# Matrix Representation of Transformations in Three-dimensional Space

• Ian O. Angell
Chapter
Part of the Macmillan Computer Science Series book series (COMPSS)

## Abstract

Transformations of co-ordinate axes in two-dimensional space were introduced in chapter 4. An extension to three-dimensional systems is an essential step before we are able to proceed to projections of three-dimensional space onto the necessarily two-dimensional graphics viewport. As in the lower dimension, there are three basic transformations: translation of origin, change of scale and axes rotation; we will ignore all other transformations such as shear. Since we have already introduced the idea of matrix representation of transformations in two dimensions, we shall move directly to a similar representation of three-dimensional transformations. It should once more be noted that certain graphics devices will have these operations in hardware. The techniques are, nevertheless, very important so a full description is given. Again the square matrices representing the transformations will be one dimension greater than the space — that is, 4×4 (remember a 4 × 4 matrix is stored in a double [5] [5] array!) — and a general point in space will be represented by, a column vector, relative to some triad of co-ordinate axes
$$\left( \begin{gathered} x \hfill \\ y \hfill \\ z \hfill \\ 1 \hfill \\ \end{gathered} \right)$$
We start with our library of functions used for creating the matrices representing three-dimensional transformations. These will be stored as a file “matrix3.c”, which in turn #includes “graphlib.c” (and hence “primitiv.c”) and two-dimensional utility functions in “utility.c”. It is given in listing 7.1.

## Keywords

Matrix Representation Actual Position Integer Pointer Actual Scene Superficial Facet
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.