Classification of Affinities

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


In this and in the following chapter we answer the natural question of how many affine maps there are. To do so we first define an equivalence relation between affine maps and study all the equivalence classes that appear. In low dimensions the problem is not too hard and is solved explicitly in this chapter. The idea is that the classification of affinities is given by the classification of endomorphisms plus a geometrical property: the invariance level.

We shall also give a geometric interpretation of the affinities of the real affine plane

The subsections are
  1. 3.1


  2. 3.2

    Similar endomorphisms

  3. 3.3

    Similar affinities

  4. 3.4

    Computations in coordinates

  5. 3.5

    Invariance level

  6. 3.6

    Classification of affinities of the line

  7. 3.7

    Classification of affinities of the real plane

  8. 3.8

    Invariance level in the real plane

  9. 3.9

    Geometrical interpretation

  10. 3.10

    Decomposition of affinities in the real plane

  11. Exercises



General Homology Characteristic Polynomial Direction Vector Invariance Level Unique Fixed Point 


  1. 8.
    Cedó, F., Reventós, A.: Geometria plana i àlgebra lineal. Collection Manuals of the Autonomous University of Barcelona, vol. 39 (2004) Google Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

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

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