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 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 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. 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. 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.

Definition

Generally, a mapping where the domain and codomain are the same set (or mathematical structure) is a projection if the mapping is idempotent, which means that a projection is equal to its composition with itself. A projection may also refer to a mapping which has a right inverse. Both notions are strongly related, as follows. Let p be an idempotent mapping from a set A into itself (thus p ∘ p = p ) and B = p(A) be the image of p. If we denote by π the map p viewed as a map from A onto B and by i the injection of B into A (so that p = i ∘ π ), then we have π ∘ i = IdB (so that π has a right inverse). Conversely, if π has a right inverse i, then π ∘ i = IdB implies that i ∘ π ∘ i ∘ π = i ∘ IdB ∘ π = i ∘ π p = i ∘ π is idempotent.

Applications

The original notion of projection has been extended or generalized to various mathematical situations, frequently, but not always, related to geometry, for example: projj , that takes an element x = (x1, ..., xj, ..., xn) of the Cartesian product X1 × ⋯ × Xj × ⋯ × Xn to the value projj(x) = xj. This map is always surjective and, when each space Xk has a topology, this map is also continuous and open. f(x) for a fixed x. The space of functions YX can be identified with the Cartesian product, and the evaluation map is a projection map from the Cartesian product. p(u) = p(p(u)) . In other words, an idempotent operator. For example, the mapping that takes a point (x, y, z) in three dimensions to the point (x, y, 0) is a projection. This type of projection naturally generalizes to any number of dimensions n for the domain and k ≤ n for the codomain of the mapping. See Orthogonal projection, Projection (linear algebra). In the case of orthogonal projections, the space admits a decomposition as a product, and the projection operator is a projection in that sense as well. r: X → X which restricts to the identity map on its image. This satisfies a similar idempotency condition r2 = r and can be considered a generalization of the projection map. The image of a retraction is called a retract of the original space. A retraction which is homotopic to the identity is known as a deformation retraction. This term is also used in category theory to refer to any split epimorphism.