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.

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

category: geometry