In mathematics, a projection is a mapping of a set into a subset, which is equal to its square for mapping composition. 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. The projection of a point is its shadow on the paper sheet. The shadow of a point on the paper sheet is this point itself. 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 Euclidean space of three dimensions 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. 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 with the plane of the line parallel to D passing through P. See for an accurate definition, generalized to any dimension.
The concept of projection in mathematics is a very old one, 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 ofperspective. 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.
Definition
In an abstract setting we can generally say that a projection is a mapping of a set which 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 map from a set A into itself and B = p 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, then we have π ∘ i = IdB. Conversely, if π has a right inverse, then π ∘ i = IdB implies that 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:
In set theory:
* An operation typified by the jth projection map, written projj, that takes an element x = of the cartesian productX1 × … × Xj × … × Xk to the value projj = xj. This map is always surjective.
* The evaluation map sends a function f to the value f for a fixed x. The space of functionsYX can be identified with the cartesian product, and the evaluation map is a projection map from the cartesian product.
For relational databases and query languages, the projection is a unary operation written as where is a set of attribute names. The result of such projection is defined as the set that is obtained when all tuples in R are restricted to the set. R is a database-relation.
In linear algebra, a linear transformation that remains unchanged if applied twice, in other words, an idempotent operator. For example, the mapping that takes a point in three dimensions to the point in the plane is a projection. This type of projection naturally generalizes to any number of dimensions n for the source and k ≤ n for the target of the mapping. See orthogonal projection, projection. 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.
In differential topology, any fiber bundle includes a projection map as part of its definition. Locally at least this map looks like a projection map in the sense of the product topology and is therefore open and surjective.
In topology, a retraction is a continuous map 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.
In category theory, the above notion of cartesian product of sets can be generalized to arbitrary categories. The product of some objects has a canonical projectionmorphism to each factor. This projection will take many forms in different categories. The projection from the Cartesian product of sets, the product topology of topological spaces, or from the direct product of groups, etc. Although these morphisms are often epimorphisms and even surjective, they do not have to be.
