Affine Spaces

  • Igor R. Shafarevich
  • Alexey O. Remizov


The theory of affine spaces and their transformations is presented. The case of affine Euclidean spaces is also considered, and their motions are investigated. For instance, it is proved that every motion (defined in the most general way, as an isometry of the affine Euclidean space as a metric space) is an affine transformation, and it can be represented as the composition of an orthogonal transformation and a translation by a vector. Finally, the theory of motions in an affine Euclidean space is interpreted by employing the notion of flags.

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Igor R. Shafarevich
    • 1
  • Alexey O. Remizov
    • 2
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.CMAPÉcole Polytechnique CNRSPalaiseau CedexFrance

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