# Affinities

Part of the Springer Undergraduate Mathematics Series book series (SUMS)

## Abstract

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.

The subsections are
1. 2.1

Introduction

2. 2.2

Definition of affinity

3. 2.3

First properties

4. 2.4

The affine group

5. 2.5

Affinities and linear varieties

6. 2.6

Equations of affinities

7. 2.7

Invariant varieties

8. 2.8

Examples of affinities

9. 2.9

Characterization of affinities of the line

10. 2.10

Fundamental Theorem of Affine Geometry

11. Exercises

## Keywords

Linear Variety Fundamental Theorem Minimal Polynomial Translation Vector Affine Space
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Bibliography

1. 8.
Cedó, F., Reventós, A.: Geometria plana i àlgebra lineal. Collection Manuals of the Autonomous University of Barcelona, vol. 39 (2004) Google Scholar
2. 20.
3. 30.
Santaló, L.: Geometría proyectiva. Eudeba, Buenos Aires (1966) Google Scholar

## Authors and Affiliations

1. 1.Department of MathematicsUniversitat Autònoma de BarcelonaBellaterra, BarcelonaSpain