ABSTRACT
This chapter wraps up the basic geometry tools. Affine maps in 3D are a primary tool for modeling and computer graphics. Figure 13.1
maps used to create realistic 3D images.
13.1 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
x′ = p+A(x− o). (13.1)
In general, we will assume that the origin of x’s coordinate system has three zero coordinates, and drop the o term:
x′ = p+Ax. (13.2)
Sketch 13.1 gives an example. Recall, the column vectors of A are the vectors a1,a2,a3. The point p tells us where to move the origin of the [e1, e2, e3]-system; again, the real action of an affine map is captured by the matrix. Thus, by studying matrix actions, or linear maps, we will learn more about affine maps.
