Projection (mathematics)

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 $D$ , onto a plane or parallel projection: The image of a point $P$ is the intersection of the plane with 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]}