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 inEuclidean geometry to denote the projection of the three-dimensionalEuclidean space onto a plane in it, like the shadow example. The two main projections of this kind are:
Theprojection from a point onto a plane orcentral projection: IfC is a point, called the centre of projection, then the projection of a pointP different fromC onto a plane that does not containC is the intersection of thelineCP with the plane. The pointsP such that the lineCP isparallel 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 (seeProjective geometry for a formalization of this terminology). The projection of the pointC itself is not defined.
Theprojection parallel to a directionD, onto a plane orparallel projection: The image of a pointP is the intersection of the plane with the line parallel toD passing throughP. SeeAffine space § Projection for an accurate definition, generalized to any dimension.[citation needed]
The concept ofprojection 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 ageometric 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]
Incartography, amap 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. The3D projections are also at the basis of the theory ofperspective.[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 ofprojective geometry.
The commutativity of this diagram is the universality of the projectionπ, for any mapf and setX.
Generally, a mapping where thedomain andcodomain are the sameset (ormathematical structure) is a projection if the mapping isidempotent, which means that a projection is equal to itscomposition with itself. A projection may also refer to a mapping which has aright inverse. Both notions are strongly related, as follows. Letp be an idempotent mapping from a setA into itself (thusp ∘p =p) andB =p(A) be the image ofp. If we denote byπ the mapp viewed as a map fromA ontoB and byi theinjection ofB intoA (so thatp =i ∘π), then we haveπ ∘i = IdB (so thatπ has a right inverse). Conversely, ifπ has a right inversei, thenπ ∘i = IdB implies thati ∘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:
An operation typified by thej-thprojection map, writtenprojj, that takes an elementx = (x1, ...,xj, ...,xn) of theCartesian productX1 × ⋯ ×Xj × ⋯ ×Xn to the valueprojj(x) =xj.[1] This map is alwayssurjective and, when each spaceXk has atopology, this map is alsocontinuous andopen.[2]
The evaluation map sends a functionf to the valuef(x) for a fixedx. The space of functionsYX can be identified with the Cartesian product, and the evaluation map is a projection map from the Cartesian product.[citation needed]
Inlinear algebra, alinear transformation that remains unchanged if applied twice:p(u) =p(p(u)). In other words, anidempotent 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 dimensionsn for the domain andk ≤n for the codomain of the mapping. SeeOrthogonal 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.[9][10][verification needed]
Intopology, aretraction is acontinuous mapr:X →X which restricts to theidentity map on its image.[11] This satisfies a similar idempotency conditionr2 =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 ishomotopic to the identity is known as adeformation retraction. This term is also used incategory theory to refer to any split epimorphism.[citation needed]
