nLab projective transformation

Idea

A homography or a projective transformation is an automorphism of a projective space: a map from a projective space $P(V)$ to itself which lifts to a linear automorphism of $V$. More generally, a homography $P(V)\to P(W)$ is a morphism of projective spaces which lift to a linear isomorphism $V\to W$.

In the synthetic approach, a homography is a finite composition of perspectivities.

For real projective spaces of dimension at least 2 this is the same as a collineation; in general collineations are slightly more general.

Literature

See also projective general linear group.

• John Bamberg, Tim Penttila, Analytic projective geometry, Cambridge Univ. Press 2023 doi
• wikipedia: homography