Much of graphics programming revolves around coordinate
transformations. Objects in the modeled scene are
specified by coordinates in *scene
space*. In order to display these objects, we
must transform their coordinates into *display
space*.

The contribution of this document is an implementation of this method using the Python language, and the extremely useful NumPy extensions to Python that make vector and matrix operations easy.

The author has found this reference to be extremely valuable in graphics work:

Foley, James D., Andries van Dam, Steven K. Feiner, and John F. Hughes. Computer graphics: principles and practice. Second edition. Addison-Wesley, 1990, ISBN 0-201-12110-7.

In the chapter on geometrical transformations, the authors
bring up homogeneous coordinates early on. This ingenous
mathematical artifice makes it possible to specify any
coordinate transformation using a small matrix: 3×3
for 2-space, 4×4 for 3-space. Points are carried as
3-element vectors for 2-space, or 4-element vectors for
3-space, and a trivial operation converts these vectors
back to * (x,y)* or

`(x,y,z)`

Moreover, you can combine these transform matrices to model the effect of multiple transforms. For example, suppose you have a model of some small rigid object like a cup, positioned in a larger scene. The coordinates of the cup are described in some standard orientation; when the cup is placed into the scene, a single transform matrix describes the transformation of those coordinates into coordinates relative to the scene. The net transformation of the cup to display coordinates can be represented as a matrix which happens to be the product of the two transform matrices: the transform that describes the cup's placement in the scene, and the transform that desribes how scene coordinates become display coordinates.

Transform matrices are easily reversible. For example, suppose the user clicks on the display, and you want to know where that click is in scene space. To convert a display coordinate to a scene coordinate requires a single matrix multiplication.

This document is structured in three general parts.

Section 3, “Interface to the Python

`homcoord`

module”: How to use this module in your Python programs.Remaining sections describe the implementation of the package itself. This starts with Section 4, “Prologue to

`homcoord.py`

”.