Perspectivity
In geometry and in its applications to drawing, a perspectivity is the formation of an image in a picture plane of a scene viewed from a fixed point. Graphics. The science of graphical perspective uses perspectivities to make realistic images in proper proportion. According to Kirsti Andersen, the first author to describe perspectivity was Leon Alberti in his "De Pictura" (1435). In English, Brook Taylor presented his "Linear Perspective" in 1715, where he explained "Perspective is the Art of drawing on a Plane the Appearances of any Figures, by the Rules of Geometry". In a second book, "New Principles  of Linear Perspective" (1719), Taylor wrote 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 formula_1 and formula_2 in a projective plane and a point "P" of that plane on neither line, the bijective mapping between the points of the range of formula_1 and the range of formula_2 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; formula_5 In this notation, to show that the center of perspectivity is "P", write formula_6 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" and "T" be  two distinct "m"-dimensional projective spaces contained in an "n"-dimensional projective space "R". Let "P" be an ("n" − "m" − 1)-dimensional subspace of "R" with no points in common with either "S" or "T". For each point "X" of "S", the space "L" spanned by "X" and "P" meets "T" in a point . This correspondence "f" is also called a perspectivity. The central perspectivity described above is the case with and . Perspective collineations. Let "S" and "T" be two distinct projective planes in a projective 3-space "R". With "O" and "O"* being points of "R" in neither plane, use  the construction of the last section to project "S" onto "T" by the perspectivity with center "O" followed by the projection of "T" back onto "S" with the perspectivity with center "O"*. This composition is a bijective map of the points of "S" 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 "S". Each point of the line of intersection of "S" and "T" 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 "S". "P" is also fixed by φ and every line of "S" 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 "S" not passing through "P" is the central perspectivity in "S" with center "P" between that line and the line which is its image under φ.