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.

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

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 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; ${\displaystyle X\doublebarwedge Y.}$ In this notation, to show that the center of perspectivity is P, write ${\displaystyle X\ {\overset {P}{\doublebarwedge }}\ Y.}$ Using the language of functions, a central perspectivity with center P is a function ${\displaystyle f_{P}\colon [\ell ]\mapsto [m]}$ (where the square brackets indicate the projective range of the line) defined by ${\displaystyle f_{P}(X)=Y{\text{ whenever }}P\in XY}$. This map is an involution, that is, ${\displaystyle f_{P}(f_{P}(X))=X{\text{ for all }}X\in [\ell ]}$.

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