Affine-projective Relationship: Applications

  • Marcel Berger
  • Pierre Pansu
  • Jean-Pic Berry
  • Xavier Saint-Raymond
Part of the Problem Books in Mathematics book series (PBM)

Abstract

We have shown, at the beginning of chapter 4 why it is necessary to go beyond the framework of affine spaces, adjoining to them points at infinity This is now possible in the following way: we associate to the affine space X its universal space \( \hat X\) (cf. 3.D), which is a vector space in which X is embedded as an affine hyperplane whose direction is a vector hyperplane of \(\hat X\). Considering now the projectivization \(\tilde X = P\left( {\hat X} \right)\) , we see that \(\tilde X\) is the disjoint union of two sets: P( X), which is canonically identified with X, and P (\(\vec X\)), which, being the space of lines in \(\vec X\), is also the space of directions of lines in X. We denote it by \({\infty _X} = P\left( {\vec X} \right)\) We write \( \tilde X = X \cup {\infty _X}\), and say that ∞x is the hyperplane at infinity in X.

Keywords

Projective Space Disjoint Union Projective Geometry Conic Section 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.

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Copyright information

© Marcel Berger 1984

Authors and Affiliations

  • Marcel Berger
    • 1
  • Pierre Pansu
    • 2
  • Jean-Pic Berry
    • 3
  • Xavier Saint-Raymond
    • 3
  1. 1.U.E.R. de Mathematique et InformatiqueUniversité Paris VIIParis, Cedex 05France
  2. 2.Centre National de la Recherche ScientifiqueParisFrance
  3. 3.P.U.K.GrenobleFrance

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