Postscript and Geometry


What is Poscript?
Basic operations
The coordinate transformation
Homogeneous coordinates
What can Postscript do?
How does Postscript render graphical information?
But why would you want to write your own code?
Let's get jiggity
Resources

What is Postscript?

A programming language whose instructions, when read by a Postscript interpreter, produce graphical output.

Typically, Postscript programs are written by other programs like Adobe Illustrator or dvips. These are never meant to be read by humans.

This talk, however, is about Postscript programs written by humans.

Postscript has two beautiful features: (1) everything (well, almost) is drawn to within the accuracy of the output device and (2) everything is easily scaled.


Basic operations

Computations are performed on a "stack"

 3 4 -1.5 add mul
 4 -1.5 add mul
 -1.5 add mul
 add mul
 mul

Graphical operations

Postscript's coordinate system

Drawing lines

 
100 100 moveto 
450 600 lineto 
stroke 

Drawing circles

 
300 300 100 0 360 arc 
stroke 

Filling circles

 
300 300 100 0 360 arc 
gsave 
  0.8 setgray 
  fill 
grestore 
10 setlinewidth 
stroke 

Drawing Bezier curves

 
200 200 moveto 
100 500 
500 300 
400 550 
curveto 
stroke 

Displaying text

 
100 100 moveto 
(g) show 

The coordinate transformation

The initial coordinate transformation

User space Device space

Translations

200 300 translate
User space Device space

Rotations

200 300 translate 
45 rotate 
User space Device space

Scaling

200 300 translate 
0.8 3 scale
User space Device space


Homogeneous coordinates

Everything produced is passed through an affine coordinate transformation from user space into device space. This transformation has the form

or in the language of linear transformations

This is typically expressed in homogeneous coordinates

In this formalism,

General transformations

200 300 translate 
[1 0 0.5 1 0 0] concat
User space Device space

General transformations

200 300 translate 
[0 1 1 0 0 0] concat
User space Device space

What do you mean everything?

Well, everything: lines, curves and text are all described in user space, then transformed into device space before rendered. To understand how this works, let's take a look under the hood.


What can Postscript do?

Apparently, not much: ultimately everything in Postscript is described in terms of lines and Bezier curves--even circular arcs:

and text

There is, however, a great benefit to be gained.


How does Postscript render graphical information?

Postscript draws a line between two points

recursively by drawing the lines between the endpoints and midpoint

and again and again

until finally

Bezier curves

The four points define the curve

where

These can be built recursively as well.

The curve splits into two other Bezier curves whose control points are easily obtained.

Just keep dividing the smaller pieces

until

So what?

This means that to transform a line or a curve, all we need to do is transform the control points. That's easy.


But why would you want to write your own code?

The Economics of Computing
After making a few widgets, the marginal cost of making more widgets is almost zero.

Postscript is a full fledged programming language with loops and other good things.

To build this:

1. Draw big triangle
2. Scale by 1/2 and draw smaller triangle
3. If not at the lowest level, repeat three times:
  • Save the coordinate transformation
  • Translate by 0 -s
  • Draw next triangle
  • Go to step 2
  • Restore coordinate transformation
  • Rotate by 120
  • From geometry

    The Elements, Book I, Proposition 1

    The Nine Point Circle

    From calculus

    Introducting Wooody, the loveable star of Toy Tale

    Transform Wooody

    Wooody emotes


    Three dimensional drawing

    Orthogonal projection

    Perspective projection

    More homogeneous coordinates


    and then

    Let's see: we need

    so we'll take

    We can accomplish this by

    Shadows are obtained in the same way


    Resources

    Adobe, creators of Postscript

    Ghostscript, a free Postscript interpreter and viewer

    Guide to Postscript Programming

    Bill Casselman's Geometry and Postscript, a text book used to teach PostScript within a Geometry course.

    Bill Casselman's Mathematical Graphics resources page.

    Peter Kleiweg's Postscript page

    P.S. All figures have been drawn in Postscript.