nLab
collineation

A collineation is a bijection from a projective space to a projective space of the same finite dimension which preserves the inclusions of subspaces. Equivalently, it is often defined by a seemingly weaker condition, which is also used in the infinite-dimensional case: a collineation is a bijection of projective spaces which sends each triple of collinear points into a triple of collinear points.

A projective collineation is a collineation such that its restriction to every projective hyperplane is a projective isomorphism.

A perspective collineation is a (projective) collineation from a projective plane to itself such that the restriction to one, and hence to any projective line is a perspectivity onto its image line.

Similarly, a correlation is a bijection from a projective space to a projective space of the same finite dimension which reverses the inclusions of subspaces.

The terms collineation and correlation have been introduced into projective geometry by Moebius in 1827.

Literature

  • Jacquelline Lelong-Ferrand, Les fondements de la géométrie
  • Coxeter, Real projective geometry
category: geometry

Last revised on May 18, 2016 at 08:12:08. See the history of this page for a list of all contributions to it.