Perspectivity

A perspectivity is a map between two (projective) lines in a plane obtained by projecting from a center.

Concretely, let $\ell ,{\ell }^{\prime }$ be two distinct lines in a (projective) plane and let $O$ be a point not lying on either line. For $P\in \ell$, define $f(P)$ to be the intersection point

Relation to projectivities

• Any perspectivity is a projectivity (in homogeneous coordinates it comes from a linear map).
• Not every projectivity is a single perspectivity; however, a projectivity between two distinct lines can be written as a composition of at most two perspectivities (and in the case of a projectivity from a line to itself, up to three may be needed).

See also:

• projectivity
• projective line