In mathematics, a projection is a mapping from a set to itselor an endomorphism of a mathematical structurethat is Idempotence, that is, equals its composition with itself. The image of a point or a subset under a projection is called the projection of .
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:
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The projection from a point onto a plane or central projection: If is a point, called the center of projection, then the projection of a point different from onto a plane that does not contain is the intersection of the line with the plane. The points such that the line 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 itself is not defined.
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The projection parallel to a direction , onto a plane or parallel projection: The image of a point is the intersection of the plane with the line parallel to passing through . See for an accurate definition, generalized to any dimension.
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 geometry 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 be an idempotent mapping from a set into itself (thus ) and be the image of . If we denote by the map viewed as a map from onto and by the injection of into (so that ), then we have (so that has a right inverse). Conversely, if has a right inverse , then implies that ; that is, 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:
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In set theory:
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An operation typified by the -th projection map, written , that takes an element of the Cartesian product to the value
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A mapping that takes an element to its equivalence class under a given equivalence relation is known as the canonical projection.
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The evaluation map sends a function to the value for a fixed . The space of functions can be identified with the Cartesian product , and the evaluation map is a projection map from the Cartesian product.
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For relational databases and , 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 in are restricted to the set .
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In spherical geometry, projection of a sphere upon a plane was used by Ptolemy (~150) in his Planisphaerium.
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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 is a projection. This type of projection naturally generalizes to any number of dimensions for the domain and 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.
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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 map and surjective.
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In topology, a retraction is a continuous map which restricts to the identity map on its image.
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The scalar resolute (or resolute) of one vector onto another.
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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 projection morphism to each factor. Special cases include the projection from the Cartesian product of sets, the product topology of topological spaces (which is always surjective and open map), or from the direct product of groups, etc. Although these morphisms are often and even surjective, they do not have to be.
Further reading
