit the output from the surrounding page.

/* -*-ePiX-*- */
#include "epix.h"      // These lines are analogous
using namespace ePiX;  // to a usepackage command.

int main()
{
  picture(P(-1,-1), P(1,1), "2 x 1in"); // corners, true size

  begin(); // ---- Figure body starts here ----

  border(Black(), "1pt"); // color, line width

  font_size("Huge");  // May be any font size, e.g. "scriptsize"
  label(P(0,0), "Hello, world!");

  end();   // ---- End figure; write output file ----
}

[Image]

Geometric Objects

Our next file uses simple objects to draw a 2-D house-and-sun scene.

int main()
{
  picture(P(0,0), P(5,2), "3.75 x 1.5in");

  begin();

  triangle(P(0.9, 1), P(3.1, 1), P(2, 1.5)); // vertices
  circle(P(4,1.5), 0.25); // center and radius

  rect(P(1,0), P(3,1)); // opposite corners

  fill(Black(0.1));     // light gray
  rect(P(2.3,0), P(2.7,0.8)); // the door
  nofill();

  bold();               // draw thicker lines
  grid(P(1.4, 0.2), P(2, 0.6), 2, 2); // corners, number of squares

  line(P(xmin(), 0), P(xmax(), 0));   // endpoints

  end();
}

[Image]

\noindent The sample file house.flx uses loops to draw gradient fills of the lawn, sky, and sun, and animates a sunset.

Function Plotting

Plotted functions must be defined in the “preamble”, before main. “High-level” elements—coordinate axes and grids, axis labels, and graphs—are drawn with mnemonically-named commands.

// double = double-precision floating point
double f(double x) { return 0.75*Sin(x) - 0.25*Sin(2*x); }

int main()
{
  picture(P(-2,-1), P(2,1), "5 x 1in"); // [-2,2] x [-1,1]

  begin();
  revolutions(); // set angle units, [0,1] = one turn

  h_axis(16); // axes w/default endpts
  v_axis(4);
  h_axis_labels(4, P(0,-4), b); // shift down 4pt, align below

  plain(Red());
  plot(f, xmin(), xmax(), 120); // use 120 intervals

  end();
}

[Image]

Multivariable Plotting

Functions of two or three variables are defined just like functions of one variable, but for plotting the return type must be a point (P), not a real number (double). The domain class specifies the set of inputs to plot.

P f(double r, double th)
{
  return P(r*Cos(th), r*Sin(th), pow(r, 3)*Cos(3*th));
}

int main()
{
  picture(P(-1,-1), P(1, 1), "2 x 1in");
  begin();
  pst_format();     // use PSTricks macros for output

  // corners ([0,1] x [0,2\pi]) and fineness
  // (8x40 rectangles, plotted at 16x120 resolution)
  domain R(P(0,0), P(1, 2*M_PI), mesh(8,40), mesh(16,120));

  camera.at(P(3,1,2)); // set the viewpoint
  arrow(P(0,0,0), P(1.25,0,0)); // coordinate axes
  arrow(P(0,0,0), P(0,1.25,0));

  plain(Blue(1.2)); // line color and width
  fill(Yellow());   // shading color
  surface(f, R);

  end();
}

[Image]

Loops and Control Structures

A function can be defined by an arbitrary algorithm, and a domain may be used to plot a family of functions for several values of one variable.

P sin_n(double x, double n)     // Taylor polynomial of sin x
{
  const int N((int) floor(n));  // convert n to an index bound
  const double sqx(-pow(x, 2)); // -x^2
  double val(x), summand(x);

  for (int i=1; i <= 2*N+1; i += 2)
    {
      summand *= (sqx/((i+1)*(i+2)));   // (-1)^i x^{2i+1}/(2i+1)!
      val += summand;
    }

  return P(x, val);    // return (x, y)
}

int main()
{
  picture(P(0, -1), P(6*M_PI, 1), "5 x 1in");

  begin();

  set_crop();
  bold(Green());
  plot(Sin, 0, xmax(), 120);

  domain R(P(0,1), P(6*M_PI, 41), mesh(60, 40), mesh(120, 40));
  for (int i=3; 0 <= i; --i)  // print in descending degree
    {
      bold(RGB(0.25*i, 0, 1-0.25*i)); // degree-dependent color
      plot(sin_n, R.slice2(5*i+1));   // plot for n = 5i+1
    }
  end();
}

[Image]

Page Layout

Page layout can be composed from sub-pages with screen objects. In the loop body below, objects are added to the “active” screen, then inset into the “canvas”, the screen representing the entire figure.

P f(double u, double v)
{
  return P((u-v)*(u+v), 2*u*v, u);
}

int main()
{
  picture(P(0,0), P(2,3), "5x7.5in"); // overall size

  begin();

  domain R(P(-1,-1), P(1,1), mesh(12,12), mesh(24,24));

  for (int i=0; i<2; ++i)
    for (int j=0; j<3; ++j)
      {
        screen my(P(-3,-3), P(3,3));
        activate(my);
        border(Red(0.6), "1pt");
        // frame-dependent viewpoint
        camera.at(sph(10, (2*j+i+1)*M_PI/8, M_PI/6));

        plot(f, R);
        arrow(P(0,0,0), 2*E_1);
        arrow(P(0,0,0), 2*E_2);
        // SW corner at (i, 2-j), padded by 0.05 on all sides
        inset(P(i+0.05,2.05-j), P(i+0.95,2.95-j));
      }

  end();
}

[Image]


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2.4 C++ Basics

An ePiX source file is a C++ program. If you’ve successfully modified and compiled the sample files, you know enough C++ to use ePiX. In the author’s experience, C grammar suffices for most applications. An excellent introduction to definitions of functions and variables, control statements, and overall program structure is Kernighan and Ritchie’s The C Programming Language, second edition KnR.


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2.4.1 File Format

Jay Belanger’s emacs mode for ePiX inserts a file template when an empty buffer is opened with the extension xp. This section explains the purposes served by the template. A few additional remarks may help you avoid basic syntax pitfalls.

A C++ file consists of “statements”, analogous to ordinary sentences. Common types include declarations (which “register” a function, variable, or type name with the compiler), definitions (which assign meaning to declared names), and function calls (which cause a named function to execute). Most statements in an ePiX input file are function calls (“commands”). Plain declarations are relatively rare in user files, since a definition serves to declare any new names that it contains.

Every statement ends with a semicolon, and conventionally a file contains at most one statement per line. The compiler ignores nearly all whitespace (spaces, tabs, and newlines), which should be used freely to make files easy to read. Other punctuation (periods, commas, (semi)colons, parentheses, braces, and quotes) dictates file parsing, and must adhere stringently to grammar.

An ePiX file always begins with the lines

#include "epix.h" // N.B. pre-processor directive, no semicolon
using namespace ePiX;

The first line is analogous to a LaTeX usepackage command: It loads the contents of the “header” file epix.h, importing the names of commands provided by ePiX. To avoid name conflicts, ePiX’s commands are enclosed in a “namespace”. For example, the label command is actually known to the compiler as ePiX::label. The second line above tells the compiler to apply the prefix tacitly.


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2.4.2 Variables and Functions

Definitions of variables and functions play the same role in a figure that macro definitions play in a LaTeX document: gathering and organizing information on which the figure depends. A variable is defined by supplying its type, name, and initial value. By far the most common data types in ePiX are double (double-precision floating point number), P, and int. The name of a variable may only consist of letters (including the underscore character) and digits, and must begin with a letter:

my_var, var2,    MY_var, aLongVariableName;  // valid
my-var, 2var, \v@@riable, $x, ${MY_VARIABLE}; // not valid

Variable names are case-sensitive, and numerous (non-universal) conventions govern the significance of capitalization. Generally, make names descriptive but not unwieldy, and avoid language keywords (such as const, true, double, class, or public) and names that begin with an underscore.

A function accepts “arguments” and “returns a value”. To define a function in C++, specify the return type, the name of the function, the types of the arguments, and the algorithm by which the value is computed from the inputs. The code block

double f(double x) 
{ 
  return sqrt(1-x*x); 
}

defines the double-valued function f of one double variable defined by the formula f(x)=\sqrt{1-x^2}.


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2.4.3 Comments

C++ has two types of comments. C-style comments, which may span several lines, are delimited by the strings /* and */. One-line comments, analogous to the LaTeX \%, are begun with //. A one-line comment may appear within a multi-line comment, but a C-style comment may not; the compiler will mistake the first */ it encounters as the end of the current multi-line comment.


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2.4.4 Program Execution

All the “action” in a C++ program occurs inside the special function main. Running a compiled C++ program is viewed by the operating system as calling the program’s main function. The return value (an int) is the program’s exit status. The contents of the output file start with begin() and terminate with end(). Intervening statements constitute the body of the file.

In C++, a function may not be defined inside another function. Variables may be defined inside main, but functions cannot be.


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2.4.5 Strings and Raw Output

In C++, a string is a sequence of characters. Most strings in ePiX input files are literals, double-quoted strings whose value is read from the input. In a string literal, backslash is an escape character; a single backslash is produced by a double backslash in the input file. Certain letters have special meanings when backslash-escaped, including “\n” (newline) and “\t” (TAB). Unlike LaTeX, C++ does not require a space to separate an escape sequence from following text; the string “\\textwidth” literally represents a LaTeX command, while “\textwidth” is read “TABextwidth” by the compiler.

Though not commonly needed, raw text can be printed to the output file. The functions write, pre_write, and post_write accept string arguments. write prints its argument where the call appears in the input file. The other functions print their arguments before or after the completed picture, respectively. These commands must be in the file body. As an application, a complete LaTeX figure environment (with caption and label) can be produced by an ePiX file, Figure See fig-captioned, .

#include "epix.h"
using namespace ePiX;

int main()
{
  picture(P(-1,-1), P(1,1), "10cm x 3in);
  begin();
  pre_write("\@float Figure[hbt]"); // comes before the picture

  post_write("\@caption{A \@code{LaTeX}\ figure.}"); // and after
  post_write("\@end float");

  < ... other ePiX commands ... >
  write("%% A comment near the end, but inside the picture.");
  end();
} // End of main()

Figure: Generating a self-contained figure in ePiX.


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2.4.6 Conditionals and Loops

An algorithm’s behavior usually depends on internal state. A conditional statement causes blocks of code to be executed according to criteria. A loop repeatedly executes a code block, usually changing the values of variables in a predictable way, so that the loop exits after finitely many traversals.


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2.5 Animation

ePiX is well-suited to the creation of mathematically accurate animations: If a figure depends suitably upon a “time” parameter, then a loop can be used to draw the entire figure for multiple time values, yielding successive “snapshots” of the figure as time progresses. The shell script flix automates the process of compiling a suitable input file into a collection of pngs and assembling these frames into a mng or gif animation. GraphicsMagick is the image-handling engine.

A flix file is an epix file with two restrictions:

Jay Belanger’s emacs mode recognizes the file extension .flx and inserts template code if an empty buffer is opened. Creation of flix files is as easy as creation of epix files. The samples directory contains a handful of flix files that may be consulted for ideas.

By default, flix creates movies with 24 frames, in which tix() runs from 0 to 1, and animates at 0.08 sec/frame. These and other parameters can be changed with command-line options.

A “typical” .flx file may take 30 seconds to a few minutes to compile, depending on the number of frames and the complexity of each frame. To facilitate debugging, elaps can be run on a flix file. elaps runs in a fraction of the time, and if elaps can’t produce a viewable image, flix will surely fail.


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2.6 Layout Tricks


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2.6.1 Stereograms

Stereograms are created by drawing a single scene twice from slightly different points of view. To create a stereogram with ePiX, write a “scene” function containing the necessary drawing commands, then set the camera appropriately and call the scene function twice, once for each frame.

Crossed-Eyes Stereograms

The sample files lorenz.xp and twisted_cubic.xp use page layout to draw crossed-eyes stereograms. For each frame, create and activate a screen, set the camera, and call the scene function. The frames are inset side by side in the final picture. For a crossed-eyes stereogram, the frame from the “rightmost” camera position is placed on the left.

Bi-Color Stereograms

The sample files cube.flx and mirrorball.flx contain animated red-cyan stereograms, suitable for viewing with 3-D glasses. The technique works best with black-and-white line drawings, but the basic approach is the same as for crossed-eyes stereograms: Create a scene function and call it twice, setting the camera appropriately. Layout is unnecessary since the frames are superposed. Pen colors should be close to red and cyan, but “optimal” choices depend on one’s eyes and 3-D glasses.

Stereographic Movies

Either form of stereogram can be animated in a .flx file, but keeping one’s eyes properly crossed requires practice while most people can instantly see depth with 3-D glasses. As always when using flix, debug the scene with elaps before compiling a movie. If a stereoscopic effect is difficult to see in a still image, it will be even harder to perceive in a movie.


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2.6.2 Inset Images

ePiX can place external images into a figure, similarly to Rolf Niepraschk’s overpic package. You provide the Cartesian center point, the file name, and optionally the true height and/or width of the image:

  label(P(a,b), "\\includegraphics[width=w,height=h]{file}");

This effect requires the graphicx package. When compiling a stand-alone graphic containing an external image, you must specify the graphicx package on the command line: elaps -p graphicx <file>


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3 Reference Manual

This chapter details ePiX’s capabilities, discussing attributes and data types, and listing available commands in the form

  label(P posn, [P offset], string text, [align]);
  circle(P ctr, double rad=1, normal=P(0,0,1));

As in Chapter See chapter-started, , function arguments are given by type (P, double, string, etc.)\ and name, or by name alone if the type is clear. Optional arguments are enclosed in square brackets. A few argument types, such as [align] above, admit only a small number of values; these cases are explained when the corresponding command is first introduced.

A name followed by an equals sign and a value indicates a default argument; if omitted in an input file, the compiler substitutes the default value. Only trailing arguments may be specified this way. For example, if the rad argument of the circle command is omitted, the normal argument must be omitted as well.

In an input file, only an argument’s value is given, not the type:

  label(P(0,0), P(0,-4), "Hello world", b);
  circle(P(0,-0.25), sqrt(2)); // use default normal

Generally, len signifies either a number (representing a length in pt) or a double-quoted string containing a number and a two-letter LaTeX length unit, such as "1.5pt" or "6cm". color represents a Color object constructed from a named primary (Blue(), Magenta(0.7), etc.), a color specification (RGB(r,g,b), CMY(c,m,y), etc.), or an operator applied to an existing Color.


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3.1 File Structure

An ePiX input file constitutes a short C++ program. When this program is compiled and run by one of the shell scripts, it creates a figure file suitable for inclusion in LaTeX.

Like a LaTeX document, an ePiX file contains a preamble, which sets up a drawing environment, and a body, which contains actual figure-generating commands. The minimal file has the form

#include "epix.h"  // N.B. no semicolon
using namespace ePiX;

int main() {
  picture(P(a1,b1), P(a2,b2), "n1 [unit1] x n2 unit2");
  begin(); // end of preamble, start of body
  end();   // end of body
}

ePiX commands are of four general types: drawing, attribute setting, definitions (of data and functions), and operations on existing objects. Except as noted below, drawing and attribute commands must appear in the body, between begin() and end(). Function definitions must come in the preamble, before main(). Data definitions may appear in the preamble or body.

Output Format

The end() command writes the output file to stdout, using eepic macros by default. The shell scripts redirect stdout to an appropriate disk file.

The attribute-setting command pst_format() causes the output file to be written using PSTricks macros when end is called. Similarly, tikz_format() causes the file to be written using tikz macros, and eepic_format() causes the file to be written using eepic macros. These commands may appear anywhere in the figure body. There is no reason for a file to contain more than one such command.

The output format may be selected on the command line, overriding any explicit request in the input file. Supplying epix, elaps, or flix with one of the options --pst, --tikz, or --eepic is tantamount to issuing a pst_format() (etc.)\ command just before the end of the file. These options are listed in decreasing precedence. If more than one is given, the “strongest” applies, regardless of the command line order.

A figure may be written directly to a specified disk file, in a format unaffected by the command line flags above. The command

  print_eepic("file.tex");

writes the figure to the named file using eepic macros. Analogous commands exist for the formats pst and tikz. These commands perform an immediate action. Consequently, the command’s location in the input file is significant, and multiple commands may appear in a single file, so long as distinct file name arguments are provided. Applications include writing the same figure in multiple formats, or creating successive “snapshots” of a lengthy computation.


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3.2 Picture Size and Aspect Ratio

The picture command specifies a figure’s logical and true sizes. In the “minimal file” snippet above, the canvas is the Cartesian rectangle [a_1,a_2]\times[b_1,b_2] whose corners are given. Either pair of opposite corners is acceptable, but confusion is less likely when the SW and NE corners are given, in this order.

The true (printed) width and height are read from picture’s third argument, a “size string” such as "4cm x 1in", containing a number and optional LaTeX length unit, an x, another number, and a mandatory length unit. Spaces may be used for legibility. The mandatory length unit becomes the LaTeX unitlength. The argument "4cm x 1in" creates a picture 4 cm wide and 1 in high, but internally converts 4 cm to inches. Recognized length units are pt (points, the default), cm (centimeters), in (inches), mm (millimeters), pc (picas), and bp (big points). (1in = 2.54cm = 72bp = 72.27pt, 1pc = 12pt.)

The logical and true sizes may be defined separately with the commands

  bounding_box(P(a1,b1), P(a2,b2));
  picture(n1, n2);
  unitlength(len); // len a length string, not a double

One way or another, the logical and true dimensions must have positive values when the begin() command is issued.

LaTeX treats the contents of a picture environment as a single box, aligned by default on its lower left corner. An offset command accepts two double arguments or a size string and shifts the page location accordingly. The command offset("0.25 x -0.5cm") shifts the picture right 0.25 cm and down 0.5 cm. If the unitlength is 1 cm, the command offset(0.25, -0.5) has the same effect.

A non-zero offset causes a picture’s contents to appear in a location where LaTeX does not expect them. This can be useful in a LaTeX document, but should be avoided when compiling a stand-alone image, since dvips may crop according to rules of its own.

The canvas’s corners are (xmin(),ymin()) and (xmax(),ymax()), while its width and height are xsize() and ysize(). The canvas is a virtual, advisory data structure; its dimensions are not directly related to the figure’s printed size, and picture elements may lie outside the canvas. Affine scaling maps the canvas to the page when the output file is written.

[Image]

The figure’s aspect ratio is controlled by sizing the canvas. The aspect ratio is “true” if the canvas and page rectangles are geometrically similar, e.g., if both boxes are 1.5 times as wide as they are tall.


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3.3 Color

ePiX provides a Color data type. Four models are implemented: RGB, CMY, CMYK, and Gray. A Color holds primary color “channels”, each carrying an intensity between 0 (no color) and 1 (full saturation). “No color” means black in RGB and Gray, white in CMY(K). RGB, CMY, and CMYK colors are written to the output file as commands in the corresponding model. Gray shades are written in RGB.

Operations on red-green-blue colors are described below. Functionally, colors are converted to RGB, operated upon, then converted back to the original model. Conversions are as described in Uwe Kern’s xcolor manual xcolor.


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3.3.1 Constructors

Each color model has a “constructor” creating a color of specified densities. The RGB and CMY models have named “primary” constructors; the density argument is optional and defaults to 1.

  // red-green-blue colors
  RGB(double r=0, double g=0, double b=0);
  Red(d=1);   Green(d=1);   Blue(d=1);
  White(d=1);   Black(d=1);
  // cyan-magenta-yellow colors
  CMY(double c=0, double m=0, double y=0);
  Cyan(d=1);   Magenta(d=1);   Yellow(d=1);
  CMY_White(d=1);   CMY_Black(d=1);

  // gray
  Gray(double d=0); // equivalent to RGB(d, d, d)

Each CMY constructor has a corresponding CMYK function, e.g.\ CyanK() or CMYK_White().

Though color densities lie between 0 and 1, ePiX’s primary color constructors take arguments mod 4, viewed as elements of [-2,2]. Consider Red(d), “red with density d”. For integer values of d, the constructor has the following meanings: Red(0) is black, Red(1) is red, Red(2)=Red(-2) is white, and Red(-1) is anti-red, or cyan. For non-integer d, the constructor interpolates between the bracketing integer values, Figure See fig-densities, . Other primary constructors work analogously.

[Image]

Figure[hbt]: The red primary constructor.

Let \delta:[-2,2]\to[0,1] be the piecewise-linear function that converts real numbers to primary color densities, \clip:{R}\to[0,1] the clipping function. The non-primary constructor RGB(r, g, b) computes the density of the red channel as \clip\bigl(\delta(r)+\delta(-g)+\delta(-b)\bigr); the green and blue channel densities are computed similarly. Every color can be created with arguments between 0 and 1, but the constructor accepts arbitrary real arguments and returns colors varying “continuously and periodically”.

Colors possess an “alpha-channel” for transparency. Except as supported by the output format, this feature is a stub.


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3.3.2 Color Operations

Colors can be scaled, inverted, blended, superposed, and filtered. In the code below, tint is a Color. The effect of each operation is described in the rgb model; the visual result is the same in all models.

  tint *= double c; // multiply channels by c, rebuild
  tint.invert();    // (r, g, b) -> (1-r, 1-g, 1-b)
  tint.blend(Color col, double t);  // (1-t)*tint + t*col
  tint.superpose(col); // add channels, then clip to [0,1]
  tint.alpha(double d); // set alpha channel to clip(d)

  tint.filter(col); // return min density in each channel

Except for filter, these operators modify their object. filter simulates the effect of viewing col through a transparent sheet of tint, and returns a new Color object having the same model as tint without modifying tint itself.


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3.4 Scene Attributes

Objects in a scene sit in 3-dimensional space. A camera maps objects to the active screen. The active screen may, in turn crop its contents. Each drawing command creates an object, “photographs” it, and adds the image to the active screen. This section describes the Camera and screen classes and a few associated concepts.


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3.4.1 Angular Mode

By default, angles are measured in radians. Two other angular modes are available: degrees and revolutions. The angular mode is set with a named command, radians(), degrees(), or revolutions(). The current angular mode affects all trigonometric functions and operations.


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3.4.2 The Camera

Three-dimensional scenes are drawn on flat paper by applying a mathematical transformation. By default, ePiX uses point projection, the technique used by art students when they trace on a window with grease pencil, Figure See fig-camera, .

[Image]

Figure[hbt]: Point projection.

ePiX depicts a Cartesian world by projecting mathematically to a screen plane, then affinely scaling to a printed page. The camera, which maps the world to the screen, consists of a body (data that determines the position and orientation of the camera), a lens (the actual mapping to the screen plane), and a filter (a color through which the scene is viewed).

Body

The camera’s spatial orientation is described by a triple of mutually perpendicular unit vectors. In memory of happy days at the beach, these vectors are called sea, sky, and eye. The screen plane is parallel to the sea-sky plane; the sea vector points horizontally to the right, sky points vertically upward. The eye is their cross product, which points directly at the viewer.

The sea-sky-eye basis is located at the viewpoint, the camera’s spatial location. The target, the origin of the screen plane, lies on the line through the viewpoint in the direction opposite the eye vector. The distance from the viewpoint to the target is the range. The camera sphere is centered at the viewpoint and passes through the target. The orientation, viewpoint, target, and range completely (and redundantly) determine the camera’s geometric situation in the world.

The Lens

A lens maps the world to the screen. ePiX comes with four lenses: shadow (the default), orthogonal, fisheye, and bubble. Each lens simulates the appearance of world objects as seen by an observer at the viewpoint. The shadow lens is point projection from the viewpoint to the screen plane. The orthogonal lens projects from infinite distance. Each of the other lenses performs radial projection to the camera sphere, then maps the sphere to the screen plane; the fisheye lens does orthogonal projection (so the entire image lies inside the disk centered at the target whose radius is the range) while the bubble lens does stereographic projection from the target’s antipode.

An input file normally uses a single camera, so a global object named camera is defined automatically. At the start of a figure, the camera looks down on the (x_1,x_2)-plane from a distant point on the x_3-axis. The resulting view, essentially projection along the axis, is suitable for 2-dimensional figures.

The camera is manipulated similarly to a real camera:

  camera.at(P posn);          // set viewpoint to posn
  camera.look_at(P targ);     // set target to targ
  camera.range(double dist);  // fix target, move viewpoint
  camera.focus(double dist);  // fix viewpoint, move target
  camera.tilt(double angle);  // rotate about the sea axis
  camera.pan(double angle);   // rotate about the sky axis
  camera.roll(double angle);  // rotate about the eye axis
  camera.clip_range(dist);    // cull scenery closer than dist

Explicitly setting the viewpoint or target makes the sky vector parallel to the projection of the x_3 axis when possible; otherwise the x_2-axis is used. Setting the range or focus moves the camera parallel along the eye line. Each command re-sizes the image; note that increasing the focus enlarges the image. The three rotation operations fix the viewpoint, but only roll fixes the target.

Filtering and Color Separation

The camera has a filter through which all scene Colors pass. For uniformity, the filter should only be set at the start of the file body, before any visible elements have been placed in the scene. The filter has two primary uses—model conversion and color separation—but can also be used for special effects.

There is a Neutral color, for which filtering has no effect at all. The camera’s default filter is Neutral. In addition, each color model has a neutral representative, RGB_Neutral, etc. Passing Colors through a neutral filter preserves their appearance but converts them to the neutral Color’s model.

Filtering can be used to “split” a Color into primary constituents. For RGB channels, the primaries themselves are suitable filters. For CMYK channels, there are special process filters, named C_Process, etc.

  camera.filter(CMY_Neutral());  // convert all colors to CMY
  camera.filter(Gray_Neutral()); // black and white photography
  camera.filter(Green());        // the green channel
  camera.filter(M_Process());    // the magenta channel
  camera.filter(Red(1.4));       // la vie en rose?

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3.4.3 Clipping

Two operations handle elements lying far from the target. Cropping culls elements based on their screen location, and is described later. Clipping removes objects whose spatial location lies outside the “clip region”.

Initially, the clip region is a very large box centered at the origin. Commands are provided to resize this box, keeping the faces parallel to coordinate planes.

  clip_box(P pt1, P pt2); // opposite corners
  clip_box(P pt);         // opposite corners pt and -pt
  clip_to (P pt);         //                  pt and P(0,0,0)
  clip_box();             // very large box

Additional “clippers” may be added manually. Individual clipping planes are specified by a point and inward-pointing normal vector. Parallel planes cutting a “slice” or “slab” are described by a location, direction, and a distance. The location lies halfway between the clipping planes, and the planes’ normals point toward the location.

  clip_face(P loc, P perp); // perp points inward
  clip_slice(loc, perp);    // very close parallel planes
  clip_slice(loc, perp, dist); // planes separated by dist
  clip_restore();  // remove manually-added clip planes

Like clip_restore, the clip_box and clip_to commands remove all user-specified half spaces.


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3.4.4 Screens and Page Layout

By default, drawing occurs in the Cartesian coordinate system of the canvas. However, insets and subfigures are most naturally composed in “local” coordinates, then placed into their final location as a unit, a process called layout. ePiX implements layout with the screen class.

Screens

A screen is a Cartesian plane containing a distinguished coordinate rectangle, its bounding box. The canvas of a figure is a screen, as is the internal representation of the printed page. At any point of a file body, some screen is active, or “open for drawing”. A new screen is created from a pair of opposite corners, but is not used until explicitly activated.

  screen subfig(P(a1,b1), P(a2,b2)); // [a1, a2] x [b1, b2]
  activate(subfig);   // open subfig for drawing
  // commands to draw in subfig
  deactivate(subfig); // re-activate the previous screen

The canvas is automatically drawn at the end of the input file. The contents of other screens must be incorporated into the canvas explicitly with an import or inset command. import affinely scales the bounding box of the “child” screen to the canvas. inset specifies corners in the “parent” where the child is placed.

  import();           // active screen to canvas
  import(subfig);     // subfig to canvas
  inset(P sw, P ne);  // active screen to specified rectangle
  inset(subfig, sw, ne); // subfig to specified rectangle
  inset(subfig);      // inset to subfig's corners

Extracting

The portion of a screen enclosed by a specified rectangle can be extracted. Each of the commands

  subfig.extract(P sw, P ne);
  canvas().extract(P sw, P ne);

returns the screen having stated corners and containing part of subfig or the canvas, respectively. Elliptical and diamond-shaped portions of a screen may be extracted. The contents occupy the ellipse or diamond inscribed in the given rectangle.

  subfig.extract_ellipse(sw, ne);
  canvas().extract_diamond(sw, ne);

The screen returned by extract (or its variants) may be used just like a manually-constructed screen, for example to construct a magnified inset.

True-Size Drawing

On occasion, it is necessary to draw at known page size, independently of the unit length. For 2-dimensional drawing in the (x_1,x_2)-plane with the camera in its default location, the coordinates (horizontal, vertical, or both) of a screen may be interpreted as true pt offsets relative to a specified point rather than as Cartesian coordinates.

  inlay(subfig, P loc);
  inlay_horizontal(subfig, loc);
  inlay_vertical(subfig, loc);

Each command maps the origin of subfig to the specified location in the active screen. The first performs affine scaling so that one Cartesian unit in subfig maps to one true pt on the page. The second and third commands perform this scaling only in the named direction.

The inlay functions are suitable only for 2-dimensional effects. For 3-dimensional true-size drawing, the function pt_to_screen converts a true length of 1 pt into Cartesian coordinates in the active screen. For example, the command line(P(0,0), P(pt_to_screen(12),0)) draws a line segment 12 pt long with its left endpoint at the Cartesian origin. True size drawing is unusual in an input file, but has definite uses in library code; ePiX’s right angle marker is a typical application.

Either type of true-size drawing works as described when drawing in the canvas, and only when drawing in a screen imported to the canvas at “true Cartesian” size (one screen unit maps to one Cartesian unit in the canvas). In other situations, the true size coordinate(s) will be scaled by additional affine transformation(s).

Cropping and Decorations

A screen has a crop mask: rectangular (default), elliptical, or diamond-shaped. If cropping is switched on in the active screen, added elements are cropped to the crop mask. For example, cropping by the default crop mask in the canvas ensures the figure lies inside the printed region allocated by LaTeX.

The crop commands below affect the active screen only. Cropping is not a “global” attribute, but must be set or unset explicitly for each screen. By default, cropping is switched off.

  set_crop();      // activate cropping
  set_crop(false); // deactivate cropping
  crop_diamond();  // <>-shaped mask, set crop state
  crop_ellipse();  // etc...
  crop();          // crop active screen's contents

Normally the crop mask is inscribed in the screen’s bounding box. The commands crop_diamond and crop_ellipse also accept a pair of P arguments, which are treated as corners of the crop box.

The active screen may be decorated with a border (outline of specified color and line width) and backing (solid fill color). The crop mask determines the shape of the border and backing.

  border(color, len);
  backing(color);

Identically-named member functions can be applied to an arbitrary screen whether or not it is active:

  scr.crop_rectangle().backing(Blue(1.8));
  canvas().crop();

Affine Maps

A plane affine map has the form T(x) = Ax+b for some invertible 2\times2 matrix A and a constant vector b. ePiX’s affine class allows affine maps to be built and applied to screens’ contents.

An affine map is uniquely determined by the images of three non-collinear points. The constructor returns the affine map sending the points (1,0), (0,1), and (0,0) to pt1, pt2, and pt0, respectively.

  affine af(P pt1, P pt2, P pt0=P(0,0));

To emphasize, the arguments are locations, not displacements, and the image of the origin comes last. There is also a void constructor (taking no arguments) which returns the identity map.

To facilitate construction of affine maps, an existing affine may be post-composed with a variety of “elementary” affine transformations. In the commands below, th is an angle (in current units), sc is a non-zero double, and ctr is a point fixed by the composing transformation. In each command, ctr defaults to (0,0).

  affine af; // the identity map
  af.shift(P arg);    // translate by arg
  af.rotate(th, [ctr]);  // counterclockwise rotation about ctr
  af.reflect(th, [ctr]); // reflect across line through ctr

  af.h_scale(sc, [ctr]); // horizontal scaling
  af.v_scale(sc, [ctr]); // vertical scaling
  af.scale(sc, [ctr]);   // dilatation

  af.h_shear(sc, [ctr]); // shear preserving horizontals
  af.v_shear(sc, [ctr]); // shear preserving verticals

  af.invert();           // the inverse
  af.postcomp(affine f); // post-compose with f
  af(f);                 // pre-compose, af not modified

A non-invertible affine can be created only by shearing or scaling with an extremely large or extremely small argument, or by supplying three collinear points to the constructor. Calling invert on a non-invertible map merely issues a warning and performs no action.

An affine may be applied to a screen’s current contents. The screen class has eight member functions (shift through shear) with syntax identical to the affine map functions. In addition, an arbitrary affine may be applied to a screen:

  scr.shift(arg); // shift scr's contents by arg; etc.
  scr.apply(f);   // apply f to scr's contents

To apply a composition of several maps to a screen, it’s best to build an affine map by composition, then apply the map. Composing affines is cheap; applying an affine is costly in proportion to the number of elements in the screen.

Applying an affine to a screen has no effect on the bounding box, border, or backing, and may move elements outside the bounding box even if cropping is active. To ensure a screen’s contents lie inside the bounding box, crop the screen after applying the affine(s).

The sample file inverse.xp uses affine maps to depict branches of inverse functions in one variable, symmetry.xp depicts the permutation group S_3 by its action on a regular hexagon.


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3.5 Drawing Attributes

ePiX maintains drawing states for filled regions, paths, and text objects.


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3.5.1 Filled Regions

Filling is either on or off. When filling is active, closed paths are filled with the current fill color.

  fill();   // turn filling on
  fill(color); // turn filling on, specify color
  nofill(); // turn filling off, same as fill(false);

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3.5.2 Paths

Paths and borders of filled regions are drawn with two pens, each described by color and line width. The line pen draws all path-like objects. The base pen does nothing unless it is wider then the line pen, in which case it draws an “underlayer” or “border” on the line pen.

  pen(len);          // set line pen width
  pen(color);        // set line color, keep width
  pen(color, len);   // set color and width

As usual, len may be either a length string or a double, interpreted as a width in pt. There are base functions with the same signatures and analogous meanings for the base pen.

The line width can be set with named declarations; the optional argument sets the line color:

  plain([color]); // 0.4pt
  bold([color]);  // 0.8pt
  bbold([color]); // 1.6pt

Path Style

By default, path-like objects are drawn with solid lines. Dashed, dotted, and free-form path style patterns are also available. (The base underlayer is always solid.) The page length of a pattern defaults to 12 pt, but can be set.

  line_style(string);
  dash_size(len=12); 

The path style is set with a WYSIWYG string of dashes, spaces, and periods, representing a pattern of dashes, gaps, and dots. In the sample styles below, the repeating units have the same page size, 12 pt.

[Image]

For brevity and uniformity, named commands are provided.

  solid(); "-"     dashed(); "-  -"    dotted(); " . "

A sequence of n dashes, spaces, and dots corresponds to a dash/dot pattern in an interval divided into n subintervals of equal length. If the ith character is a dash or space, the ith subinterval is drawn solid or empty, respectively. If the ith character is a period, a dot is placed at the midpoint of the ith subinterval.

This pattern is applied to a path-like object as follows. The page length of each edge is divided by the current dash_size and the ceiling (next largest integer) taken. This many copies of the current path style are scaled onto the edge. The first and last characters are adjacent in repeated units.

There are three inequivalent ways to adjust the dash length/dot spacing in a non-line path: Change the dash_length, create a path with a different number of points, or use a longer, repetitive pattern. For best results, the style string should not be longer than about a dozen characters.

If you need several dashed/dotted line styles in multiple figures, it’s best to define a custom header instead of hard-coding line styles. See Section See section-extensions, for guidance.


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3.5.3 Text Objects

Two types of textual element may appear in a file: labels (text boxes) and markers (LaTeX symbols). A marker occupies a box of zero size, and is placed at a specified Cartesian location. A label has typographical size, and is usually offset from its Cartesian location. For placement, an alignment point is attached to each label, and Cartesian coordinates position the alignment point.

Labels

A label is printed as a LaTeX box. By default, the alignment point is its reference point, the intersection of the left edge and the baseline, which is used by LaTeX to position the box on the page:[Image]

The alignment point may be offset manually by a specified number of pt. Additionally, a label’s location with respect to the alignment point can be chosen with an optional LaTeX-style alignment option. This scheme allows labels to be placed easily where they will not overlap other parts of the figure, and ensures labels stay properly positioned when the size or aspect ratio of a figure changes.

  label(P posn, P offset, string msg, [align]);
  label(posn, msg);

The first two components of the offset argument are numbers of pt to shift the alignment point right and up. The optional align argument may be one—or an appropriate pair—of t, b, r, or l (top, bottom, right, left), or c (center). These alignment options specify the position of the label relative to the Cartesian location posn, namely they work opposite to the way they work in LaTeX.

[Image]\qquad [Image]

The msg argument is usually a snippet of LaTeX code enclosed in double quotes. C++ treats “\” as an escape character, so a double backslash is needed in the source to get a single backslash in the output. For example,

  label(P(0,0), P(2,-1), "$\\rho = \\sin\\theta$", br);

positions an alignment point 2pt right and 1pt below the (Cartesian) origin, and typesets the equation \rho=\sin\theta below and to the right.

Labels can be rotated; the (counterclockwise) angle is set in current angle units with the command label_angle(theta). For example, a rotation angle of 90 degrees prints labels along a vertical axis. Though label rotation has legitimate uses, it can make labels more difficult to read, defeating their purpose. Do not use rotated labels merely because they are available. As a practical matter, if an output file contains rotated labels, the enclosing document must use the rotating package. elaps automatically loads this package.

An affine behaves moderately intelligently when acting on a marker or label. The mapping is applied to the label’s location, and the “linear part” is used to adjust the offset and label angle. No attempt is made to treat alignment. For best results, if a screen will have an affine applied to it, label positions should be fine-tuned only with offsets, not alignment arguments. In all situations, the font itself is unchanged; no attempt is made to print sheared, scaled, or reflected text. The sample file inverse.xp illustrates the effect of affine maps on labels.

Fonts and Type Size

By default, the font in an ePiX figure is that of the enclosing document. The font size and face are changed with “declaration-style” commands such as

font_size("Large");
font_face("sc");

The argument of font_size is a valid LaTeX size. If no argument is given, normalsize is understood. font_face accepts a two-letter string, appended to the string “text” to give a LaTeX font declaration command (“textsc” above). Finer-grained control is accomplished by placing LaTeX commands into the label text.

Label Attributes and Masked Labels

Each label command has a corresponding “mask” version (masklabel) that draws an opaque rectangle under the label text. The mask size exceeds the label’s size by an amount of padding, and the mask is itself surrounded by a rectangular border, of specified color and width.

  label_color(color);         // set label text color
  label_mask(color=White());  // set mask color
  label_pad(string len);

  label_border(color, [len]); // set color (and line width)
  label_border(len);          // set line width only
  no_label_border();          // turn off label borders

Markers

ePiX markers are obtained with marker(P pt, <MARKER TYPE>); (Table See table-marker, .) Several “dot-like” marker types are available by name: [Image]

  spot(P pt);  dot(P pt);   ddot(P pt);
     ---       box(P pt);   bbox(P pt);  
  ring(P pt);  circ(P pt);     ---

A circ is filled with the current mask_color, and a ring is “hollow”. Each dot-like marker can also be called with label syntax, generating a labeled marker with one command.

  dot(P posn, offset=P(0,0), msg="", align=none); // etc.

By default, spot and ring are 4 pt in diameter; dot, box, and circ are 3 pt in diameter; ddot and bbox are 2 pt in diameter. At arbitrary diameter, a spot is 4/3 the diameter of a dot and a ddot is 2/3 the diameter. The command dot_size(diam=3) sets the diameter of a dot, and hence the size of all dot-like markers.

[Image]

Table[tb]: ePiX’s marker types.

Reminders

When constructing and placing a label,


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3.5.4 Color Declarations

Old-style color declarations set the fill color, line color, and text color.

  rgb(r, g, b);      cmyk(c, m, y, k);
  rgb(P);            cmyk(P);  // for function-controlled colors
  red(d);  // similarly for other primaries

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3.6 Creating and Drawing Objects

Scene elements include geometric objects, coordinate grids, axis labels, and function plots. A few commands create an object (a point, line segment, circle, sphere, or plane) that can be used in subsequent computations, but most drawing commands automatically create, draw, and discard objects.


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3.6.1 Geometric Data Structures

Points

The simplest object in the world, and by far the most common named data structure, is P, an ordered triple of real numbers (double-precision floats). The function P(x1,x2,x3) creates the point (x_1,x_2,x_3). If only two arguments are provided, x_3=0 by default. This convention allows ePiX to treat 2- and 3-dimensional figures uniformly. The standard basis is available: E_1=P(1,0,0), etc.

Depending on context, a P may represent either a location (point) or a displacement (vector). Almost all ePiX functions treat a P as a point. However, algebraic operators and commands that plot vector fields treat P arguments as displacements.

Polar, cylindrical, and spherical coordinate P constructors are sensitive to the current angular mode.

  P pt=polar(r,t);   // (r*Cos(t), r*Sin(t), 0)
  P pt=cis(t);       // (Cos(t), Sin(t), 0) = polar(1, t)
  P pt=cyl(r,t,z);   // (r*Cos(t), r*Sin(t), z)
  P pt=sph(r,t,phi); // polar(r,t)*Cos(phi) + (0,0,r*Sin(phi))

Algebraic operations—addition/subtraction, scalar multiplication; scalar, cross, and componentwise products; orthogonalization—can be performed on Ps. In compound expressions, the binary operators below should be enclosed in parentheses, and scalars must be collected at left, Ps at right.

  double u=pt.x1();   // first coordinate of pt, etc.
  P(a,b,c)|P(x,y,z);  // scalar product, ax+by+cz
  P(a,b,c)&P(x,y,z);  // componentwise product (ax, by, cz)
  P(a,b,c)*P(x,y,z);  // cross product (bz-cy, cx-az, ay-bx)
  J(p);               // quarter turn about the x3-axis
  p%q;                // orthogonalization, p (mod q)

Explicitly, p\%q is the unique vector p+k*q perpendicular to q.

P operations express mathematical relationships, and therefore imbue a figure with logical structure, making the input file easier to read, modify, and maintain. Commonly, a file preamble will define a few named points with hard-coded coordinates, then define additional points of interest using P operators.

Complex Numbers

The Complex class represents a complex number, specified by its real and imaginary parts. Arithmetic operators (addition, subtraction, multiplication, division) are provided, as well as exponential, logarithmic, circular, and hyperbolic functions (sensitive to angle units), the latter with C appended to their names. A complex number implicitly converts to a point with third coordinate zero. The sample file cubic_cutaway.xp illustrates usage, including how to define and plot functions and color-shade Riemann surfaces.

  Complex u, z(4,3);  // u = 0, z = 4 + 3i
  double len(norm(z));// len = 5
  double th(Arg(z));  // th = atan2(3, 4);
  P pt(z);            // pt = (4, 3, 0)
  Complex w(expC(z)); // w = e^{4 + 3i}
  u = LogC(z,k=0);    // u = Log(z) + 2*pi*ki
  u = sqrtC(z,k=0);   // sqrt(z)*expC(pi*ki)
  u = rootC(z,n,k=0); // (z^{1/n})*expC(2*pi*ki/n)
  u = powC(z,n);      // z^n

Other Geometric Classes

In addition to P, objects of type Circle, Plane, Segment, and Sphere can be used for Euclidean geometry constructions. Simple affine operations are supplied for each type, as is a draw() function, which represents the object as a path in the screen.

  obj.shift(P arg);    // translate by arg
  obj.move_to(P arg);  // move center to arg
  obj.scale(double c); // scale about center by c
  obj.draw();

A Segment’s “center” is its midpoint. A Plane has no center; move_to translates the Plane to pass through arg, and scale has no effect.

A Circle data structure consists of a center, radius, and a perpendicular unit vector. Three constructors are provided:

  Circle(center=P(0,0,0), double rad=1, normal=E_3);
  Circle(P center, P point);
  Circle(P p1, P p2, P p3);

The second constructor creates the Circle parallel to the (x_1,x_2) plane, with given center, and radius equal to the distance between the arguments. (A warning is printed if the second argument does not lie on the circle, namely, if the arguments do not lie in a plane parallel to the (x_1,x_2) plane.) The third returns the Circle passing through the given points; the points must not be collinear.

The data defining a Circle are recovered with member functions named center(), radius(), and perp().

A Plane is specified by a point and normal vector, or by three non-collinear points. The draw() function clips the plane and draws the resulting polygon. Unless the clip box has been set manually, the clipped polygon’s vertices will have large coordinates.

A Segment is constructed from its endpoints. The member function midpoint() returns the center.

A Sphere is specified by a point and a radius—by default the origin and unity, or by the center and a point on the sphere. Member functions center() and radius() return the defining data. Capabilities specific to geography and spherical geometry are described below, pp. non-euclff.

The draw() function of a Sphere draws the horizon visible from the current viewpoint. While this horizon is a circle in object space, its image in the screen is generally an ellipse. Antipodal points are not generally mapped to points symmetrically placed with respect to the center of this ellipse. These effects are most pronounced when the viewpoint is close to the Sphere and the center is not close to the target.

Intersection

To facilitate geometric computation, ePiX’s Circle, Plane, Segment, and Sphere classes can be intersected with the * operator.

A Circle has a center, radius, and unit normal; a Plane has a distinguished point and unit normal; a Segment has two endpoints; a Sphere has a center and radius. An object is malformed if these conditions are not met. The constructors return well-formed objects with two exceptions: Circle and Plane create malformed objects if called with three collinear points. The operator * returns a malformed object if either argument is malformed, or if the operands are disjoint, tangent, or coincident. Malformedness is benign: Calling draw() on a malformed object does nothing.

Orthonormal Frames

A frame comprises three mutually perpendicular unit vectors. The constructor takes three vectors. The frame’s third vector e_3 is positively proportional to v3, the second vector e_2 is positively proportional to v2\%v3, and the first is the cross product, e_1=e_2\times e_3. Thus, a frame is right-handed, and does not depend on v1.

The elements of a frame are named sea, sky, and eye, just as for the camera. A frame can be rotated through an arbitrary angle about any of its elements.

  frame();        // the standard basis {E_1, E_2, E_3}
  frame fr(v1, v2, v3); // orthonormalize {v1, v2, v3}
  fr.sea();       // the first element of fr, etc.
  fr.rot1(theta); // rotate fr through theta about sea, etc.

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3.6.2 Path-Like Elements

Basic path-like objects are drawn with named commands. Arguments of polygon commands are endpoints/vertices. Except for line and Line, the following are subject to filling.

  line(P p1, P p2, [double expand]);
  Line(p1, p2);  // draw line through p1, p2 (crop required)
  triangle(P p1, P p2, P p3);
  rect(P p1, P p2);
  quad(P p1, P p2, P p3, P p4); // quadrilateral
  circle(ctr=P(0,0,0), rad=1, normal=E_3);
  circle(ctr, pt);
  circle(pt1, pt2, pt3);

The optional line argument is an expansion parameter: line(p1,p2,t); draws a segment centered at the midpoint of p1 and p2, with length scaled by 2^{t/100}. (Setting t=100 doubles the length, while t=-100 halves the length.) The arguments of rect() must lie in a plane parallel to a coordinate plane. The arguments to circle commands are the same as for Circle constructors.

Quadratic and cubic splines are described by their control points. A list of P is drawn as a “natural” spline (the C^2 piecewise cubic curve with vanishing second derivatives at the endpoints); the number of points per cubic segment must be specified. Circular and elliptical arcs are given by center, a basis, angular range, and an optional number of intervals.

  spline(P p1, P p2, P p3, [int n]);       // quadratic
  spline(P p1, P p2, P p3, P p4, [int n]); // cubic
  spline(vector<P>, int n);                // natural spline

  arc(P ctr, rad, t_min, t_max);  // parallel to (x1,x2)-plane
  ellipse(P ctr, P v1, P v2);     // in plane spanned by v1, v2
  ellipse(P ctr, P v1, P v2, t_min, t_max, [int n]);

If parameter bounds are omitted in an ellipse command, the entire ellipse is drawn. When the angular range subtends one or more full turns in an arc or ellipse the curve is subject to filling.

Commands for planar (half-)ellipses remain from ePiX’s early days:

  ellipse(P ctr, P radius);
  ellipse_left(P ctr, P radius);
  ellipse_right(P ctr, P radius);
  ellipse_top(P ctr, P radius);
  ellipse_bottom(P ctr, P radius);

If radius is P(a,b), these commands draw all or half of the ellipse with given center in the (x_1,x_2) plane, axes parallel to the coordinate axes, and axis lengths 2a and 2b.

Two commands are available to mark off right angles or a subtended angle. Each accepts a spatial location and two non-zero directions, and draws a scene element in the plane spanned by the vectors.

  right_angle(P loc, P v1, P v2, scale=8);
  arc_measure(P loc, P v1, P v2, scale=8);
  arc_measure(P loc, P v1, P v2, offset, text, align, scale=8);

The right_angle command does not check its arguments for perpendicularity. The arc_measure commands mark the small angle subtended by the directions; the label form places a label at the midpoint of the arc. The (optional) scale argument is the true size in pt of the marker.

Arrows

Line segments, splines, and arcs can be drawn with arrowheads at one end. In profile, an arrowhead’s width is 3pt, and its height is 5.5 times the width. The actual printed height depends on the head’s orientation with respect to the camera. By default, an arrowhead is a filled triangle. The shape and size are adjusted with declarations:

  arrow_width(w=3);   // width in pt
  arrow_ratio(r=5.5); // height-to-width
  arrow_inset(c=0);   // base indent as frac of ht

[Image]\qquad [Image]\qquad [Image] \\ The inset must be between -1 and 1. Each arrow command accepts an optional scale argument, which scales the arrowhead.

  arrow(P tail, P tip, [scale]);
  arrow(P p1, P p2, P p3, [scale]);   // spline arrows
  arrow(P p1, P p2, P p3, P p4, [scale]);
  arrow(P ctr, P v1, P v2, t_min, t_max, [scale]); // ellipse

A few “special-purpose” commands are supplied:

  dart (P p1, P p2);  // same as arrow(p1, p2, 0.5);
  aarrow(P p1, P p2); // double-headed arrow <--->
  arc_arrow(ctr, rad, t_min, t_max, [scale]);

If an arc_arrow is too short, nothing is drawn.


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3.6.3 Coordinate Axes and Labels

ePiX provides an axis class for coordinate axes. Labels are generated automatically in a variety of styles: decimal, scientific notation, fraction, and trigonometric fraction. Logarithmic axes and labels are available. Axis and labeling commands from Version 1.0 have been retained.

The axis Class

An axis consists of a line segment divided into a specified number of equal-length subintervals by “major” (long) tick marks. Each subinterval may be divided further by “minor” (short) tick marks. Minor ticks may be equally-spaced (“Cartesian”) or logarithmically placed. Finally, a label is written at each major tick mark with specified offset and alignment. Labels are generated automatically from the endpoints, so the line should be parallel to a coordinate axis. Label attributes (masking, borders, font size and face, rotation angle) are determined by the current label style, not by the axis.

The command

  axis(P tail, P head, int n, P offset=P(0,0), [align = none]);

creates an axis joining tail to head, divided into n segments of equal length, with a major tick mark and label at each division point and endpoint. The offset and align arguments have the same meaning as for ordinary labels.

The number and length of minor ticks, and the alignment of ticks, are controlled with member functions:

  axis Ax(P(a,c), P(b,c), n);
  Ax.subdivide(n);  // put n-1 minor ticks in each axis segment
  Ax.tick_ratio(r); // minor length = r*major length
  Ax.align(AL=c);   // align all ticks; AL = t, b, l, r, or c

By default, major ticks are 6pt long and twice the length of minor ticks. The global declaration tick_size(len) sets the major length, subject to “reasonable stylistic limits”. For visual consistency, tick lengths should not be changed casually.

Labels on an axis may be drawn in several styles, selected with member functions:

  Ax.dec();  // decimals (default)
  Ax.frac(); // fractions, e.g. 0.5 -> \frac{1}{2}
  Ax.trig(); // fractional multiples of \pi
  Ax.sci();  // scientific notation, k\times base^N

  Ax.unmark(double); // remove label at selected location
  Ax.precision(p);   // set number of digits for decimal labels
  Ax.align_labels(AL); // re-align labels

Arbitrary textual labels depending on one coordinate are obtained by writing a string-valued function of double and “registering” it:

  // f represents x as a string in given precision and base
  std::string f(double x, unsigned int prec, unsigned int base);
  Ax.label_rep(f);

For example, textual tags can be printed instead of numeric labels.

By default, an axis is Cartesian. The member function log(int b=10) converts an axis to “log mode” with specified base; this affects both tick marks and labeling. If b is at least 3, minor tick marks appropriate for a logarithmic axis base b are drawn. Second, labels are written in decimal or scientific notation appropriately for a logarithmic axis; that is, “k\times b^N” (or its decimal value) is written at location N+\log_b k.

Minor ticks of a log axis may be labeled individually; again, this is controlled with member functions:

  Ax.tag(d);   // labels at N+log_b(d)
  Ax.tag235(); // tag 2, 3, and 5 if b=10
  Ax.tags();   // tag 1, ..., b-1
  Ax.untag();  // remove all tags, including 1
  Ax.untag(double);  // remove one tag, e.g. Ax.untag(9);

For convenience, an axis object along an edge of the active screen can be created with a named command:

  top_axis(n, offset, align);
  bottom_axis(n, offset, align);
  left_axis(n, offset, align);
  right_axis(n, offset, align);

The ticks automatically point into the bounding box.

The functions above create objects or set axis attributes, but write no output. Tick marks and labels can be printed separately, or at once:

  Ax.draw();        // axis, tick marks, and labels
  Ax.draw_ticks();  // axis and ticks only
  Ax.draw_labels();

Other Axis-Drawing Commands

ePiX formerly supplied commands for drawing simple axes and their labels. These commands have been kept for compatibility. Horizontal axes are generated with

  h_axis(p1, p2, n, align=c);    // n subintervals (n+1 ticks)
  h_log_axis(p1, p2, n, align=c, base=10);

For vertical axes, use v_axis. The style of tick mark is appropriate for an axis of the given type. Horizontal axis tick marks may be aligned t (above the axis) or b (below). Similarly, vertical axis ticks may be aligned r or l.

The endpoint arguments of a coordinate axis may be omitted, in which case they default to p_1=(x_min,0) and p_2=(x_max,0) for a horizontal axis, or to p_1=(0,y_min) and p_2=(0,y_max) for a vertical axis. If the bounding box has integer width and/or height, omitting the number of points draws tick marks one unit apart.

Labels for a horizontal Cartesian or logarithmic axis are generated with the commands

  h_axis_labels(P p1, P p2, int n, P offset, [align]);
  h_axis_masklabels(p1, p2, n, offset, [align]);
  h_axis_log_labels(p1, p2, [n], offset, [align], base=10);
  h_axis_log_masklabels(p1, p2, [n], offset, [align], base=10);

Labels for a vertical axis are generated with v_axis_labels, etc. The labels are automatically generated to match their horizontal location. The first puts (n+1) evenly-spaced labels on the segment joining p1 and p2. As with ordinary labels, the offset is in pt, and the optional LaTeX-style alignment option positions the labels using their corners. The second command draws masked labels according to the current label masking attributes. The third writes labels in exponential notation, using the Cartesian coordinate as exponent.

As for coordinate axes, the initial and final points may be omitted in an axis_[mask]labels command, with the same defaults. The offset and number of labels must always be specified.

Broken Axes

Broken axes are best drawn using page layout, especially if axis labels are to be drawn. To accomplish the task, create a screen for each piece of axis, using appropriate Cartesian coordinates for the corners, then inset the screens so the axis pieces are nearly end to end. A zig-zag glyph signifies the break:

  axis_break(P, P, scale=12);

The P arguments are the screen coordinates of the ends to be joined; the optional third argument is the true height and width in pt of the glyph. The sample file coord_tricks.xp uses layout and axis breaks.

Coordinate Grids

Cartesian grids fill a coordinate rectangle, and have a specified number of lines in each direction. A polar grid has specified radius, and numbers of rings and sectors.

  grid(n1, n2);         // fills the bounding box
  grid(p1, p2, n1, n2); // fills the box with corners p1, p2
  polar_grid(r, n1, n2);

Log and semi-log grids are drawn with analogous syntax:

  log_grid (p1, p2, n1, n2, [base1], [base2]); // log-log
  log1_grid(p1, p2, n1, n2, [base]);           // log-lin
  log2_grid(p1, p2, n1, n2, [base]);           // lin-log

The n_i arguments dictate the number of squares, namely the number of orders of magnitude spanned in the logarithmic direction(s). The (optional) base arguments default to 10, and control the number of lines drawn per order of magnitude. As with Cartesian grids, the corners are optional; if omitted, the grid fills the bounding box.

Graph paper may be created by superimposing grids:

  pen(0.25);
  grid(10*xsize(), 10*ysize());
  pen(0.5);
  grid(2*xsize(), 2*ysize());
  pen(1);
  grid(xsize(), ysize());

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3.6.4 The Path Class

A path data structure is an ordered list of points that can be cropped, clipped, mapped, concatenated, and drawn. The first four constructors mirror polygon-drawing commands above. Function graphs and parametric paths are built from a real- or P-valued function f of one variable.

  path(p1, p2, expand=0);     // line (endpoints)
  path(p1, p2, p3, [n]);      // quadratic spline
  path(p1, p2, p3, p4, [n]);  // cubic spline
  path(p1, v1, v2, t_min, t_max, [n]); // ellipse
  path(f, t_min, t_max, [n]);

The member function pt() accepts a P or three (or two) doubles and appends the specified point to a path. This snippet creates a regular n-gon:

  path ngon; // declare new path
  for (int i=0; i<=n; ++i)
    ngon.pt(cis(i*full_turn()/n)); // works in all angle modes

Compound paths may be built by concatenation. If path1 and path2 share an endpoint, the commands

  path1 += path2;
  path1 -= path2;

replace path1 with the result of traversing path1 “forward”, then following path2 in the forward or reverse direction (respectively). For expected results, the first (or last) point on path2 should be the last point of path1. The notation suggests 1-dimensional homology chains. The sample file contour.xp illustrates path creation and manipulation.

A path is a data structure, and must be drawn explicitly to create visible output. By default a path is not a closed loop (even if the first and last points are the same), and is not filled when drawn. Member functions perform these tasks. Continuing the n-gon snippet above,

  ngon.close(); // mark path as closed
  ngon.fill();  // draw filled region if filling is active
  ngon.draw();  // print to the screen

path::close() adds a closing edge if necessary. Once a path is closed, no more points can be added. A closed path clips and crops differently than an open path with the same data. path::fill() has no effect on an unclosed path.


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3.6.5 Function Plotting

The noun “map” refers to a C++ function that accepts one or more double arguments and returns a double or a P. Mathematically, a map can be depicted in two ways: as a graph (which retains information about the domain), or as a parametrized curve or surface (which discards domain information). ePiX assumes that double-valued maps are graphed and P-valued maps are drawn parametrically. Either sort of depiction is called a “plot”. ePiX plots are either “wire mesh”, produced by a plot command, or “shaded”, produced by a surface command.

Basic Plotting

For the moment, “function” means “function of one variable” (precisely, a double-valued function of a double variable). A function graph depends on the domain and the number of points to use. Each of the commands

  plot(f, t_min, t_max, n);
  polarplot(f, t_min, t_max, n);
  shadeplot(f, t_min, t_max, n);

graphs the function f on the interval [t_min, t_max] by dividing the interval into n subintervals of equal length. The first gives a Cartesian plot, the second a polar plot with bounds in current angular units, the third shades the region between the graph and the horizontal axis. If two functions are given to shadeplot, the region between their graphs is shaded.

Domains and Wiremesh Plots

An ePiX domain is a coordinate box of dimension one, two, or three, specified by a pair of opposite corners and two meshes (“coarse” and “fine”, respectively) which specify the amount of data to be plotted. Plotting is explained in detail below.

  // [a1,a2] x [b1,b2]: n1 x n2 rectangles, m1 x m2 intervals
  domain R2(P(a1,b1), P(a2,b2), mesh(n1,n2), [mesh(m1,m2)]);

  // [a1,a2] x [b1,b2] x [c1,c2] divided analogously
  domain R3(P(a1,b1,c1), P(a2,b2,c2),
            mesh(n1,n2,n3), [mesh(m1,m2,m3)]);

If unspecified, the fine mesh is the same as the coarse mesh. For expected behavior, the coarse mesh should “divide” the fine mesh, in that m_i should be a (usually small) integer multiple of n_i for each i.

A domain may be resized in any coordinate for which the thickness is positive, and can be sliced by setting one variable to a constant. The result of slicing is a domain whose dimension is one smaller than the original. Finally, “slices” operators return the list of domains obtained by setting one variable to evenly-spaced constants. By default, the number of slices is specified by the coarse mesh. An optional argument specifies the number of slices. This argument need not be related to the coarse mesh.

  R2.resize2(a,b); // [a1,b1] x [a,b]
  R2.slice1(t);    // set x1 = t
  R3.slices3([n]); // (n+1) domains with x3 = const

[Image]

\smallskip When possible, resizing preserves grid square sizes. Generally, though, integer truncation occurs: If R=domain(P(0,0), P(1,1), mesh(10,6)), then R.resize1(0,0.25) is the rectangle [0,0.25]\times[0,1] subdivided into 2\times6 subrectangles, since 10\div4=2 in integer arithmetic. For expected behavior, choose mesh sizes to avoid integer truncation.

The arguments of a plot command are a map, followed by either a domain or its logical equivalent.

  double f(double t) { return t*t; }
  P F(double u, double v) { return P(u, v, exp(u)*Sin(v)); }
  P G(double u, double v, double w) { return P(v*w, u*w, u*v); }

  plot(f, a, b, n);       // f:[a,b] -> R, using n intervals
  plot(F, R2);            // graph of exp(u)*Sin(v)
  plot(G, R3.slice2(0.5); // G: R^3 -> R^3 restricted to y=0.5

By (compiler-enforced) convention, plot commands involving a P-valued map accept a domain argument, as in the second and third commands above. To plot a double-valued function, by contrast, supply the logical equivalent of a domain, usually the endpoints and the number of intervals, as in the first plot command above.

Resizing and slicing allow a map F to be plotted selectively over parts of its domain. This can be used to emphasize parts of the image, layer scene elements, patch surfaces together, and so forth. Resize and slice(s) commands may be used directly in a plot command:

  plot(F, R2.resize1(0,0.5));
  plot(F, R2.slices1());

Meshes and Plotting

The P arguments of a domain are a pair of opposite corners. The first mesh argument, the coarse mesh, specifies the number of subdivisions in each coordinate direction. The second mesh, the fine mesh, determines the number of points used in each direction when plotting.

Separating the roles of coarse and fine meshes allows a plot to conform closely to a surface without using a fine grid of curves. Both parts of Figure See fig-plot, are drawn with a 6\times20 coarse mesh. In the first picture, the fine mesh is also 6\times20, while in the second, the fine mesh is 12\times60.

[Image] [Image]

Figure[hbt]: Coarse and fine meshes.

The coarse mesh is significant only for domains of dimension at least 2. The coarse mesh’s size determines the number of curves or surfaces plotted perpendicularly to a coordinate direction, while the fine mesh’s size determines the number of segments used along that direction. For predictable results, the fine mesh should be a small multiple of the coarse mesh.

Plotting works analogously for 3-dimensional domains and maps depending on three variables: The “one-dimensional skeleton” of the domain’s image is drawn. A P-valued map of three variables can be plotted over a 1- or 2-dimensional domain. (The effect may be unexpected unless the domain arises by slicing, however.) A map depending on one or two variables cannot be plotted over a 3-dimensional domain.

Shaded Surfaces

ePiX’s shaded surface plotting implements a degree of hidden surface removal. The algorithm breaks a surface into mesh fragments, sorts them in decreasing (approximate) distance to the camera, and prints them. If filling is active, a mesh fragment is shaded according to the angle between the normal vector and the direction to the camera, simulating constant ambient lighting; otherwise, the current fill color is used. This technique works fairly well for surfaces without intersection, and even acceptably handles intersecting surfaces for which mesh elements intersect only along boundaries.

The syntax of a surface command is identical to a plot command when only a single surface is drawn. For example,

  surface(F, R, cull=0);

plots the P-valued function F over the domain R, the shaded equivalent of the corresponding plot command. The optional cull argument removes elements that point toward (cull=-1) or away from (cull=1) the camera. Naturally, orientation of mesh elements depends on the parametrization F, not merely on the surface. Culling reduces the output file size, but is useful mostly for closed, convex surfaces.

There are special commands for surfaces of rotation; each accepts a final cull argument.

  surface_rev(f, t_min, t_max, n_lats, n_longs);
  surface_rev(f, g, t_min, t_max, n_lats, n_longs=24);
  surface_rev(f, g, R, frame coords);

The first revolves the graph of f about the x-axis, the second uses the parametric curve t\mapsto\bigl(f(t),g(t)\bigr) as profile. In each case, the parameter interval [t_min,t_max] is divided into n_lats equal-length subintervals, n_longs copies of the profile curve are drawn, and the complete surface (one full turn) is drawn.

The third form uses a domain to control the range of longitudes, and draws a surface of rotation in the Cartesian coordinate system defined by the orthonormal basis coords, by default the standard basis. The arguments f and g define a parametric curve in the plane spanned by the first two elements of coords, and the first element is the axis of rotation.

As in wire mesh plotting, the fine mesh is used to draw the boundaries of surface patches; this tends to make surfaces look smoother for modest-sized coarse meshes. If the coarse mesh is too coarse, however, two visually undesirable effects can occur. First, adjacent regions of the surface may be shaded very differently, since shading is constant over patches defined by the coarse mesh. Second, a patch nearly tangent to a line of sight may be drawn badly if the patch bends back on itself, since the boundary of the patch is drawn, not the visible edge of the mathematical surface. See samples/artifacts.xp.

Multiple Domains and/or Maps

A scene containing two or more shaded surfaces cannot generally be built up one surface at a time. Instead, multiple surfaces must be assembled into a single data structure before they can be drawn. Multiple surfaces are built from one or more maps and one or more 2-dimensional domains. In the code snippets below, F and G are P-valued functions of 3 variables, and R is a 3-dimensional domain.

To plot the images of several domains under a single map, assemble the domains into a list if necessary, then issue a surface command:

  surface(F, R.slices3(), cull=0);

  domain_list DL(R.slice1(0));  // build domain list
  DL.add(R.slice2(0.5));        // add a domain, etc.
  surface(G, DL, cull=0);       // draw

For multiple maps, ePiX provides the scenery class. Conceptually, scenery is an agglomeration of shaded surfaces, built one surface at a time from maps and 2-dimensional domains. The add function accepts two arguments—a map, and either a domain or a list of domains—and contributes its data to the scenery rather than plotting immediately. Completed scenery is drawn manually.

  scenery S(F, R.slice3(0.25)); // S contains one surface
  S.add(F, R.slice2(0));        // S contains two surfaces
  S.add(G, R.slices1(3));       // S contains six surfaces
  S.draw(cull=0);

Complete examples are included in the samples directory: spherical.xp and minkowski.xp.

In principle, a scene may contain arbitrarily many surfaces. However, figures that contain many objects tend to tax LaTeX’s internal stacks. Frequent color changes exacerbate the problem. Even if you use hugelatex (or increase LaTeX’s memory), a figure containing more than a few thousand mesh elements is unlikely to compile. At moderate resolution, a surface can easily contain 1000 patches. Each shell script has a command-line option to invoke hugelatex; your mileage may vary.

User-Specified Color Shading

By default, a surface or scenery is colored according to the current fill color. For finer control, each surface, surface_rev, and scenery command accepts an optional position-dependent color specification.

  surface(F, R, color, cull=0);
  surface_rev(f, [g], t_min, t_max, n_lats, n_longs, color);
  surface_rev(f, g, R, color, [coords]);

  scenery S(F, R, color);
  S.add(F, R, color);

The color argument is a P-valued function of two or three doubles whose output is interpreted as a set of RGB densities. If color takes two arguments, they are domain coordinates, and the surface is colored according to parameter values. If color takes three arguments, they are Cartesian coordinates, and the surface is colored according to spatial location. Please see the sample files surface_shade.xp and S2_harmonics.xp.


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3.6.6 Calculus Plotting

ePiX provides high-level commands for plotting derivatives and definite integrals, Riemann sums, tangent lines, slope- and vector fields, and solutions of planar and spatial systems of differential equations.

In this section, f and g are double-valued functions of one variable.

Utility Functions

  sup(f, a, b);     // max/min of f on [a,b]
  inf(f, a, b);
  newton(f, g, x0); // find approximate crossing point

Newton’s method returns the crossing point of the given functions, starting from the specified seed, which should be reasonably close to the expected solution. If a critical point is hit or 5 iterations pass, a warning is issued and the current result (probably incorrect) is returned. The second function g defaults to the zero function if omitted.

Derivatives and Integrals

The classes Deriv and Integral are used to calculate values of derivatives and integrals, and to plot these functions.

  Deriv df(f); // function object: df(x) = f'(x)
  df.eval(t);  // return f'(t)
  df.left(t);  // deriv from left at t:  (f(t)-f(t-dt))/dt
  df.right(t); // deriv from right at t: (f(t+dt)-f(t))/dt

  Integral prim(f,a); // function object: prim(x) = int_a^x f
  prim.eval(b);  // numerical integral of f over [a,b]
  double val(Integral(f).eval(1));   // val = \int_0^1 f

The lower limit on an integral is 0 by default. Derivs and Integrals can be used directly in a plot command:

  plot(Deriv(f), a, b, n);  // plot f' over [a,b]
  plot(Integral(f, x0), a, b, n);
  riemann_sum(f, a, b, n, TYPE);

The second graphs the definite integral x\mapsto\int_{x_0}^x f(t)\,dt over [a,b]. As above, x_0 defaults to 0. The third draws rectangles or trapezoids whose area approximates the definite integral of f over [a,b]. The TYPE may be UPPER, LOWER, LEFT, RIGHT, MIDPT, or TRAP.

Tangent lines and envelopes (families of tangent lines) are drawn with

  tan_line(f, t);                // f real- or vector-valued
  envelope(f, t_min, t_max, n);  // family of tangent lines
  tan_field(f, g, t_min, t_max, n); // field of tangents

The sample files conic.xp and lissajous.xp illustrate these features.

Systems of Differential Equations

Let F be a P-valued function of two or three variables.

  ode_plot(F, p_0, t_min, t_max, n); 
  flow(F, p_0, t_max, n);

The first plots the solution curve of the initial-value problem \dot{x}=F(x), x(0)=p_0, over the specified time interval. If t_min is omitted, its value is 0, so the curve starts at p_0. With manual calculation to rotate a planar field a quarter turn, ode_plot can be used to draw level curves (isobars) of a function of two variables; see the sample file dipole.xp. The flow function returns the result of starting at p_0 and flowing by F for time t_max, using Euler’s method with n time steps. This is useful for placing markers or arrowheads precisely along a flow line.

A planar or spatial vector field itself may be plotted over a domain R in three styles:

  vector_field(F, R, [scale]); //  true length
  dart_field  (F, R, [scale]); // const length
  slope_field (F, R, [scale]); // const length

The field is sampled at the grid points of the coarse mesh. If the domain is 2-dimensional, the plot is a planar slice of the field, even if the field depends on three variables. If the domain is 3-dimensional, the field is drawn in successive slices x_3={const}, starting at the height of the first corner of R and ending at the height of the second corner.

The optional final argument, which defaults to 1, scales the arrowheads in a vector field, and scales the (constant) length of field elements for slope and dart fields. The sample files layout2.xp, lorenz.xp, slopefield.xp, and vfield.xp illustrate usage.

In each field-plotting command, the domain argument may be replaced by two points, representing corners of a coordinate rectangle, and two integers, the number of grid intervals in the selected coordinate directions. Only planar slices of a vector field can be plotted using the alternative syntax.


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3.6.7 Non-Euclidean Geometry

ePiX provides limited features for spherical and hyperbolic geometry: the ability to draw lines in the half-plane and Poincar\’e disk models of the hyperbolic plane, and to draw latitudes, longitudes, great circle arcs, spherical triangles, regular polyhedra, and parametrized curves on a sphere.

Hyperbolic line segments are specified by their endpoints in the upper half space or ball (Poincar\’e) models. In each case there is no output if either endpoint lies outside the model.

  hyperbolic_line(p, q);
  disk_line(p, q);

For compatibility with 2-dimensional hyperbolic space, the half-space model is the set \{(x_1,x_2,x_3)\mid x_2>0\}.

A frame determines geographical coordinates on a Sphere: the first element points toward longitude 0 on the equator, the third element points to the north pole. A latitude line depends on a Sphere, a frame, the numerical latitude, and a range of longitudes. A longitude line is described similarly.

  latitude(lat, long_min, long_max, Sphere S, frame coords);
  longitude(lngtd, lat_min, lat_max, Sphere S, frame coords);

By default, coords is the standard frame and S is the unit sphere. These commands draw only the portion of the curve that is visible from the current viewpoint. The function back_latitude draws the invisible portion of a latitude line.

Spherical arcs and triangles are described by their endpoints. Only the direction vector from the center of the sphere to an endpoint is significant; if a sphere is scaled or moved, the same function call will draw the corresponding object on the new sphere.

The following draw the visible (front) portions of great circle arcs:

  front_arc(p1, p2, S);  // short arc on S from p1 to p2
  front_arc2(p1, p2, S); // arc from p1 to -p1 through p2
  front_line(p1, p2, S); // great circle through p1 and p2

Triangles and regular (Platonic) polyhedra are provided. The sample file sample/polyhedra.xp illustrates usage.

  front_triangle(p1, p2, p3, S); // spherical triangle
  front_tetra(S, coords);  // regular tetrahedron
  front_cube(S, coords);   //    hexahedron
  front_octa(S, coords);   //    octahedron
  front_dodeca(S, coords); //    dodecahedron
  front_icosa(S, coords);  //    icosahedron

Each function has a back version, which draws the hidden portion. The tetrahedron, cube, and octahedron are (up to scale) inscribed in the cube of side length 2 centered at the origin whose sides are parallel to frame. The point (1,1,1) is a vertex of the tetrahedron.

Up to scale, the icosahedron’s vertices lie on the golden rectangle with vertices (\pm\gamma,0,\pm1) and its images under cyclic permutation of coordinates. The dodecahedron is dual to the icosahedron.

Spherical Geometry

The Sline class represents geodesic arcs on the unit sphere, and provides intersection and reflection operators.

  Sline L1(tail, head); // non-proportional vectors
  L1.pole();            // pole of equator
  L1.cosine();          // subtended arc
  L1.reflect(arg);      // reflect arg across L1
  L1.reflect(L2);       // reflect L2 across L1
  L1.collinear(L2);     // test for collinearity
  L1.cuts(L2);          //   or crossing
  L1.draw();            // arc
  L1.draw_front();      // front arc (same for back)
  L1.draw_line();       // draw entire line
  L1.line_front();      // front line (same for back)

Spherical Plotting

Parametrized paths on a Sphere S (by default the unit sphere) can be specified either by radial projection of a P-valued curve, or by stereographic projection of a plane curve given as a pair of double-valued functions:

  frontplot_R(phi, t_min, t_max, n, [S]);    // radial
  frontplot_N(f1, f2, t_min, t_max, n, [S]); // from north pole
  frontplot_S(f1, f2, t_min, t_max, n, [S]); // from south pole

Attempts to perform radial projection on a path through the origin will generate division-by-zero errors. Stereographic projection maps the equatorial plane \{x_3=0\} to the unit sphere by projection from the corresponding pole: N=(0,0,1), S=(0,0,-1).

Each spherical plot command has a back version that prints only the portion of the path invisible from the current viewpoint. Because of the way ePiX layers output, it is generally best to put hidden portions of the input before visible portions, with line width and/or style that suggests hidden lines.


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3.6.8 Data Plotting

Thanks in large part to code and ideas from Marcus Hanwell, files of numerical data can be created, manipulated, analyzed, plotted (paths, scatter plots, and histograms), read, and written. The format for a data file is one or more floating-point numbers per line, with the same number of entries per line. Anything that appears on a line after the LaTeX comment character \% is a comment.

ePiX provides two plot commands for file data. The first facilitates plotting selected columns; the second simplifies plotting the first two columns with one or both scales logarithmic. Either form can be used to plot selected columns logarithmically.

The general commands read numbers from two or three columns of a specified file, pass them as arguments to a P-valued function F, and plot the resulting points:

  plot("filename", STYLE, [i_1], [i_2], [i_3], [F]);
  plot("filename", STYLE, F, [i_1], [i_2], [i_3]);

The first argument is the name of the data file. The STYLE may be PATH, which joins the points in the order they appear, or any of the marker types in Table See table-marker, . The integers i_k specify columns from which to extract data; these default, respectively, to the first column, second column, and null (a column of zeroes). If the “coordinate system” F is omitted in the first command, it defaults to the Cartesian point constructor. The function F is mandatory in the second form; useful choices include log_log, log_lin, and lin_log, which plot the corresponding coordinate logarithmically.

Data Files

For more elaborate analysis, the data_file class presents an interface to a file as an ordered list of columns. There are two general ways to create a data_file: read in an external file, or generate data (up to three columns) using specified double-valued functions. In the constructors below, each function fi is a double-valued function of one variable.

  data_file DF("my_data"); // read data from disk file
  data_file DF(f1, t_min, t_max, num_pts); // values of f1
  data_file DF(f1, f2, t_min, t_max, num_pts);
  data_file DF(f1, f2, f3, t_min, t_max, num_pts);
  data_file DF(3);  // create empty data_file with 3 columns
  DF.read("file1"); // read a disk file

Columns of a data_file can be transformed by a user-specified function, averaged, correlated, extracted (for use by other code), scatter plotted, and written to a disk file at specified precision. Below, the function f is a double-valued function of one variable and F is a P-valued function of two or three variables, whose components are written back to the selected columns.

  DF.transform(f, i);    // apply f to selected column(s)
  DF.transform(F, i=1, j=2);
  DF.transform(F, i, j, k);

Basic statistical operations on columns are provided.

  DF.dot(i,j);        // dot product of columns i, j
  DF.avg(i);          // mean of column i
  DF.var(i);          // population variance
  DF.covar(i,j);      // covariance
  DF.regression(i,j); // plot regression line

A data_file is scatter plotted using syntax as described above. % The output is cropped automatically to the bounding box.

  DF.plot(STYLE, [i1], [i2], [i3], [F]);
  DF.plot(STYLE, F, [i1], [i2], [i3]);

Histograms and bar charts are described below.

A data_file can be written to a disk file as raw data, or in specified format. Below, fmt denotes a string-valued formatting function of two variables and myfile is the name of the disk file to be written.

  DF.precision(4);    // set to 4 significant figures
  DF.write("myfile"); // write as tab-separated columns
  DF.write("myfile", fmt, [i1], [i2]); // apply fmt to cols

A column can be extracted as a C++ vector for use by another function.

  DF.column(i);    // i-th column
  DF.column(f, i); // i-th column, transformed by f

Data Containers

ePiX provides a data_mask class for culling data from a file according to the values in a specified column, and a data_bins class for sorting data by value.

A data_mask consists of an interval of numbers and a “filter” function. The (closed, open, or half-open) interval is given as a string in standard mathematical notation, or by its endpoints (for a closed interval). The filter is a double-valued function of double, by default the identity, f(x)=x.

  data_mask dm("[a, b]", [f]);
  data_mask dm(a, b, [f]);

A data_mask “passes” inputs x if f(x) lies in the interval. The member function reverse() inverts this logical test.

The data_file class has prune functions to cull rows for which a specified column’s entry satisfies a data_mask’s criterion.

  DF.prune(dm, i);   // remove row if i-th column entry fails
  DF.prune(a, b, i); // remove row if i-th column outside [a,b]

A data_bins object models an interval divided at specified locations into “bins”, not necessarily of equal length. Numerical data is read in and the number of points in each bin counted. The lifetime of a data_bins object has two stages. First, “cuts” (endpoints of subintervals) are added. Then, once data is read, the cuts are “locked” and cannot subsequently be changed.

  // [xmin, xmax] divided into n equal intervals, 1 by default
  data_bins db(xmin, xmax, [n]);
  db.cut(x);  // add a cut at x (if x is in bounds)
  db.read(vector<double>); // read data, lock bins

A data_bins object can be plotted as a histogram (rectangles’ area is proportional to the bin population), bar chart (rectangles’ height is proportional to the bin population), or spline interpolation of a bar chart.

  db.histogram(c=1);  // c = vertical scale factor
  db.bar_chart(c=1);
  db.plot(c=1);

By default (c=1), the height of a bar chart rectangle is the fraction of the total population contained in the bin; thus, the height is always between 0 and 1. For a histogram, the height of a rectangle is the fraction of the population per horizontal unit in the bin; thus, the total area over an interval [a,b] does not depend on how [a,b] has been subdivided. The sample files dataplot.xp and histogram.xp illustrate use.

Statistical convention dictates cuts be chosen distinctly from data values; that is, values should all fall strictly within a bin, not at a boundary point. With large, unknown data sets, this convention may be difficult to uphold. ePiX attempts to handle anomalous data intelligently, by keeping counts of values “out-of-range” or “on-cut”.

In detail, if x<a or x>b is a data value, it is counted as out-of-range and does not contribute to the histogram population. If x=a or x=b, the value counts as both out-of-range and on-cut but is added to the population of the lowest or highest bin, respectively. Any other cut appearing as a data input is flagged as on-cut, and increments the population of each adjacent bin by one-half. When a histogram or bar chart is written, ePiX prints a warning message summarizing the number of anomalous data seen.

Error Bars

Simple horizontal and vertical error bars are provided. The final (optional) argument is the true height or width (respectively) in pt.

  h_error_bar(P location, double error, <mark type>, ht=6);
  v_error_bar(P location, double error, <mark type>, wd=6);

To create more complex elements, such as asymmetric bars, whisker plots, labeled error bars, and the like, write a custom function using true-size drawing, see page truesize. For example, a fillable, labeled, asymmetric, vertical rectangular error bar can be implemented (entirely in ePiX) like this:

  void error_bar(P loc, double lo, double hi, P offset,
                 std::string text, align, double wd=6)
  {
    const double width(pt_to_screen(0.5*wd)); // converts to 3pt
    rect(loc - P(width, lo), loc + P(width, hi));
    line(loc - P(width,  0), loc + P(width,  0));
    label(loc + P(width, 0), offset, text, align);
  }

For stylistic uniformity, functions such as this should be put into a library and used systematically. Section See section-extensions, outlines the process of writing, compiling, and using a custom library.


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3.6.9 Legends

A legend systematically labels different parts of a plot by associating visual “keys” with explanatory text. This tends to be most useful for plots containing several distinct but conceptually related graphs requiring contrast.

Visually, a legend is represented as an aligned list of rows, each containing a box (the key), a gap (the label skip), and some text. These rows are printed in a (usually large) masked label. By default, keys are 12pt squares bordered in black, the label skip is 6pt, the background is white, and the border is plain black. These attributes are controlled (simultaneously for all items) with member functions. Parameters of type double represent lengths in pt.

  legend L;
  L.backing(color);          // set background
  L.border(color, [double]); // set border color [and width]
  L.border(double);          // border width

  L.item_border(color, [double]);
  L.item_border(double);     // same, for item borders

  L.label_skip(double);
  L.key_size(double);

Legend Keys

There are three types of legend key, representing filled regions, paths, and markers. Fill and path keys get their visual attributes from the current drawing state. A mark key must be told the marker type. Each is created by specifying the item text.

  L.fill_item(text);
  L.path_item(text);
  L.mark_item(<mark type>, text);

Keys in the printed legend appear in the same order they are created in the input file.

Creating a Legend

Normally, a legend is defined near the start of a file, and an item is added at the point in the file where the corresponding object or plot is drawn, so that the item receives the correct attributes. A legend is placed into the figure with the draw function. The arguments have the same meaning as for ordinary labels.

  L.draw(P location, P offset, align);

All three arguments are mandatory. “Global” legend settings (border, backing, etc.) may be changed anywhere between the legend’s creation and draw function. The sample file legend.xp revisits the example on page taylor_poly, including a trig-labeled axis and a legend. The file shadeplot.xp contains filled keys, and dataplot.xp contains a legend with items of mixed type.


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3.7 More About C++

A textbook or similarly detailed reference is essential for serious study of C or C++. The C Programming Language, second edition, by Kernighan and Ritchie KnR, is an excellent, manageable resource for the basics of procedural programming. C++ Primer Plus, by Stephen Prata Prata, clearly lays out the extensive details of C++. Marshall Cline’s C++ FAQ Lite Cline engagingly discusses common points of confusion and furnishes tips on good design and programming.

C++ is a powerful, complex language whose syntax is similar to that of C, or to the scripting languages of Maple and Mathematica. An ePiX input file is source code for a C++ program that writes a LaTeX picture as output. ePiX may be viewed as an extension to C++; in the same way that LaTeX furnishes a high-level interface to TeX, ePiX provides a high-level bridge between the computational power of C++ and the LaTeX picture environment.

Like all high-level programming languages, C++ provides variables, functions, and control structures. Variables hold pieces of data such as numerical values and geometric locations, while functions operate on data. A control structure, such as a loop or conditional statement, affects the program’s course according to the program’s current state. A source file is composed primarily of “statements”, which perform actions ranging from defining variables and functions to setting figure attributes, performing calculations, and writing objects to the output file.


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3.7.1 Names and Types

Names of variables and functions may consist (only) of letters, digits, and the underscore character. The first character of a name must not be a digit, and the language standard reserves names starting with underscore for library authors. Names are case-sensitive, but it’s usually a bad idea to use a single name capitalized and uncapitalized in a single file. Numerous capitalization conventions are used informally; this document uses uncapitalized words separated by underscores for variables and functions, and occasionally uses all capitals for constants. As with names of LaTeX macros, primary considerations are clarity (of meaning), readability, and consistency.

Every variable in C++ has a “type”, such as integer (int), double-precision floating point (double), or Boolean (bool, true or false). ePiX provides additional types, the most common of which is P, for point. The construct P(x,y,z) creates (x,y,z), while P(x,y) gives (x,y,0), which is effectively the pair (x,y). A variable is defined by giving its type, its name, and an initializing expression.

In C and C++, a pointer variable holds the memory address of another variable. Pointers are useful for manipulating (possibly large) data structures through “handles” of fixed small size. C++ also provides references, which bind an additional name to an existing object and allow the object to be manipulated through this alias. The statements

  double  x=1;   // ordinary variable definition
  double& rx=x;  // bind a reference, signified by &

define a variable x having the value 1, and bind a reference variable rx to it. As long as rx exists, it refers to x. If the value of x changes, the value of rx does as well. Conversely, the value of x can be altered by assigning to rx. However, rx is the size of a pointer, regardless of the size of x, so rx can be passed efficiently in a function call.


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3.7.2 Functions

In a programming language, the term “function” refers to a block of code that is executable by name. A C++ function takes a list of “arguments”, and has a “return value”. This information, together with the function’s name, must be provided when a function is defined. A function may not be defined inside another function. However, a function may call other functions (including itself) as part of its execution:

  int factorial(unsigned int n)
  {
    if (n == 0) return 1;
    else return n*factorial(n-1);
  }

The special type void represents a “null type”. A function that performs an action but does not return a value has return type void. A function that takes no arguments may be viewed as taking a single void argument.

Every C++ program has a special function main(), which is called by the operating system when the program is run. The arguments of main() are command-line arguments, and the return type is an integer that signals success or failure. User-specified functions must be defined before the call to main() or in a separately-compiled file.

Functions in C++ may be as simple as an algebraic formula or as complex as an arbitrary algorithm. Greatest common divisors, finite sums, numerical derivatives and integrals, solutions of differential equations, recursively generated fractal curves, and curves of best fit are a few applications in ePiX. Several sample files contain user-level algorithms, which do not require knowledge of ePiX’s internal data structures. The source file functions.cc contains simple functions defined by algorithms, and functions.h illustrates the use of C++ templates. Other source files, such as plots.cc, may be consulted for Simpson’s rule, Euler’s method, and the like.


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3.7.3 Mathematical Functions

C++ knows several familiar mathematical functions by name:

  sqrt   exp   log   log10   ceil   floor   fabs

(fabs is the absolute value for a floating-point argument.) ePiX provides trig and inverse trig functions sensitive to angular mode:

  Cos       Sin       Tan
  Sec       Csc       Cot
  Acos      Asin      Atan

The inverse functions are principle branches.

The function pow(x,y) returns x^y when x>0, and atan2(y,x) (N.B. argument order) returns {Arg}(x+iy)\in(-\pi,\pi], the principle branch of arg. C++ knows many constants to 20 decimal places, such as M_PI, M_PI_2, and M_E for \pi, \pi/2, and e respectively. ePiX defines a few additional functions:

  sgn   zero   sinx   cb

sgn is the signum function; zero is the constant function; sinx is the function x\mapsto \sin(x)/x with the discontinuity removed; cb (for “Charlie Brown”) is the period-2 extension of the absolute value function on [-1,1].

The GNU C++ library defines other functions, including inverse hyperbolic functions (acosh, etc.), log and exp with base 2, 10, or arbitrary b (log2, etc.), the error and gamma functions (erf and tgamma [sic], respectively), and Bessel functions of first and second kind: j0, j1, y0, etc. Use, e.g., jn(5, ) to get higher indices. The GNU C library reference manual GNUC describes these and other functions in detail.

Functions may be used in subsequent definitions, and functions of two (or more) variables are defined in direct analogy to functions of one variable:

double f(double t) { return t*t*log(t*t); } // t^2 \ln(t^2)
double g(double s, double t) { return exp(2*s)*Sin(t); }

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3.7.4 Basics of Classes

Unlike C, C++ supports “object-oriented programming”. In a nutshell, a class is an abstraction in computer code of some concept, such as a point, a sphere, a mapping that can be plotted, or a camera. Classes allow a programmer to separate an object’s interface (the set of meaningful operations) from its implementation (the data structures and algorithms that realize the interface).

A class implementation consists of members (named data elements) and member functions (functions that belong to the class and have free access to members). C++ classes enforce access permissions on their members, protecting data from being manipulated except as promised by the interface.

An ideal interface looks like a black box: It hides the implementation completely. In order to cooperate, two classes need only know each other’s interfaces. This separation of form and function modularizes a program, and facilitates debugging, code reuse, and overall maintainability, particularly in large programs.

In simple programming, classes may be treated like built-in types. Each class object has its own member functions, whose call syntax differs from standard function calls:

  Circle C1(P(1,0), 1.5); // circle of given center and radius
  C1.draw();              // member function Circle::draw();

Naturally, this call draws the circle C1. Generally, a member function call consists of a class object’s name, a period, and the name of the member function. Arguments, if any, go in the parentheses after the member function name, just as in a regular function call.

A few short paragraphs cannot do more than scratch the surface of classes and object-oriented programming. For more details, please consult a book, such as Prata Prata or Stroustrup S, or Cline’s on-line FAQ Cline.


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3.7.5 References and Function Arguments

C and C++ are “call by value” languages. Variables are not passed to a function; instead a copy of the value is made, and the function operates only on the copy. Though this feature causes occasional inconvenience, it prevents an object from being altered unexpectedly by a function call in a different part of the program. Calling by value helps localize the logic of a program, and circumvents easy-to-write but extremely hard-to-find bugs.

In C++, a function may accept reference arguments. Passing an object by reference grants the calling function access to the object itself, not to a copy. There are two common applications: The object is a large data structure for which copying is “expensive”, or the function needs to modify its arguments (e.g., a function swap(x,y) that exchanges the values of x and y).

For the first situation, C++ provides the const keyword, which ensures the function does not modify its arguments, but accepts a reference merely for efficiency. Any attempt to modify a const argument will be caught by the compiler. Most ePiX commands accept const reference arguments.

The ability to pass function arguments by reference is sometimes touted as a feature in C++ texts. However, the technique circumvents the data encapsulation of calling by value, and should be avoided unless absolutely necessary. If a function merely “updates” the value of a variable, probably the variable should be of class type, and the update should be performed by a member function.

A function declaration must indicate that its arguments are references. The declarations below have the indicated idiomatic meanings.

  class matrix;
  double det(matrix); // call by value, perhaps inefficient
  matrix& transpose(matrix&); // probably changes its argument
  double trace(const matrix&); // does not change its argument

Unlike pointer arguments, reference arguments impose no syntactic burden on the user. If A is a matrix, then transpose(A); and trace(A); will compile. You need not declare explicit reference variables and pass those to the function.


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3.7.6 Overloading

C++ provides “overloading”: Multiple functions can be given the same name, so long as the number and/or type of their arguments differ. (It is not enough for the return types alone to differ. The compiler must be able to select a function from its calling syntax.) To the user, the appearance is that a single function intelligently handles multiple argument lists. Naturally, overloaded names should refer to functions that are conceptually related.


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3.7.7 Scope

A C++ statement ends with a semicolon. A collection of statements enclosed by curly braces is a “code block”, and may be viewed as a single logical statement. Curly braces determine a “scope”, inside which variable names may be re-used without ambiguity. Function bodies are code blocks, as are the alternatives associated to control statements. A variable defined between curly braces is said to be local to the scope in which it is defined; its value cannot be used out of scope. Variables should be declared in the smallest scope possible.

The compiler is not picky about spaces, tabs, and newlines, so an input file should be formatted to make local scopes visually apparent. emacs automatically indents code to reflect scope, though the default behavior does not please all users. As with variable naming, clarity and consistency are paramount.


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3.7.8 Headers and Pre-Processing

A C++ source file is compiled in multiple stages that occur transparently to the user. The first step, pre-processing, involves simple text replacement for file inclusion, macro expansion and conditional compilation. Next, the source is compiled and assembled: Human-readable language instructions are parsed, then represented in assembly language. Finally, the object files are linked: Function calls are resolved to hard-coded file offsets, possibly involving external library files, and the program instructions are packaged into an executable binary that the operating system can run.

Pre-processing is used much less in C++ than in C; the language itself supports safer and more featureful alternatives to macros, such as const variables and inline functions. File inclusion and conditional compilation are the chief uses of the pre-processor. Lines of the form

  #include <iostream>
  #include "epix.h"

cause the contents of a header file to be read into the source file. A header file contains variable and function declarations, statements that specify types and names but do not define actual data. Declarations tell the compiler just enough to resolve expressions and function calls without knowing specific values or function definitions.

Conditional compilation is similar to conditional LaTeX code. For example, a file might produce either color or monochrome output as follows:

#ifdef COLOR
  ...  // code for generating color figure
#endif /* COLOR */
#ifndef COLOR
  ...  // monochrome code
#endif /* undef COLOR */

The “compiler symbol” COLOR is an ordinary C++ name. To control compilation, either put a \#define COLOR line in the file, or (better) supply the flag on the command line: \\ epix -DCOLOR <file.xp> \\ Every \#ifdef must have a matching \#endif. Commenting the \#endif is a good habit; in a realistic file, the start and end of a conditional block may be separated by more than one screen.


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3.7.9 Comparison with LaTeX Syntax

As a programming language, C++ provides certain features common to all languages (such as LaTeX, MetaPost, Perl, Lisp\ldots) and adheres to rules of grammar. Salient differences between LaTeX and C++ include:

  1. Every C++ statement and function call must end with a semicolon. An omitted semicolon may result in a cryptic error message from the compiler. Pre-processor directives, which start with a \#, do not end with a semicolon.
  2. Backslash is an escape character in C++:
      // Put label $y=\sin x$ at (2,1)
      // Note single  ^ backslash in output
      label(P(2,1), P(0,0), "$y=\\sin x$"); 
      //       Double backslash ^^ in source
    
  3. Variable and function names may contain letters (including underscore) and digits only, are case sensitive, and must begin with a letter.
  4. Variables in C++ must have a declared type, such as int (integer) or double (double-precision floating point). If a variable has global scope and its value does not change, the definition should probably come in the preamble or at the beginning of main. Local variables should be defined in the smallest possible scope. Unlike C, C++ allows variables to be defined where they first appear.
  5. C++ requires explicit use of * to denote multiplication; juxtaposition is not enough. C++ does not support the use of ^ for exponentiation, e.g., t^2 is invalid. Instead, use t*t or pow(t,2).
  6. C++ has single- and multi-line comments. Everything between a double slash and the next newline is ignored, while the strings /* and */ delimit multi-line comments. A single-line comment may appear within a multi-line comment, but the compiler does not nest multi-line comments.

Between them, C and C++ have about 100 reserved keywords which cannot be used as function or variable names.


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3.8 Attribute Quick Reference

In the body of an input file, the “drawing state” determines the figure’s appearance. Attributes are declarations, set by commands that accept arguments of the stated type.

A len argument is a double-quoted string containing a number and a two-letter LaTeX length unit, such as "1.5pt" or "6cm". A color argument is a named primary (Red(), Cyan(), White(), etc.), a Color specified by densities (RGB(...), CMYK(...), etc.), or a Color object. Using Neutral() as a Color argument generally turns off the corresponding attribute.

Do not confuse Color constructors with the similarly-named (deprecated) lowercase attribute-setting commands, rgb(r,g,b), cmyk(c,m,y,k), red(d), etc. These commands affect text, paths, and filled regions. For example, the single command red() has the same effect as the three commands pen(Red()), fill(Red()), and label_color(Red()).


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4 Advanced Topics

This chapter covers ad hoc tricks and open-ended techniques that require relatively more programming sophistication. You will almost surely need an external C++ reference if you do not speak the language.


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4.1 Hidden Object Removal

ePiX writes the output file in the same order that objects appear in the input. The order is significant because PostScript builds a figure in layers: Objects are drawn over objects that come earlier in the file. Shaded polygons can be used to obtain surprisingly effective hidden object removal in surface meshes. This section describes the data structures defined in the source files surface.*.

The basic idea is to create a shaded polygon class that knows its approximate distance to the camera. For computational simplicity, a mesh “facet” is treated as a quadrilateral, located at the arithmetic mean of its vertices. A facet’s boundary is created from a map and a domain by tracing a fine mesh rectangle counterclockwise.

To draw a parametrized surface, facets are stored in a C++ vector, sorted in decreasing order of distance to the camera, and printed to the output file. If filling is active, the gray density of a facet depends on the cosine of the angle between the normal vector and the vector from the camera to the element.

This simple algorithm works surprisingly well when mesh elements intersect at most along complete edges. To incorporate line-like elements (e.g., coordinate axes, wire-mesh plots) with shaded surfaces, the best technique is often to order high-level scene elements manually, breaking up shaded surfaces (for example, with domain resizing or clipping) as necessary. The sample file saddle.xp illustrates possible techniques.

Shaded surfaces can be decorated with a bit of hackery. For example, the facet::draw function in facet.cc can be modified easily to draw line elements, tangents, or normal vectors along with the facet itself. The sample file decorate.xp contains a couple of ideas. (The decorations are activated by compiler flags; please consult the file itself for information on compiling.)


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4.2 Extensions

Thanks to a suggestion of Andrew Sterian, ePiX is extensible. User extensions span a spectrum, from header files that require only basic knowledge of C++ to separately compiled libraries that add substantial new features. The structure of the source code is outlined in Section See section-programmers, .


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4.2.1 Header Files

A C++ header file conventionally has suffix .h, as in myheader.h. To use this custom header, put a line \#include "myheader.h" in your source file.

User definitions can be easily and robustly implemented with “inline functions”. Inline functions are superficially similar to macros, but are far more safe and featureful (since they are handled by the compiler rather than by the pre-processor). Examples are

  inline void Bold() { pen(1.6); }
  inline void purple() { rgb(0.5, 0, 0.7); }
  inline void draw_square(double s) { rect(P(-s,-s),P(s,s)); }
  inline double cube(double x) { return pow(x,3); // x^3 }

The keyword void signifies a function that does not return a value, or (when used as an implicit parameter) a function that does not accept arguments. Inline function definitions are syntactically identical to ordinary function definitions, but must occur in a header file or in the source file where they are used. The examples above might be used in an input file as follows:

  Bold();  
  draw_square(cube(1.25));

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4.2.2 Compiling

The next few sections outline the creation of a “static library” on GNU/Linux, and explain how to incorporate custom features at runtime. The extensively-commented sample files std_F.cc and std_F.h illustrate the techniques described below, and may be used for guidance and experimentation.

A small library is usually written as a header file, which contains class and function declarations (also called “prototypes”), and a source file, which contains the actual code. Conventionally (under *nix), these files have extension .h and .cc respectively. Header and source files may “include” other header files, to incorporate additional functionality.

/* my_code.h */
#ifndef MY_CODE
#define MY_CODE
#include <cmath>   // standard library math header
#include "epix.h"  // ePiX header
using ePiX::P;

namespace Mine {   // to avoid name conflicts
  // functions for special relativity
  double lorentz_norm(const P&);
  bool   spacelike(const P&);
} // end of namespace
#endif /* MY_CODE */

This file exhibits two “safety features”. The three MY_CODE lines prevent the file from being included multiple times. In a file of this size, inclusion protection is overkill, but as your code base grows and the number of header files increases, this protection is essential. Second, the header introduces a “Mine” namespace. Inside this namespace, two functions are declared as prototypes, giving the function’s return type, name, and argument type(s). A header file should be commented fairly liberally, so that a year or two from now you’ll be able to decipher the file’s contents. For a longer file, version and contact information, an overall comment describing the file’s features, and license information are appropriate.

Next, the corresponding source file; definitions are also placed into the namespace, and must match their prototypes from the header file exactly.

/* my_code.cc */
#include "my_code.h"
using namespace ePiX;

namespace Mine {
  double lorentz_norm(const P& arg)
  {
    double x(arg.x1()), y(arg.x2()), z(arg.x3()); // extract coords
    return (y-x)*(y+x) + z*z; // -x^2 + y^2 + z^2
  }
  bool spacelike(const P& arg)
  {
    return (lorentz_norm(arg) > 0); // true if inequality is
  }
} // end of namespace

Copies of these files are included with the source code so you can experiment with them. Next, the source file must be “compiled”, “archived”, and “indexed”. In the commands below, the percent sign is the prompt.

% g++ -c my_code.cc
% ar -ru libcustom.a my_code.o
% ranlib libcustom.a

Please see your system documentation for details on command options and what each step does. For linking (below), the name of the library file must begin “lib” and have the extension .a. Once these steps are successfully completed, put the library libcustom.a and header file my_code.h in your project directory. You’re ready to use the code in an ePiX figure.


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4.2.3 Runtime Linking

The script epix allows input files to be linked with external libraries at run time, when the input file is compiled into a temporary executable.

epix recognizes command line options and passes them verbatim to the compiler. The most commonly used options are those of the form

   -I<include>     -L<libdir>     -l<lib>

For example, to link figure.xp against mylibs/libcustom.a, run the command

   epix -Lmylibs -lcustom figure

The options -I. -L. tell the compiler to look in the current directory for header and library files. Compiler options may appear in any order, but must come before the name of the input file(s).

Compiler options may be placed in the configuration file \$HOME/.epixrc, with syntax as above. A line in the config file that contains a pound sign (\#) is a comment, no matter where in the line the \# appears. If any non-comment line fails to start with a dash, the rest of the file is silently discarded. Command-line options are read before the config file.


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4.2.4 Using Multiple Versions

The script epix links by default against the C math library libm.a and the ePiX library libepix.a. The command option --no-defaults clears the header and include paths and removes libepix.a from the link list. The script may therefore be used with multiple versions of ePiX, a potentially useful feature if you regularly need to compile old source files, or simply prefer the syntax of an older version.

To install and use (say) Version 1.0.0, build the package according to its INSTALL instructions, but do not use the makefile to install. Instead, manually install the header and library only, using their version number:

# install -m 644 epix.h /usr/local/include/epix-1.0.h
# install -m 644 libepix.a /usr/local/lib/libepix-1.0.a

A non-system directory may be used instead of /usr/local. To use the old version, a source file must include the appropriate header file (which is identified by its version number). To compile, issue a command such as

epix --no-defaults -I/usr/local -L/usr/local -lepix-1.0 file.xp

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4.3 Programmer’s Guide

This section briefly surveys ePiX’s implementation, and is intended for (potential) programmers. The source code is divided into modules with small, well-defined responsibilities, but the user interface is mostly compatible with the syntax of Version 1.0. These constraints demanded a degree of implementation hiding. For example, the user-visible classes defined in Color.h, path.h, and screen.h contain only a pointer to the implementation class, and style data is hidden behind global commands.

Functionally, the code consists of the user interface; implementation class\-es comprising drawing attributes, spatial objects, screens and representations of their elements, and output; and miscellaneous utility functions. The headers in each group, and their contents, are described in Section See section-interface, .

The user interface headers are assembled into a single file, epix.h, and installed in /usr/local/include by default. In normal use, the shell scripts read only the user interface header. The individual headers, including the components of epix.h, are installed in /usr/local/include/epix. These are provided for authors of external libraries, who may need access to implementation details.


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4.3.1 External Packages

ePiX harnesses the computational power of C++ to the typographical capabilities of LaTeX. Consequently, ePiX should be viewed in part as a framework for expressing numerical data visually.

In the course of your work, you may develop specialized code filling a gap in ePiX’s functionality. If your code seems likely to be of interest to other users, please consider bundling it as an external package and notifying the ePiX community so your work can be linked from the project pages and distributed to interested users.

There are no formal requirements for external packages, but in the interest of uniformity contributed code should follow the GNU Coding Standards GNUCoding. At a minimum, an external package should build with the standard ./configure; make; make install commands, and the configure script should accept an option --with-epix for the user to specify a non-default ePiX install directory.

If an external package builds a static library, it should provide a single header containing all the package’s entry points, and enclose its interface in a namespace. Entry points should not collide with ePiX functions. Naming the package “epix-<...>” is a good idea, but not essential. For example, a package providing textual nodes and diagram layout might be named epix-nodes. Input files would use the package with the lines

  #include "epix-nodes.h"
  using namespace ePiX-nodes;

and be compiled with

  epix -lepix-nodes <file>

The user and internal interfaces of ePiX-1.2 are not likely to change. Still, it’s prudent to rely only on the user interface in contributed code whenever possible. Doing so also simplifies your work as an author; your library can simply include the user header file, and deal only with high-level objects and drawing attributes.


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4.3.2 User Interface

These files (in order) comprise the global header epix.h.

enums.h Marker, alignment, Riemann integral, and vector field types.

length.h Physical lengths, conceptually a number and two-letter LaTeX length unit.

interval.h Closed, open, and half-open interval ranges for data culling.

triples.h The P class.

Complex.h The Complex class.

functions.h Angle-sensitive trig functions, miscellaneous utility functions, the Deriv and Integral classes.

pairs.h Screen locations and displacements, with complex arithmetic operations.

Color.h The Color class interface, named primaries and constructors.

state.h Angle mode; clipping and cropping; label styles; filling; arrow head style; dot and tick sizes, dash length; line style; line and base pen attributes; color-setting commands.

frame.h Orthonormal bases.

domain.h Coordinate boxes for function plotting.

camera.h The camera.

screen.h The screen class.

picture.h Dimension-setting, offset, layout, decoration, verbatim text, and output format commands.

markers.h Point markers, axis labels, and coordinate axes.

axis.h Coordinate axes and labels in various styles.

legend.h Plot legends.

path.h The path class.

curves.h Polygons, arrows, ellipses, arcs, splines, coordinate grids, and recursive fractal curves.

circle.h The Circle shape object class.

plane.h The Plane shape object class.

segment.h The Segment shape object class.

sphere.h The Sphere shape object class.

intersections.h Shape object intersection operators.

plots.h Plotting commands.

surface.h Shaded surface plots.

data_mask.h Helper class for data pruning.

data_file.h Class for storing and representing data.

data_bins.h Class for sorting and counting data.

geometry.h Latitudes and longitudes; spherical plotting, arcs, polygons, and polyhedra; hyperbolic arcs.

Sline.h The Sline (spherical line) class.


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4.3.3 Implementation Classes

Color_Base.h The Color implementation interface.

Color_CMY.h The CMY color model.

Color_CMYK.h The CMYK color model.

Color_Gray.h The Gray color model.

Color_Neutral.h Each color class has a “Neutral” member that converts colors to that model by filtering. This file defines the unique “model-less” Neutral color for which filtering performs no action.

Color_RGB.h The RGB color model.

Color_Sep.h Classes for CMYK separation.

active_screen.h Simple manipulator for the active screen.

picture_data.h Picture implementation: two screen s (representing the canvas and the output page), true dimensions and offsets, pointer to output format, list of colors, and lists of verbatim text to write before and after printing the picture environment in the output file. For simplicity, the screens and dimensions are public; encapsulation from the user results from “hiding” this header.

Style Attributes

Declaration-style attributes are maintained with functions returning static references: the_angle_style(), the_arrowhead_style(), the_label_style(), the_mark_size(), the_paint_style(), and the_path_style(). Each function is declared in the analogously-named header.

angle_units.h Angular modes: radians , revolutions, and degrees.

arrow_style.h Arrow head style data: width, ratio, and inset.

label_style.h Text object style: Label and mask colors, padding, border color and width, alignment, font size, font face, and angle.

marker_style.h Dot and tick sizes.

paint_style.h Line and base pens, fill color.

path_style.h Solid, dashed, dotted lines.

pen_data.h The pen class.

Objects

arrow_data.h Arrow representation.

facet.h Shaded surface elements.

label_data.h Text (label and marker) objects.

legend_item.h Items for legend s.

path_data.h Path implementation.

spline.h The natural spline class.

spline_data.h Templates for quadratic and cubic splines.

Screen Representation

Elements in a screen are represented polymorphically as “tiles”, of which there are six types: glyph (textual elements), pen_arrow (arrows), pen_fill (filled regions), pen_line (path-like elements), verbatim (raw text), and legend_tile (legends).

Border and background shape are dictated by the screen_mask class. The cropping algorithm assumes the contour of a screen mask is convex.

affine.h Affine maps.

cropping.h Screen mask cropping.

glyph.h Markers and labels.

legend_tile.h Screen representation of a legend .

mask_diamond.h Screen diamond mask.

mask_ellipse.h Screen elliptical mask.

mask_rectangle.h Screen rectangular mask.

pen_arrow.h Arrows.

pen_fill.h Filled regions.

pen_line.h Paths.

screen_data.h Screen implementation class.

screen_mask.h Screen mask interface.

tile.h Screen element interface.

verbatim.h Text in output stream.

Output

Output is divided into a couple of high-level operations and several “atomic” low-level operations. To create a new output format, one need only implement the format interface for the desired file type. Paths and filled regions may be implemented however the output type dictates.

eepic.h eepic macros.

fmt_template.h “skeleton” header for new output formats.

format.h The output interface.

pst.h PSTricks macros.

tikz.h tikz macros.

Utilities

Color_Utils.h Functions for setting color channel densities.

clipping.h The clip box.

constants.h Global constants: Line widths; dot, tick, and dash sizes; arrowhead parameters; miscellaneous numerical constants and internal parameters.

crop_algorithms.h Path and loop clipping templates.

deriv.h Finite difference template.

edge_data.h Path element representation template.

errors.h Warning and error messages.

frac.h Rational numbers and operators.

halfspace.h Halfspace cutting.

hatching.h Filling regions in eepic.

lens.h Camera lenses.

map.h Wrappers for templated plotting.

plot_algorithms.h Plotting templates.

screen_crop.h Crop paths and loops.

utils.h Truncation, date and time, line breaking.


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Appendix A Software Freedom

Academics in general, and mathematicians in particular, depend on free exchange of information. We prove theorems or establish experimental results, write up formal accounts, place preprints on public file servers, and submit papers to peer-reviewed journals. If accepted, the results—data, techniques, methods of reasoning, citations, and conclusions—are published in print and become part of the public record, governed by copyright law. Libraries purchase journal subscriptions, but researchers and scholars may use ideas from the literature merely by giving appropriate citations in their own work. “Theft” arises from false claims of authorship.

Carried over to software, the academic process would guarantee rights similar to those provided by the GNU General Public License (GPL):

In reality, attitudes toward software differ markedly. Most academics work on a proprietary platform, use proprietary software for research and teaching, and share information with colleagues and students in proprietary, even obfuscated, data formats. Contrary to the academic ethic, proprietary software licenses restrict access to information: preventing users from learning how a program works internally (“reverse engineering”), limiting the number of users who may run a piece of software, and forbidding users from running (or sometimes even installing) a purchased copy on multiple machines.

Restrictions on use aside, if one cannot examine a program’s source code, one cannot fully trust the output, any more than one can trust (for purposes of scientific publication) results of a commercial testing lab.

Consider a hypothetical future world in which scholarly results are disseminated like software. Instead of subscriptions, journals sell licenses granting readership to a specified number of individuals. Photocopying an article for a class or research seminar constitutes “piracy”, though if the institution has purchased a sufficiently large site license the teacher or speaker may bring the physical volume to class and project the pages onto a screen.

Of course, reading an article is scarcely enlightening. Mathematics papers contain only the statements of theorems. Merely opening the journal binds the reader to a lengthy legal agreement, stating that theorems be used only for specific purposes and threatening serious legal consequences for attempting to discover the author’s proofs.

The actions of a single student, employee, or faculty member can expose an institution to a costly “journal audit” from the Mathematical Society of America (MSA), with the institution responsible for legal costs if the audit reveals license violations anywhere in the organization.

Mathematicians who long for the Old Days when papers contained proofs and were shared freely are dismissed as idealistic cranks or labeled anti-business communists. Common knowledge asserts the obvious superiority of proprietary journals, and the necessity of licenses for keeping mathematicians gainfully employed.

Back in our world, some vendors have attempted to placate opponents of closed source with “shared source” licenses, under which one may sign a non-disclosure agreement and subsequently examine source code. In the future world analogy, a shared source agreement would allow journal licensees to sign an NDA, then see the proofs of theorems. Readers could thereby correct errors in proofs (benefitting the publisher by improving the reliability of the journal), but would be legally forbidden from using the ideas elsewhere (denying benefit to other mathematicians).

Legally and conceptually codifying software as a commodity ignores a fundamental reality: Like an idea or recipe, software can be copied without loss of the original. The perception of “theft” by copying arises from an artificial belief that software has an owner who must be monetarily compensated each time a person acquires a copy. The nature of software does not enforce the “sale” model in the way services and physical commodities do. It therefore seems philosophically inappropriate to treat software as a commodity, and perilous to conform the legal system to the enforcement of such a model.

At its best, software enhances our productivity and creativity. Sharing software, like sharing ideas, benefits a larger number of people without detriment to existing users. I hope this modest program is, in conjunction with the much larger efforts of others (especially Donald Knuth, Richard Stallman, and the many people who have contributed to the authorship of LaTeX and its packages), useful to you in your mathematical work.

Please visit the Free Software Foundation, at http://www.fsf.org, to learn more about free software and how you can contribute to its development and adoption.


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Appendix B Acknowledgments

ePiX is built on the work of many people, many of whom I am unaware. The following people have contributed, sometimes unknowingly but always generously:

Infrastructure Donald Knuth, Conrad Kwok, Leslie Lamport, Tim Morgan, Piet van Oostrum, Sunil Podar, Richard Stallman, Till Tantau, Herbert Vo\ss, Timothy van Zandt

Enhancements Jay Belanger, Robin Blume-Kohout, Julian Gilbey, Marcus Hanwell, Yvon Henel, Svend Daugaard Pedersen, Andrew Sterian

Porting and packaging Youn\‘es Hafri (Crux), Julian Gilbey (Debian); Tsuguru Kato (FreeBSD); Markus Dittrich, Danny van Dyk, Christian Faulhammer, Olivier Fisette, Chris Gianelloni, Michael Hanselmann, Marcus Hanwell, David Holm, Peter Johanson, Patrick Kursawe, Tobias Scherbaum, Markus Ullmann (Gentoo); Guido Gonzato (RPM); Rene Rebe (T2)

Debugging, advice, and other assistance Maik Beckmann, Jay Belanger, Felipe Paulo Guazzi Bergo, Karl Berry, Robin Blume-Kohout, Aran Clauson, Patrick Cousot, Stephen Gibson, Julian Gilbey, Dov Grobgeld, Bob Grover, Jim Hefferon, Jacques L’helgoualc’h, Yvon Henel, Hartmut Henkel, Herng-Jeng Jou, Walter Kehowski, Paul Kornman, Kevin McCormick, Ross Moore, Mike Protts, Thorsten Riess, Jean-Michel Sarlat, Alan Sill, Neel Smith, Michael Somos, Andrew Sterian, Ryszard Tanas, Ben Tillman, Kai Trukenmueller, Torbjorn Vik, Wenguang Wang, Gabe Weaver, Mariusz Wodzicki


Next: , Previous: , Up: Top   [Contents][Index]

Function Index

Jump to:   A   B   C   D   E   F   G   H   I   J   L   M   N   O   P   Q   R   S   T   U   V   W   Y   Z  
Index Entry  Section

A
aarrow: Path-Like Elements
Acos: Mathematical Functions
activate: Screens and Page Layout
affine: Screens and Page Layout
apply: Screens and Page Layout
arc: Path-Like Elements
arc_arrow: Path-Like Elements
arc_measure: Path-Like Elements
Arg: Geometric Data Structures
arrow: Path-Like Elements
arrow_inset: Path-Like Elements
arrow_ratio: Path-Like Elements
arrow_width: Path-Like Elements
Asin: Mathematical Functions
Atan: Mathematical Functions
axis: Coordinate Axes and Labels
axis.align: Coordinate Axes and Labels
axis.align_labels: Coordinate Axes and Labels
axis.dec: Coordinate Axes and Labels
axis.draw: Coordinate Axes and Labels
axis.draw_labels: Coordinate Axes and Labels
axis.draw_ticks: Coordinate Axes and Labels
axis.frac: Coordinate Axes and Labels
axis.label_rep: Coordinate Axes and Labels
axis.precision: Coordinate Axes and Labels
axis.sci: Coordinate Axes and Labels
axis.subdivide: Coordinate Axes and Labels
axis.tag: Coordinate Axes and Labels
axis.tag235: Coordinate Axes and Labels
axis.tags: Coordinate Axes and Labels
axis.tick_ratio: Coordinate Axes and Labels
axis.trig: Coordinate Axes and Labels
axis.unmark: Coordinate Axes and Labels
axis.untag: Coordinate Axes and Labels
axis_break: Coordinate Axes and Labels

B
backing: Screens and Page Layout
backplot_N: Non-Euclidean Geometry
backplot_R: Non-Euclidean Geometry
backplot_S: Non-Euclidean Geometry
back_arc: Non-Euclidean Geometry
back_arc2: Non-Euclidean Geometry
back_cube: Non-Euclidean Geometry
back_dodeca: Non-Euclidean Geometry
back_icosa: Non-Euclidean Geometry
back_latitude: Non-Euclidean Geometry
back_line: Non-Euclidean Geometry
back_longitude: Non-Euclidean Geometry
back_octa: Non-Euclidean Geometry
back_tetra: Non-Euclidean Geometry
back_triangle: Non-Euclidean Geometry
bbold: Paths
bbox: Text Objects
begin: File Structure
Black: Constructors
Blue: Constructors
bold: Paths
border: Screens and Page Layout
bottom_axis: Coordinate Axes and Labels
bounding_box: Picture Size and Aspect Ratio
box: Text Objects

C
Camera.at: The Camera
Camera.clip_range: The Camera
Camera.focus: The Camera
Camera.look_at: The Camera
Camera.pan: The Camera
Camera.range: The Camera
Camera.roll: The Camera
Camera.tilt: The Camera
canvas: Screens and Page Layout
cb: Mathematical Functions
circ: Text Objects
Circle: Geometric Data Structures
circle: Path-Like Elements
cis: Geometric Data Structures
clip_box: Clipping
clip_face: Clipping
clip_restore: Clipping
clip_slice: Clipping
clip_to: Clipping
CMY: Constructors
CMYK: Constructors
cmyk: Color Declarations
CMY_Black: Constructors
CMY_White: Constructors
collinear: Non-Euclidean Geometry
Color.alpha: Color Operations
Color.blend: Color Operations
Color.filter: Color Operations
Color.invert: Color Operations
Color.superpose: Color Operations
Complex.conj: Geometric Data Structures
Complex.im: Geometric Data Structures
Complex.re: Geometric Data Structures
Cos: Mathematical Functions
CosC: Geometric Data Structures
CoshC: Geometric Data Structures
cosine: Non-Euclidean Geometry
Cot: Mathematical Functions
CotC: Geometric Data Structures
CothC: Geometric Data Structures
crop: Screens and Page Layout
crop_diamond: Screens and Page Layout
crop_ellipse: Screens and Page Layout
Csc: Mathematical Functions
CscC: Geometric Data Structures
CschC: Geometric Data Structures
cuts: Non-Euclidean Geometry
Cyan: Constructors
cyl: Geometric Data Structures

D
dart: Path-Like Elements
dart_field: Calculus Plotting
dashed: Paths
dash_size: Paths
data_bins: Data Plotting
data_bins.bar_chart: Data Plotting
data_bins.cut: Data Plotting
data_bins.histogram: Data Plotting
data_bins.plot: Data Plotting
data_bins.read: Data Plotting
data_file: Data Plotting
data_file.avg: Data Plotting
data_file.column: Data Plotting
data_file.covar: Data Plotting
data_file.dot: Data Plotting
data_file.plot: Data Plotting
data_file.precision: Data Plotting
data_file.read: Data Plotting
data_file.regression: Data Plotting
data_file.transform: Data Plotting
data_file.var: Data Plotting
data_file.write: Data Plotting
data_mask: Data Plotting
data_mask.prune: Data Plotting
ddot: Text Objects
deactivate: Screens and Page Layout
Deriv: Calculus Plotting
Deriv.eval: Calculus Plotting
Deriv.left: Calculus Plotting
Deriv.right: Calculus Plotting
disk_line: Non-Euclidean Geometry
domain: Function Plotting
domain.resize: Function Plotting
domain.slice: Function Plotting
domain.slices: Function Plotting
domain_list: Function Plotting
domain_list.add: Function Plotting
dot: Text Objects
dotted: Paths
dot_size: Text Objects
draw: Geometric Data Structures
draw: Non-Euclidean Geometry
draw_back: Non-Euclidean Geometry
draw_front: Non-Euclidean Geometry
draw_line: Non-Euclidean Geometry

E
ellipse: Path-Like Elements
ellipse: Path-Like Elements
ellipse_bottom: Path-Like Elements
ellipse_left: Path-Like Elements
ellipse_right: Path-Like Elements
ellipse_top: Path-Like Elements
end: File Structure
envelope: Calculus Plotting
expC: Geometric Data Structures
extract: Screens and Page Layout
extract_diamond: Screens and Page Layout
extract_ellipse: Screens and Page Layout

F
fill: Filled Regions
flow: Calculus Plotting
font_face: Text Objects
font_size: Text Objects
frame: Geometric Data Structures
frontplot_N: Non-Euclidean Geometry
frontplot_R: Non-Euclidean Geometry
frontplot_S: Non-Euclidean Geometry
front_arc: Non-Euclidean Geometry
front_arc2: Non-Euclidean Geometry
front_cube: Non-Euclidean Geometry
front_dodeca: Non-Euclidean Geometry
front_icosa: Non-Euclidean Geometry
front_line: Non-Euclidean Geometry
front_octa: Non-Euclidean Geometry
front_tetra: Non-Euclidean Geometry
front_triangle: Non-Euclidean Geometry

G
Gray: Constructors
Green: Constructors
grid: Coordinate Axes and Labels
grid: Coordinate Axes and Labels

H
hyperbolic_line: Non-Euclidean Geometry
h_axis: Coordinate Axes and Labels
h_axis_labels: Coordinate Axes and Labels
h_axis_log_labels: Coordinate Axes and Labels
h_axis_log_masklabels: Coordinate Axes and Labels
h_axis_masklabels: Coordinate Axes and Labels
h_error_bar: Data Plotting
h_log_axis: Coordinate Axes and Labels
h_scale: Screens and Page Layout
h_shear: Screens and Page Layout

I
import: Screens and Page Layout
include: File Format
inf: Calculus Plotting
inlay: Screens and Page Layout
inlay_horizontal: Screens and Page Layout
inlay_vertical: Screens and Page Layout
inset: Screens and Page Layout
Integral: Calculus Plotting
Integral.eval: Calculus Plotting

J
J: Geometric Data Structures

L
label: Inset Images
label: Text Objects
label_border: Text Objects
label_color: Text Objects
label_mask: Text Objects
label_pad: Text Objects
latitude: Non-Euclidean Geometry
left_axis: Coordinate Axes and Labels
legend: Legends
legend.backing: Legends
legend.border: Legends
legend.draw: Legends
legend.fill_item: Legends
legend.item_border: Legends
legend.key_size: Legends
legend.label_skip: Legends
legend.mark_item: Legends
legend.path_item: Legends
line: Path-Like Elements
Line: Path-Like Elements
line_back: Non-Euclidean Geometry
line_front: Non-Euclidean Geometry
line_style: Paths
log1_grid: Coordinate Axes and Labels
log2_grid: Coordinate Axes and Labels
logC: Geometric Data Structures
log_grid: Coordinate Axes and Labels
longitude: Non-Euclidean Geometry

M
Magenta: Constructors
move_to: Geometric Data Structures

N
newton: Calculus Plotting
nofill: Filled Regions
norm: Geometric Data Structures
no_label_border: Text Objects

O
ode_plot: Calculus Plotting

P
P: Geometric Data Structures
P.x1: Geometric Data Structures
P.x2: Geometric Data Structures
P.x3: Geometric Data Structures
path: The Path Class
path.close: The Path Class
path.draw: The Path Class
path.fill: The Path Class
pen: Paths
pen: Coordinate Axes and Labels
picture: File Structure
picture: Picture Size and Aspect Ratio
plain: Paths
Plane: Geometric Data Structures
plot: Function Plotting
plot: Function Plotting
plot: Function Plotting
plot: Calculus Plotting
plot: Data Plotting
polar: Geometric Data Structures
polarplot: Function Plotting
polar_grid: Coordinate Axes and Labels
pole: Non-Euclidean Geometry
postcomp: Screens and Page Layout
post_write: Strings and Raw Output
powC: Geometric Data Structures
pre_write: Strings and Raw Output
pt_to_screen: Screens and Page Layout

Q
quad: Path-Like Elements

R
rect: Path-Like Elements
Red: Constructors
reflect: Screens and Page Layout
reflect: Non-Euclidean Geometry
RGB: Constructors
rgb: Color Declarations
riemann_sum: Calculus Plotting
right_angle: Path-Like Elements
right_axis: Coordinate Axes and Labels
ring: Text Objects
rootC: Geometric Data Structures
rotate: Screens and Page Layout

S
scale: Screens and Page Layout
scale: Geometric Data Structures
scenery: Function Plotting
scenery: Function Plotting
scenery.add: Function Plotting
scenery.add: Function Plotting
scenery.draw: Function Plotting
Sec: Mathematical Functions
SecC: Geometric Data Structures
SechC: Geometric Data Structures
Segment: Geometric Data Structures
set_crop: Screens and Page Layout
sgn: Mathematical Functions
shadeplot: Function Plotting
shift: Screens and Page Layout
shift: Geometric Data Structures
Sin: Mathematical Functions
SinC: Geometric Data Structures
SinhC: Geometric Data Structures
sinx: Mathematical Functions
Sline: Non-Euclidean Geometry
slope_field: Calculus Plotting
solid: Paths
sph: Geometric Data Structures
Sphere: Geometric Data Structures
spline: Path-Like Elements
spot: Text Objects
sqrtC: Geometric Data Structures
sup: Calculus Plotting
surface: Function Plotting
surface: Function Plotting
surface: Function Plotting
surface_rev: Function Plotting
surface_rev: Function Plotting

T
Tan: Mathematical Functions
TanC: Geometric Data Structures
TanhC: Geometric Data Structures
tan_field: Calculus Plotting
tan_line: Calculus Plotting
top_axis: Coordinate Axes and Labels
triangle: Path-Like Elements

U
unitlength: Picture Size and Aspect Ratio

V
vector_field: Calculus Plotting
v_axis: Coordinate Axes and Labels
v_axis_labels: Coordinate Axes and Labels
v_axis_log_labels: Coordinate Axes and Labels
v_axis_log_masklabels: Coordinate Axes and Labels
v_axis_masklabels: Coordinate Axes and Labels
v_error_bar: Data Plotting
v_log_axis: Coordinate Axes and Labels
v_scale: Screens and Page Layout
v_shear: Screens and Page Layout

W
White: Constructors
write: Strings and Raw Output

Y
Yellow: Constructors

Z
zero: Mathematical Functions

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Previous: , Up: Top   [Contents][Index]

Concept Index

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Index Entry  Section

A
Affine map: Screens and Page Layout
Affine map!action on labels: Text Objects
Angle!mode: Geometric Data Structures
Angle!mode: Mathematical Functions
Angle!units: Angular Mode
Angle!units: Text Objects
Animation: Animation
Animation: Animation
Arrow: Path-Like Elements
Aspect ratio%: Picture Size and Aspect Ratio
Axes: Coordinate Axes and Labels
Axes!broken: Coordinate Axes and Labels
Axes!labeling: Coordinate Axes and Labels
Axes!labeling: Coordinate Axes and Labels
Axes!logarithmic: Coordinate Axes and Labels

B
Backslash character%: Strings and Raw Output
Bar Charts: Data Plotting
Bounding box: Screens and Page Layout

C
Camera: The Camera
Camera: The Camera
Camera: Hidden Object Removal
Camera!color separation: The Camera
Camera!lens: The Camera
Camera!manipulation: The Camera
Camera!manipulation: The Camera
Camera!viewpoint: The Camera
canvas%: The Drawing Model
Circle: Geometric Data Structures
Classes: Basics of Classes
Classes: Basics of Classes
Clipping: Clipping
Clipping%: Clipping
Color: Color
Color: Color Declarations
Color!blencing%: Color Operations
Color!declaration: Color Declarations
Color!density%: Constructors
Color!density%: Constructors
Color!primitive: Constructors
Color!separation: The Camera
Color!surface plotting: Function Plotting
Color!transparency: Constructors
Color%: Reference Manual
Complex number: Geometric Data Structures
Complex number: Geometric Data Structures
Conditional statement: More About C++
Cplusplus@C++: C++ Basics
Cplusplus@C++: Conditionals and Loops
Cplusplus@C++: More About C++
Cplusplus@C++: Comparison with LaTeX Syntax
Cropping: Screens and Page Layout

D
Data plotting: Data Plotting
Data plotting: Data Plotting
Domain: Function Plotting
Domain: Function Plotting
Drawing style: Paths
Drawing style: Paths

E
Emacs@emacs: Software Dependencies
Emacs@emacs: Running ePiX
Emacs@emacs: File Format
Emacs@emacs: Animation
Error bars: Data Plotting

F
Filling: Filled Regions
Fonts: Text Objects
Free software: Introduction
Free software: Software Freedom
Free software: Software Freedom
Function: More About C++
Function: Functions
Function: Overloading
Function!call syntax: References and Function Arguments
Function!class member: Basics of Classes
Function!complex: Geometric Data Structures
Function!mathematical: Mathematical Functions
Function!mathematical: Mathematical Functions
Function!overloaded: Overloading
Function!returning void: Functions

G
Graph paper: Coordinate Axes and Labels
Graphical interface: Tutorial

H
Header file listing: User Interface
Header file listing: Implementation Classes
Hidden object removal: Hidden Object Removal
Hidden object removal: Hidden Object Removal
Histograms: Data Plotting

I
Input file!comment in: Comparison with LaTeX Syntax
Input file!conditional compilation: Headers and Pre-Processing
Installation: Software Dependencies
Installation: Installation
Intersection: Geometric Data Structures

L
Label: Text Objects
Label: Text Objects
Label: Comparison with LaTeX Syntax
Label!alignment: Text Objects
Label!alignment: Text Objects
Label!alignment: Text Objects
Label!axis: Coordinate Axes and Labels
Label!axis: Coordinate Axes and Labels
Label!backslash in: Text Objects
Label!double quotes: Text Objects
Label!fonts in: Text Objects
Label!masked%: Text Objects
Label!offset%: Text Objects
Label!rotated: Text Objects
Layout: Layout Tricks
Layout: Screens and Page Layout
Legends: Legends
Length%: Reference Manual
Length%: Picture Size and Aspect Ratio
Line style: Paths
Line style: Paths

M
Mac OS X: Software Dependencies
Marker: Text Objects
Marker!types of: Text Objects
Marker!types of: Text Objects

O
Output file!writing directly to%: Strings and Raw Output
Output format: File Structure

P
Path: Filled Regions
Path: Paths
Path!class: The Path Class
Path!filled: Filled Regions
Path!style: Paths
Picture!aspect ratio: Picture Size and Aspect Ratio
Picture!bounding box: Picture Size and Aspect Ratio
Picture!offset%: Picture Size and Aspect Ratio
Picture!size%: Picture Size and Aspect Ratio
Plane: Geometric Data Structures
Plotting: Function Plotting
Plotting: Calculus Plotting
Plotting!calculus: Calculus Plotting
Plotting!calculus: Calculus Plotting
Plotting!data: Data Plotting
Plotting!data: Data Plotting
Plotting!spherical: Non-Euclidean Geometry
Plotting!spherical: Non-Euclidean Geometry
Plotting!surface: Function Plotting
Plotting!surface: Function Plotting
Point: Geometric Data Structures
Point: Geometric Data Structures
Pointer: Names and Types
Preamble: File Structure

S
Screen: The Camera
Screen: The Camera
Screen: Screens and Page Layout
Screen: Screens and Page Layout
Screen!active: Screens and Page Layout
Segment: Geometric Data Structures
Self-contained figure%: Strings and Raw Output
Sline: Non-Euclidean Geometry
Sphere: Geometric Data Structures
Stereograms: Stereograms
string%: Strings and Raw Output

V
Variable!local: Scope
Variable!names: Names and Types
Variable!names: Names and Types
Variable!pointer: Names and Types
Variable!reference: References and Function Arguments
Variable!type of: Names and Types
Viewpoint: The Camera

W
Windows operating system: Setting up an Environment Under Windows
Windows operating system: Setting up an Environment Under Windows
Writing directly to output file%: Strings and Raw Output

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