Perspectivity

Projective geometry

In projective geometry the points of a line are called a projective range, and the set of lines in a plane on a point is called a pencil.

Given two lines ${\displaystyle \ell }$ and ${\displaystyle m}$ in a projective plane and a point P of that plane on neither line, the bijective mapping between the points of the range of ${\displaystyle \ell }$ and the range of ${\displaystyle m}$ determined by the lines of the pencil on P is called a perspectivity (or more precisely, a central perspectivity with center P). A special symbol has been used to show that points X and Y are related by a perspectivity; ${\displaystyle X⩞Y.}$ In this notation, to show that the center of perspectivity is P, write ${\displaystyle X\stackrel{P}{⩞}Y.}$

The existence of a perspectivity means that corresponding points are in perspective. The dual concept, axial perspectivity, is the correspondence between the lines of two pencils determined by a projective range.

Projectivity

The composition of two perspectivities is, in general, not a perspectivity. A perspectivity or a composition of two or more perspectivities is called a projectivity (projective transformation, projective collineation and homography are synonyms).

There are several results concerning projectivities and perspectivities which hold in any pappian projective plane:

Theorem: Any projectivity between two distinct projective ranges can be written as the composition of no more than two perspectivities.

Theorem: Any projectivity from a projective range to itself can be written as the composition of three perspectivities.

Theorem: A projectivity between two distinct projective ranges which fixes a point is a perspectivity.

Higher-dimensional perspectivities

The bijective correspondence between points on two lines in a plane determined by a point of that plane not on either line has higher-dimensional analogues which will also be called perspectivities.

Let S_m and T_m be two distinct m-dimensional projective spaces contained in an n-dimensional projective space R_n. Let P_n−m−1 be an (n − m − 1)-dimensional subspace of R_n with no points in common with either S_m or T_m. For each point X of S_m, the space L spanned by X and P_n-m-1 meets T_m in a point Y = f_P(X). This correspondence f_P is also called a perspectivity. The central perspectivity described above is the case with n = 2 and m = 1.

Perspective collineations

Let S₂ and T₂ be two distinct projective planes in a projective 3-space R₃. With O and O* being points of R₃ in neither plane, use the construction of the last section to project S₂ onto T₂ by the perspectivity with center O followed by the projection of T₂ back onto S₂ with the perspectivity with center O*. This composition is a bijective map of the points of S₂ onto itself which preserves collinear points and is called a perspective collineation (central collineation in more modern terminology). Let φ be a perspective collineation of S₂. Each point of the line of intersection of S₂ and T₂ will be fixed by φ and this line is called the axis of φ. Let point P be the intersection of line OO* with the plane S₂. P is also fixed by φ and every line of S₂ that passes through P is stabilized by φ (fixed, but not necessarily pointwise fixed). P is called the center of φ. The restriction of φ to any line of S₂ not passing through P is the central perspectivity in S₂ with center P between that line and the line which is its image under φ.