nLab perspectivity

Definition

Given a projective space of dimension $n$ and two projective subspaces $E,F$ of dimension $m\lt n$, a (bijective) map $f:E\to F$ is called a central perspectivity if there exist a point $O$ which is on neither of the two subspaces such that for each $e\in E$ the points $e,f(e)$ and $O$ are colinear. Point $O$ is then called the center of the perspectivity.

A finite composition of perspectivities is called a projectivity or projective transformation.

Two triangles in a projective space (not necessarily in the same plane) are said to be in a central perspective if there exist a point $O$ and a bijection $f$ among their vertices such that for each vertex $x$, the points $x$, $O$ and $f(x)$ are colinear.

In a projective plane, if $A$, $B$ are two points, consider the pencils of all lines $a$ through $A$ and $b$ through $B$. A (bijective) map $f:a\to b$ is an axial perspectivity if there exist a line $p$ not incident with any of the two points $A,B$ such that for each line $q\in a$ through $A$ intersection $p\cap q = p\cap f(q)$.

Two triangles in a projective space are said to be in an axial perspective if there exist a bijection $f$ among the sides $p_1,p_2,p_3$ and $f(p_1),f(p_2),f(p_3)$ of the two triangles such that the three intersections $p_i\cap f(p_i)$ are colinear.

Properties

Every projectivity among lines in a real plane can be presented as a composition of at most 2 perspectivities.

Desargues's theorem states that two triangles in a projective space over a field are in a central perspective iff they are in an axial perspective.

Literature

• John Bamberg, Tim Penttila: Analytic projective geometry, Cambridge Univ. Press (2023) [doi:10.1017/9781009260626]

category: geometry