Overview

Maps mapping affine spaces into affine spaces are of particular interest if they preserve the affine structure, i.e., if they preserve affine combinations. Such maps are called affine and are also characterized by the property that they induce a linear map between the underlying vector spaces. Like linear maps, affine transformations have a matrix representation. Examples of affine maps are the parallel projections in Sections 4.1 4.3.

Literature: Baker, Berger, Schaal

11.1 Barycentric Representation

Consider two affine spaces and of dimensions n and m, respectively, and a map which leaves affine combinations invariant. The transformation ? is called an affine map. Let p ₀, , p _n form a barycentric coordinate system of and let q ₀, ..., q _n be its image in , as illustrated in Figure 11.1. Since ? is affine, a point x = p ₀ x ₀ + ? + p _nx _n of is mapped onto the point

Figure 11.1: Definition of an affine map.

of , i.e., one has

An immediate consequence of this equation is the following theorem:

An affine map is uniquely determined by the images of dim affinely independent points of .

Note that the images q ₀, ..., q _n neither have to be affinely independent nor do they have to span .

In the discussion above the points q ₀, ..., q