# 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).^{[1]} 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".^{[2]} In a second book, *New Principles of Linear Perspective* (1719), Taylor wrote

- When Lines drawn according to a certain Law from the several Parts of any Figure, cut a Plane, and by that Cutting or Intersection describe a figure on that Plane, that Figure so described is called the
*Projection*of the other Figure. The Lines producing that Projection, taken all together, are called the*System of Rays*. And when those Rays all pass thro’ one and same Point, they are called the*Cone of Rays*. And when that Point is consider’d as the Eye of a Spectator, that System of Rays is called the*Optic Cone*^{[3]}

## 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 and in a plane and a point *P* of that plane on neither line, the bijective mapping between the points of the range of and the range of determined by the lines of the pencil on *P* is called a **perspectivity** (or more precisely, a *central perspectivity* with center *P*).^{[4]} A special symbol has been used to show that points *X* and *Y* are related by a perspectivity; In this notation, to show that the center of perspectivity is *P*, write Using the language of functions, a central perspectivity with center *P* is a function (where the square brackets indicate the projective range of the line) defined by .^{[5]} This map is an involution, that is, .

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:^{[6]}

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.^{[7]} The central perspectivity described above is the case with *n* = 2 and *m* = 1.

### Perspective collineations

Let *S*_{2} and *T*_{2} be two distinct projective planes in a projective 3-space *R*_{3}. With *O* and *O** being points of *R*_{3} in neither plane, use the construction of the last section to project *S*_{2} onto *T*_{2} by the perspectivity with center *O* followed by the projection of *T*_{2} back onto *S*_{2} with the perspectivity with center *O**. This composition is a bijective map of the points of *S*_{2} onto itself which preserves collinear points and is called a *perspective collineation* (*central collineation* in more modern terminology).^{[8]} Let φ be a perspective collineation of *S*_{2}. Each point of the line of intersection of *S*_{2} and *T*_{2} 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*_{2}. *P* is also fixed by φ and every line of *S*_{2} 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*_{2} not passing through *P* is the central perspectivity in *S*_{2} with center *P* between that line and the line which is its image under φ.

## Notes

- ↑ Kirsti Andersen (2007) The Geometry of an Art, page 1,Springer ISBN 978-0-387-25961-1
- ↑ Andersen 1992, p. 75
- ↑ Andersen 1992, p. 163
- ↑ Coxeter 1969, p. 242
- ↑ Pedoe 1988, p. 281
- ↑ Fishback 1969, pp. 65-66
- ↑ Pedoe 1988, pp. 282–3
- ↑ Young 1930, p. 116

## References

- Andersen, Kirsti (1992),
*Brook Taylor's Work on Linear Perspective*, Springer, ISBN 0-387-97486-5 - Coxeter, Harold Scott MacDonald (1969),
*Introduction to Geometry*(2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-50458-0, MR 123930 - Fishback, W.T. (1969),
*Projective and Euclidean Geometry*, John Wiley & Sons - Pedoe, Dan (1988),
*Geometry/A Comprehensive Course*, Dover, ISBN 0-486-65812-0 - Young, John Wesley (1930),
*Projective Geometry*, The Carus Mathematical Monographs (#4), Mathematical Association of America

## External links

- Christopher Cooper Perspectivities and Projectivities from Macquarie University.
- James C. Morehead Jr. (1911) Perspective and Projective Geometries: A Comparison from Rice University.
- John Taylor Projective Geometry from University of Brighton.