Collineations

A collineation is a bijection of a projective or affine space that preserves incidence, in particular sending lines to lines and preserving which points lie on which lines. Collineations are the symmetries of the geometric structure that maintain the notion of collinearity.

In projective geometry, a projective space PG(n, F) has as its collineations those bijections that map lines to lines. Such maps preserve the incidence relations between points and lines. The group of all projective collineations is often described as the projective general semilinear group PΓL(n+1, F). If one considers only maps induced by nonsingular linear transformations on the underlying (n+1)-dimensional vector space, modulo scalar multiplication, the group reduces to the projective general linear group PGL(n+1, F). Allowing field automorphisms leads to the semilinear extension PΓL(n+1, F); when the field has no nontrivial automorphisms this coincides with PGL(n+1, F).

In affine geometry, a collineation is a bijection of the affine space AG(n, F) that preserves lines. Such maps preserve parallelism and incidence, and over a field they are typically described by affine semilinear transformations. In the standard case, ordinary affine maps have the form x ↦ Ax + b with A an invertible matrix and b a fixed vector; more generally, collineations can be written as x ↦ φ(Ax) + b where A is invertible and φ is a field automorphism.

Collineations play a central role in the study of incidence structures and symmetry groups of geometries. They are distinct from metric or angle-preserving maps, as collineations focus solely on the preservation of straight lines and incidence relations.