In this chapter we introduce affinities, the most natural maps to consider between affine spaces. The definition of affine map, or affinity, is so natural that we shall see that affinities are simply those maps that take collinear points to collinear points.
We shall also see that there are enough affine maps. In fact, in an affine space of dimension n, given two subsets of n+1 points, there exists an affine map such that takes the points of the first subset to the points of the second.
In the Exercises at the end of the chapter we verify the axioms 4 and 5 of Affine Geometry given in the Introduction.
Definition of affinity
The affine group
Affinities and linear varieties
Equations of affinities
Examples of affinities
Characterization of affinities of the line
Fundamental Theorem of Affine Geometry
KeywordsLinear Variety Fundamental Theorem Minimal Polynomial Translation Vector Affine Space
- 8.Cedó, F., Reventós, A.: Geometria plana i àlgebra lineal. Collection Manuals of the Autonomous University of Barcelona, vol. 39 (2004) Google Scholar
- 20.Lang, S.: Algebra, 3rd edn. Addison-Wesley, Reading (1971) Google Scholar
- 30.Santaló, L.: Geometría proyectiva. Eudeba, Buenos Aires (1966) Google Scholar