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 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]
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; that is, p = i ∘ π is idempotent.[citation needed]
The original notion of projection has been extended or generalized to various mathematical situations, frequently, but not always, related to geometry, for example:
Exercise A.32. Suppose X 1 , … … --> , X k {\displaystyle X_{1},\ldots ,X_{k}} are topological spaces. Show that each projection π π --> i : X 1 × × --> ⋯ ⋯ --> × × --> X k → → --> X i {\displaystyle \pi _{i}:X_{1}\times \cdots \times X_{k}\to X_{i}} is an open map.
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