Geometries are more than just a set of objects, like lines and points. A geometry is also characterized by properties of these objects under the maps of the geometry. Thus Euclidean geometry is characterized by Euclidean maps, which leave lengths and angles unchanged. Affine geometry is characterized by affine maps, which leave the ratio of three collinear points unchanged. Finally, projective geometry is characterized by projective maps which leave the cross ratio of four collinear points unchanged. In this chapter, we shall discuss those aspects of projective maps that will later be relevant for NURBS.

2.1 Perspectivities

We have mentioned projections of three-space onto a plane before. At a lower level, one might try to project from a line onto another line, as illustrated in Fig. 2.1. Let two points p ₀, p ₁ be given on a line L and two points q ₀, q ₁ on another line M. If c is a point on neither L nor M, then we may define a projection of L onto M through c in the following way: if x is on L, then its image y on M under a projection ? is defined by

Figure 2.1: Perspectivities x = ? x is defined as a projection of x through c onto M.

(2.1)

Such maps are called perspectivities. The point c is called the