Collineation

Collineation is a term in geometry describing a bijection that preserves the incidence structure of points and lines. In projective geometry, a collineation of a projective space P is a bijection that sends lines to lines and preserves which points lie on which lines. Equivalently, collineations are the automorphisms of the projective space.

In the usual setting over a field F, every collineation of the projective space P^n(F) is induced by a semilinear isomorphism of the underlying vector space F^{n+1}, defined up to a nonzero scalar. The group of all collineations of P^n(F) is called the projective semilinear group, denoted PΓL(n+1, F). The subgroup arising from linear maps without field automorphisms is the projective linear group PGL(n+1, F). In concrete terms, a collineation can be realized by a nonsingular (n+1)×(n+1) matrix acting on homogeneous coordinates, with two matrices representing the same collineation if they differ by a nonzero scalar multiple.

In affine and Euclidean geometry, a collineation is a bijection that preserves lines, i.e., it maps any line to a line and preserves incidences among points and lines. In many contexts, such collineations are affine or semiaffine transformations, described by maps x ↦ Ax^σ + b, where A is invertible, b is a translation vector, and σ is a field automorphism; the precise form depends on the underlying algebraic structure.

Collineations preserve fundamental projective invariants such as incidence and the cross-ratio of quadruples of collinear points. They form groups under composition and are central objects of study in projective and algebraic geometry.