TABLE OF CONTENTS Transformations which map lines into lines and also preserve parallelism and ratios are called affine due to Leonid Euler (1707 1783). The parallel projections and scalings discussed in Chapter 4 are examples of affine maps.
In his inaugural address at the University of Erlangen in 1872, the famous Erlangener Programm, Felix Klein (1849 1925) distinguished the different geometries by the properties and theorems which remain valid under certain groups of transformations. Affine geometry consists of all propositions left invariant under affine maps.
Many concepts, tools, and objects in geometric design, including the notions of smoothness, tangents, and control points, linear interpolation, and quadrics, belong to or possess an affine structure. The corresponding constructions, e.g., of points and tangents, subdivision algorithms, etc., are often also invariant under affine maps. Such invariance makes these constructions very valuable in practical computer applications.
