Practical Linear Algebra, 3rd Edition

Chapter 10

Affine Maps in 3D

Affine maps in 3D are a primary tool for modeling and computer graphics. Figure 10.1 illustrates the use of various affine maps. This chapter goes a little further than just affine maps by introducing projective maps—the maps used to create realistic 3D images.

Figure 10.1

Affine maps in 3D: fighter jets twisting and turning through 3D space.

10.1 Affine Maps

Linear maps relate vectors to vectors. Affine maps relate points to points. A 3D affine map is written just as a 2D one, namely as

$\begin{array}{cc}{x}^{\prime }=p+A(x-o),& (10.1)\end{array}$

where x, o, p, x′ are 3D points and A is a 3 × 3 matrix. In general, we will assume that the origin of x’s coordinate ...