# Projection (mathematics)

## Mapping equal to its square under mapping composition / From Wikipedia, the free encyclopedia

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In mathematics, a **projection** is an idempotent mapping of a set (or other mathematical structure) into a subset (or sub-structure). In this case, idempotent means that projecting twice is the same as projecting once. The restriction to a subspace of a projection is also called a *projection*, even if the idempotence property is lost.
An everyday example of a projection is the casting of shadows onto a plane (sheet of paper): the projection of a point is its shadow on the sheet of paper, and the projection (shadow) of a point on the sheet of paper is that point itself (idempotency). The shadow of a three-dimensional sphere is a closed disk. Originally, the notion of projection was introduced in Euclidean geometry to denote the projection of the three-dimensional Euclidean space onto a plane in it, like the shadow example. The two main projections of this kind are:

- The
**projection from a point onto a plane**or**central projection**: If*C*is a point, called the**center of projection**, then the projection of a point*P*different from*C*onto a plane that does not contain*C*is the intersection of the line*CP*with the plane. The points*P*such that the line*CP*is parallel to the plane does not have any image by the projection, but one often says that they project to a point at infinity of the plane (see Projective geometry for a formalization of this terminology). The projection of the point*C*itself is not defined. - The
**projection parallel to a direction**or*D*, onto a plane**parallel projection**: The image of a point*P*is the intersection with the plane of the line parallel to*D*passing through*P*. See Affine space § Projection for an accurate definition, generalized to any dimension.^{[citation needed]}

The concept of **projection** in mathematics is a very old one, and most likely has its roots in the phenomenon of the shadows cast by real-world objects on the ground. This rudimentary idea was refined and abstracted, first in a geometric context and later in other branches of mathematics. Over time different versions of the concept developed, but today, in a sufficiently abstract setting, we can unify these variations.^{[citation needed]}

In cartography, a map projection is a map of a part of the surface of the Earth onto a plane, which, in some cases, but not always, is the restriction of a projection in the above meaning. The 3D projections are also at the basis of the theory of perspective.^{[citation needed]}

The need for unifying the two kinds of projections and of defining the image by a central projection of any point different of the center of projection are at the origin of projective geometry. However, a projective transformation is a bijection of a projective space, a property *not* shared with the *projections* of this article.^{[citation needed]}

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