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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).^{ [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] }

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

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.

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:^{ [5] }

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.

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.^{ [6] } The central perspectivity described above is the case with *n* = 2 and *m* = 1.

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).^{ [7] } 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.

- ↑ 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
- ↑ Fishback 1969 , pp. 65–66
- ↑ Pedoe 1988 , pp. 282–3
- ↑ Young 1930 , p. 116

In mathematics, a **projective plane** is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus *any* two distinct lines in a projective plane intersect in one and only one point.

In Euclidean geometry, an **affine transformation**, or an **affinity**, is a geometric transformation that preserves lines and parallelism.

In mathematics, a **plane** is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point, a line and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with one of a room's walls, infinitely extended, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.

In geometry, the **stereographic projection** is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles at which curves meet. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.

In mathematics, **homogeneous coordinates** or **projective coordinates**, introduced by August Ferdinand Möbius in his 1827 work *Der barycentrische Calcul*, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix.

In mathematics, **projective geometry** is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points to Euclidean points, and vice-versa.

A **3D projection** is a design technique used to display a three-dimensional (3D) object on a two-dimensional (2D) surface. These projections rely on visual perspective and aspect analysis to project a complex object for viewing capability on a simpler plane. This concept of extending 2D geometry to 3D was mastered by Heron of Alexandria in the first century. Heron could be called the father of 3D. 3D Projection is the basis of the concept for Computer Graphics simulating fluid flows to imitate realistic effects. Lucas Films 'ILM group is credited with introducing the concept. In 1982 the first all digital computer generated sequence for a motion picture file was in: Star Trek II: Wrath of Khan. A 1984 patent related to this concept was written by William E Masters, "Computer automated manufacturing process and system" US4665492A using mass particles to fabricate a cup. The process of particle deposition is one technology of 3D Printing.

A **vanishing point** is a point on the image plane of a perspective drawing where the two-dimensional perspective projections of mutually parallel lines in three-dimensional space appear to converge. When the set of parallel lines is perpendicular to a picture plane, the construction is known as one-point perspective, and their vanishing point corresponds to the oculus, or "eye point", from which the image should be viewed for correct perspective geometry. Traditional linear drawings use objects with one to three sets of parallels, defining one to three vanishing points.

In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) **duality** is the formalization of this concept. There are two approaches to the subject of duality, one through language and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a * duality*. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry.

In painting, photography, graphical perspective and descriptive geometry, a **picture plane** is an image plane located between the "eye point" and the object being viewed and is usually coextensive to the material surface of the work. It is ordinarily a vertical plane perpendicular to the sightline to the object of interest.

In geometry, the notion of **line** or **straight line** was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, which are often described in terms of two points or referred to using a single letter.

In projective geometry, a **homography** is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include **projectivity**, **projective transformation**, and **projective collineation**.

**Three-dimensional space** is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension.

The **Lambert azimuthal equal-area projection** is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772. "Zenithal" being synonymous with "azimuthal", the projection is also known as the **Lambert zenithal equal-area projection**.

In projective geometry, a **collineation** is a one-to-one and onto map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus an *isomorphism* between projective spaces, or an automorphism from a projective space to itself. Some authors restrict the definition of collineation to the case where it is an automorphism. The set of all collineations of a space to itself form a group, called the **collineation group**.

In projective geometry, a **correlation** is a transformation of a *d*-dimensional projective space that maps subspaces of dimension *k* to subspaces of dimension *d* − *k* − 1, reversing inclusion and preserving incidence. Correlations are also called **reciprocities** or **reciprocal transformations**.

**Two-dimensional space** is a geometric setting in which two values are required to determine the position of an element. The set ℝ^{2} of pairs of real numbers with appropriate structure often serves as the canonical example of a two-dimensional Euclidean space. For a generalization of the concept, see dimension.

In projective geometry, the **harmonic conjugate point** of an ordered triple of points on the real projective line is defined by the following construction:

The **Steiner conic** or more precisely **Steiner's generation of a conic**, named after the Swiss mathematician Jakob Steiner, is an alternative method to define a non-degenerate projective conic section in a projective plane over a field.

In geometry, **smooth projective planes** are special projective planes. The most prominent example of a smooth projective plane is the real projective plane . Its geometric operations of joining two distinct points by a line and of intersecting two lines in a point are not only continuous but even *smooth*. Similarly, the classical planes over the complex numbers, the quaternions, and the octonions are smooth planes. However, these are not the only such planes.

- 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 0123930 - 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

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

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