LaTeX/Producing Mathematical Graphics

Most people use LaTeX for typesetting their text. But as the non content and structure oriented approach to authoring is so convenient, LaTeX also offers a, if somewhat restricted, possibility for producing graphical output from textual descriptions. Furthermore, quite a number of LaTeX extensions have been created in order to overcome these restrictions. In this chapter, you will learn about a few of them.

Overview

The picture environment allows programming pictures directly in LaTeX. On the one hand, there are rather severe constraints, as the slopes of line segments as well as the radii of circles are restricted to a narrow choice of values. On the other hand, the picture environment of LaTeX2e brings with it the \qbezier command, "q" meaning quadratic. Many frequently used curves such as circles, ellipses, or catenaries can be satisfactorily approximated by quadratic Bézier curves, although this may require some mathematical toil. If, in addition, a programming language like Java is used to generate \qbezier blocks of LaTeX input files, the picture environment becomes quite powerful.

Although programming pictures directly in LaTeX is severely restricted, and often rather tiresome, there are still reasons for doing so. The documents thus produced are "small" with respect to bytes, and there are no additional graphics files to be dragged along.

Packages like epic, eepic or pstricks help to eliminate the restrictions hampering the original picture environment, and greatly strengthen the graphical power of LaTeX.

While the former two packages just enhance the picture environment, the pstricks package has its own drawing environment, pspicture. The power of pstricks stems from the fact that this package makes extensive use of PostScript possibilities. In addition, numerous packages have been written for specific purposes. One of them is XY-pic, described at the end of this chapter. A wide variety of these packages is described in detail in The LaTeX Graphics Companion (not to be confused with The LaTeX Companion).

Perhaps the most powerful graphical tool related with LaTeX is MetaPost, the twin of Donald E. Knuth’s METAFONT. MetaPost has the very powerful and mathematically sophisticated programming language of METAFONT. Contrary to METAFONT, which generates bitmaps, MetaPost generates encapsulated PostScript files, which can be imported in LaTeX. For an introduction, see A User’s Manual for MetaPost. A very thorough discussion of LaTeX and TEX strategies for graphics (and fonts) can be found in TEX Unbound.

The picture Environment

Basic Commands

A picture environment is available in any LaTeX distribution, without the need of loading any external package. This environment is created with one of the two commands

\begin{picture}(x, y) ... \end{picture}

or

\begin{picture}(x, y)(x0, y0) ... \end{picture}

The numbers x, y, x0, y0 refer to \unitlength, which can be reset any time (but not within a picture environment) with a command such as

\setlength{\unitlength}{1.2cm}

The default value of \unitlength is 1pt. The first pair, $(x,y)$, effects the reservation, within the document, of rectangular space for the picture. The optional second pair, $(x_0,y_0)$, assigns arbitrary coordinates to the bottom left corner of the reserved rectangle.

Most drawing commands have one of the two forms

\put(x, y){object}

or

\multiput(x, y)(dx, dy){n}{object}

Bézier curves are an exception. They are drawn with the command

\qbezier(x1, y1)(x2, y2)(x3, y3)

Line Segments

Line segments are drawn with the command:

\put(x, y){\line(x1, y1){length}}

The \line command has two arguments:

1. a direction vector,
2. a length.

The components of the direction vector are restricted to the integers (−6, −5, ... , 5, 6) and they have to be coprime (no common divisor except 1). The figure below illustrates all 25 possible slope values in the first quadrant. The length is relative to \unitlength. The length argument is the vertical coordinate in the case of a vertical line segment, the horizontal coordinate in all other cases.

\setlength{\unitlength}{5cm} \begin{picture}(1,1) \put(0,0){\line(0,1){1}} \put(0,0){\line(1,0){1}} \put(0,0){\line(1,1){1}} \put(0,0){\line(1,2){.5}} \put(0,0){\line(1,3){.3333}} \put(0,0){\line(1,4){.25}} \put(0,0){\line(1,5){.2}} \put(0,0){\line(1,6){.1667}} \put(0,0){\line(2,1){1}} \put(0,0){\line(2,3){.6667}} \put(0,0){\line(2,5){.4}} \put(0,0){\line(3,1){1}} \put(0,0){\line(3,2){1}} \put(0,0){\line(3,4){.75}} \put(0,0){\line(3,5){.6}} \put(0,0){\line(4,1){1}} \put(0,0){\line(4,3){1}} \put(0,0){\line(4,5){.8}} \put(0,0){\line(5,1){1}} \put(0,0){\line(5,2){1}} \put(0,0){\line(5,3){1}} \put(0,0){\line(5,4){1}} \put(0,0){\line(5,6){.8333}} \put(0,0){\line(6,1){1}} \put(0,0){\line(6,5){1}} \end{picture}

Arrows

Arrows are drawn with the command

\put(x, y){\vector(x1, y1){length}}

For arrows, the components of the direction vector are even more narrowly restricted than for line segments, namely to the integers (−4, −3, ... , 3, 4). Components also have to be coprime (no common divisor except 1). Notice the effect of the \thicklines command on the two arrows pointing to the upper left.

\setlength{\unitlength}{0.75mm} \begin{picture}(60,40) \put(30,20){\vector(1,0){30}} \put(30,20){\vector(4,1){20}} \put(30,20){\vector(3,1){25}} \put(30,20){\vector(2,1){30}} \put(30,20){\vector(1,2){10}} \thicklines \put(30,20){\vector(-4,1){30}} \put(30,20){\vector(-1,4){5}} \thinlines \put(30,20){\vector(-1,-1){5}} \put(30,20){\vector(-1,-4){5}} \end{picture}

Circles

The command

\put(x, y){\circle{diameter}}

draws a circle with center (x, y) and diameter (not radius) diameter. The picture environment only admits diameters up to approximately 14mm, and even below this limit, not all diameters are possible. The \circle* command produces disks (filled circles). As in the case of line segments, one may have to resort to additional packages, such as eepic or pstricks.

\setlength{\unitlength}{1mm} \begin{picture}(60, 40) \put(20,30){\circle{1}} \put(20,30){\circle{2}} \put(20,30){\circle{4}} \put(20,30){\circle{8}} \put(20,30){\circle{16}} \put(20,30){\circle{32}} \put(40,30){\circle{1}} \put(40,30){\circle{2}} \put(40,30){\circle{3}} \put(40,30){\circle{4}} \put(40,30){\circle{5}} \put(40,30){\circle{6}} \put(40,30){\circle{7}} \put(40,30){\circle{8}} \put(40,30){\circle{9}} \put(40,30){\circle{10}} \put(40,30){\circle{11}} \put(40,30){\circle{12}} \put(40,30){\circle{13}} \put(40,30){\circle{14}} \put(15,10){\circle*{1}} \put(20,10){\circle*{2}} \put(25,10){\circle*{3}} \put(30,10){\circle*{4}} \put(35,10){\circle*{5}} \end{picture}

There is also a possibility within the picture environment. If one is not afraid of doing the necessary calculations (or leaving them to a program), arbitrary circles and ellipses can be patched together from quadratic Bézier curves. See Graphics in LaTeX2e for examples and Java source files.

Text and formulas

As this example shows, text and formulas can be written environment with the \put command in the usual way:

\setlength{\unitlength}{0.8cm} \begin{picture}(6,5) \thicklines \put(1,0.5){\line(2,1){3}} \put(4,2){\line(-2,1){2}} \put(2,3){\line(-2,-5){1}} \put(0.7,0.3){$A$} \put(4.05,1.9){$B$} \put(1.7,2.95){$C$} \put(3.1,2.5){$a$} \put(1.3,1.7){$b$} \put(2.5,1.05){$c$} \put(0.3,4){$F= \sqrt{s(s-a)(s-b)(s-c)}$} \put(3.5,0.4){$\displaystyle s:=\frac{a+b+c}{2}$} \end{picture}

\multiput and \linethickness

The command

\multiput(x, y)(dx, dy ){n}{object}

has 4 arguments: the starting point, the translation vector from one object to the next, the number of objects, and the object to be drawn. The \linethickness command applies to horizontal and vertical line segments, but neither to oblique line segments, nor to circles. It does, however, apply to quadratic Bézier curves!

\setlength{\unitlength}{2mm} \begin{picture}(30,20) \linethickness{0.075mm} \multiput(0,0)(1,0){26}% {\line(0,1){20}} \multiput(0,0)(0,1){21}% {\line(1,0){25}} \linethickness{0.15mm} \multiput(0,0)(5,0){6}% {\line(0,1){20}} \multiput(0,0)(0,5){5}% {\line(1,0){25}} \linethickness{0.3mm} \multiput(5,0)(10,0){2}% {\line(0,1){20}} \multiput(0,5)(0,10){2}% {\line(1,0){25}} \end{picture}

Ovals

The command

\put(x, y){\oval(w, h)}

or

 
\put(x, y){\oval(w, h)[position]}

produces an oval centered at (x, y) and having width w and height h. The optional position arguments b, t, l, r refer to "top", "bottom", "left", "right", and can be combined, as the example illustrates. Line thickness can be controlled by two kinds of commands: \linethickness{length} on the one hand, \thinlines and \thicklines on the other. While \linethickness{length} applies only to horizontal and vertical lines (and quadratic Bézier curves), \thinlines and \thicklines apply to oblique line segments as well as to circles and ovals.

\setlength{\unitlength}{0.75cm} \begin{picture}(6,4) \linethickness{0.075mm} \multiput(0,0)(1,0){7}% {\line(0,1){4}} \multiput(0,0)(0,1){5}% {\line(1,0){6}} \thicklines \put(2,3){\oval(3,1.8)} \thinlines \put(3,2){\oval(3,1.8)} \thicklines \put(2,1){\oval(3,1.8)[tl]} \put(4,1){\oval(3,1.8)[b]} \put(4,3){\oval(3,1.8)[r]} \put(3,1.5){\oval(1.8,0.4)} \end{picture}

Multiple Use of Predefined Picture Boxes

A picture box can be declared by the command

\newsavebox{name}

then defined by

\savebox{name}(width,height)[position]{content}

and finally arbitrarily often be drawn by

 
\put(x, y)\usebox{name}

The optional position parameter has the effect of defining the "anchor point" of the savebox. In the example it is set to "bl" which puts the anchor point into the bottom left corner of the savebox. The other position specifiers are top and right.

The name argument refers to a LaTeX storage bin and therefore is of a command nature (which accounts for the backslashes in the current example). Boxed pictures can be nested: In this example, \foldera is used within the definition of \folderb. The \oval command had to be used as the \line command does not work if the segment length is less than about 3 mm.

\setlength{\unitlength}{0.5mm} \begin{picture}(120,168) \newsavebox{\foldera} \savebox{\foldera} (40,32)[bl]{% definition \multiput(0,0)(0,28){2} {\line(1,0){40}} \multiput(0,0)(40,0){2} {\line(0,1){28}} \put(1,28){\oval(2,2)[tl]} \put(1,29){\line(1,0){5}} \put(9,29){\oval(6,6)[tl]} \put(9,32){\line(1,0){8}} \put(17,29){\oval(6,6)[tr]} \put(20,29){\line(1,0){19}} \put(39,28){\oval(2,2)[tr]} } \newsavebox{\folderb} \savebox{\folderb} (40,32)[l]{% definition \put(0,14){\line(1,0){8}} \put(8,0){\usebox{\foldera}} } \put(34,26){\line(0,1){102}} \put(14,128){\usebox{\foldera}} \multiput(34,86)(0,-37){3} {\usebox{\folderb}} \end{picture}

Quadratic Bézier Curves

\setlength{\unitlength}{0.8cm} \begin{picture}(6,4) \linethickness{0.075mm} \multiput(0,0)(1,0){7} {\line(0,1){4}} \multiput(0,0)(0,1){5} {\line(1,0){6}} \thicklines \put(0.5,0.5){\line(1,5){0.5}} \put(1,3){\line(4,1){2}} \qbezier(0.5,0.5)(1,3)(3,3.5) \thinlines \put(2.5,2){\line(2,-1){3}} \put(5.5,0.5){\line(-1,5){0.5}} \linethickness{1mm} \qbezier(2.5,2)(5.5,0.5)(5,3) \thinlines \qbezier(4,2)(4,3)(3,3) \qbezier(3,3)(2,3)(2,2) \qbezier(2,2)(2,1)(3,1) \qbezier(3,1)(4,1)(4,2) \end{picture}

As this example illustrates, splitting up a circle into 4 quadratic Bézier curves is not satisfactory. At least 8 are needed. The figure again shows the effect of the \linethickness command on horizontal or vertical lines, and of the \thinlines and the \thicklines commands on oblique line segments. It also shows that both kinds of commands affect quadratic Bézier curves, each command overriding all previous ones. Let $P_1=(x_1,y_1)$, $P_2=(x_2,y_2)$ denote the end points, and $m_1$, $m_2$ the respective slopes, of a quadratic Bézier curve. The intermediate control point $S=(x,y)$ is then given by the equations

$\{\begin{array}{c}x=\frac{m_2{x}_2-m_1{x}_1-(y_2-y_1)}{m_2-m_1}\\ y=y_i+m_i(x-x_i);(i=1,2)\end{array}$

See Graphics in LaTeX2e for a Java program which generates the necessary \qbezier command line.

Catenary

\setlength{\unitlength}{1cm} \begin{picture}(4.3,3.6)(-2.5,-0.25) \put(-2,0){\vector(1,0){4.4}} \put(2.45,-.05){$x$} \put(0,0){\vector(0,1){3.2}} \put(0,3.35){\makebox(0,0){$y$}} \qbezier(0.0,0.0)(1.2384,0.0) (2.0,2.7622) \qbezier(0.0,0.0)(-1.2384,0.0) (-2.0,2.7622) \linethickness{.075mm} \multiput(-2,0)(1,0){5} {\line(0,1){3}} \multiput(-2,0)(0,1){4} {\line(1,0){4}} \linethickness{.2mm} \put( .3,.12763){\line(1,0){.4}} \put(.5,-.07237){\line(0,1){.4}} \put(-.7,.12763){\line(1,0){.4}} \put(-.5,-.07237){\line(0,1){.4}} \put(.8,.54308){\line(1,0){.4}} \put(1,.34308){\line(0,1){.4}} \put(-1.2,.54308){\line(1,0){.4}} \put(-1,.34308){\line(0,1){.4}} \put(1.3,1.35241){\line(1,0){.4}} \put(1.5,1.15241){\line(0,1){.4}} \put(-1.7,1.35241){\line(1,0){.4}} \put(-1.5,1.15241){\line(0,1){.4}} \put(-2.5,-0.25){\circle*{0.2}} \end{picture}

In this figure, each symmetric half of the catenary $y=\cosh x-1$ is approximated by a quadratic Bézier curve. The right half of the curve ends in the point (2, 2.7622), the slope there having the value m = 3.6269. Using again equation (*), we can calculate the intermediate control points. They turn out to be (1.2384, 0) and (−1.2384, 0). The crosses indicate points of the real catenary. The error is barely noticeable, being less than one percent. This example points out the use of the optional argument of the \begin{picture} command. The picture is defined in convenient "mathematical" coordinates, whereas by the command

\begin{picture}(4.3,3.6)(-2.5,-0.25)

its lower left corner (marked by the black disk) is assigned the coordinates (−2.5,−0.25).

Rapidity in the Special Theory of Relativity

\setlength{\unitlength}{0.8cm} \begin{picture}(6,4)(-3,-2) \put(-2.5,0){\vector(1,0){5}} \put(2.7,-0.1){$\chi$} \put(0,-1.5){\vector(0,1){3}} \multiput(-2.5,1)(0.4,0){13} {\line(1,0){0.2}} \multiput(-2.5,-1)(0.4,0){13} {\line(1,0){0.2}} \put(0.2,1.4) {$\beta=v/c=\tanh\chi$} \qbezier(0,0)(0.8853,0.8853) (2,0.9640) \qbezier(0,0)(-0.8853,-0.8853) (-2,-0.9640) \put(-3,-2){\circle*{0.2}} \end{picture}

The control points of the two Bézier curves were calculated with formulas (*). The positive branch is determined by $P_1=(0,0)$, $m_1=1$ and $P_2=(2,\tanh 2)$, $m_2=1/{\cosh }^{2}2$. Again, the picture is defined in mathematically convenient coordinates, and the lower left corner is assigned the mathematical coordinates (−3,−2) (black disk).

XY-pic

xy is a special package for drawing diagrams. To use it, simply add the following line to the preamble of your document:

\usepackage[options]{xy}

where options is a list of functions from XY-pic you want to load. These options are primarily useful when debugging the package. I recommend you pass the all option, making LaTeX load all the XY commands.

XY-pic diagrams are drawn over a matrix-oriented canvas, where each diagram element is placed in a matrix slot:

\begin{displaymath} \xymatrix{A & B \\ C & D } \end{displaymath}

The \xymatrix command must be used in math mode. Here, we specified two lines and two columns. To make this matrix a diagram we just add directed arrows using the \ar command.

\begin{displaymath} \xymatrix{ A \ar[r] & B \ar[d] \\ D \ar[u] & C \ar[l] } \end{displaymath}

The arrow command is placed on the origin cell for the arrow. The arguments are the direction the arrow should point to (up, down, right and left).

\begin{displaymath} \xymatrix{ A \ar[d] \ar[dr] \ar[r] & B \\ D & C } \end{displaymath}

To make diagonals, just use more than one direction. In fact, you can repeat directions to make bigger arrows.

\begin{displaymath} \xymatrix{ A \ar[d] \ar[dr] \ar[drr] & & \\ B & C & D } \end{displaymath}

We can draw even more interesting diagrams by adding labels to the arrows. To do this, we use the common superscript and subscript operators.

\begin{displaymath} \xymatrix{ A \ar[r]^f \ar[d]_g & B \ar[d]^{g’} \\ D \ar[r]_{f’} & C } \end{displaymath}

As shown, you use these operators as in math mode. The only difference is that that superscript means "on top of the arrow", and subscript means "under the arrow". There is a third operator, the vertical bar: | It causes text to be placed in the arrow.

\begin{displaymath} \xymatrix{ A \ar[r]|f \ar[d]|g & B \ar[d]|{g’} \\ D \ar[r]|{f’} & C } \end{displaymath}

To draw an arrow with a hole in it, use \ar[...]|\hole. In some situations, it is important to distinguish between different types of arrows. This can be done by putting labels on them, or changing their appearance

\shorthandoff{"} \begin{displaymath} \xymatrix{ \bullet\ar@{->}[rr] && \bullet\\ \bullet\ar@{.<}[rr] && \bullet\\ \bullet\ar@{~)}[rr] && \bullet\\ \bullet\ar@{=(}[rr] && \bullet\\ \bullet\ar@{~/}[rr] && \bullet\\ \bullet\ar@{^{(}->}[rr] && \bullet\\ \bullet\ar@2{->}[rr] && \bullet\\ \bullet\ar@3{->}[rr] && \bullet\\ \bullet\ar@{=+}[rr] && \bullet } \end{displaymath} \shorthandon{"}

Notice the difference between the following two diagrams:

\begin{displaymath} \xymatrix{ \bullet \ar[r] \ar@{.>}[r] & \bullet } \end{displaymath}
\begin{displaymath} \xymatrix{ \bullet \ar@/^/[r] \ar@/_/@{.>}[r] & \bullet } \end{displaymath}

The modifiers between the slashes define how the curves are drawn. XY-pic offers many ways to influence the drawing of curves; for more information, check XY-pic documentation.

The picture environment and gnuplot

The powerful scientific plotting package "gnuplot" has the capability to output directly to a LaTeX picture environment. It is often far more convenient to plot directly to LaTeX, since this saves having to deal with potentially troublesome postscript files. Plotting scientific data (or, indeed, mathematical figures) this way gives much greater control, and of course typesetting ability, than is available from other means (such as postscript). Such pictures can then be added to a document by an \include{} command.

N.B. gnuplot is a powerful piece of software with a vast array of commands. A full discussion of gnuplot lies beyond the scope of this note.