Matrix Representation of Transformations in Three-dimensional Space

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 — and a general point in space will be represented, by a column vector, relative to some triad of co-ordinate axes

$$\left( {\begin{array}{*{20}{c}}x \\y \\z \\1 \end{array}} \right)$$

We start with our library of routines used for creating the matrices representing three-dimensional transformations.