Projection (mathematics)

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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 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 (see Projective geometry for a formalization of this terminology). 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 Affine space § Projection for an accurate definition, generalized to any dimension.^{[citation needed]}

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

Generally, a mapping where the

• In set theory:
  • An operation typified by the j^th
    open.^[2]
  • A mapping that takes an element to its
    canonical projection.^[3]
  • The evaluation map sends a function f to the value f(x) for a fixed x. The space of functions Y^X can be identified with the Cartesian product ${\textstyle \prod _{i\in X}Y}$, and the evaluation map is a projection map from the Cartesian product.^[citation needed]
• For relational databases and query languages, the projection is a unary operation written as ${\displaystyle \Pi _{a_{1},\ldots ,a_{n}}(R)}$ where ${\displaystyle a_{1},\ldots ,a_{n}}$ 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 ${\displaystyle \{a_{1},\ldots ,a_{n}\}}$.^{[4][5][6][verification needed]} R is a database-relation.^{[citation needed]}
• In
  one-point compactification for the plane when a point at infinity is included to correspond to C, which otherwise has no projection on the plane. A common instance is the complex plane where the compactification corresponds to the Riemann sphere. Alternatively, a hemisphere is frequently projected onto a plane using the gnomonic projection.^{[citation needed}
  ]
• In
  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.^{[9][10][verification needed}
  ]
• In
  open and surjective.^{[citation needed}
  ]
• In
  deformation retraction. This term is also used in category theory to refer to any split epimorphism.^{[citation needed}
  ]
• The
  vector onto another.^{[citation needed}
  ]
• In
  open), or from the direct product of groups, etc. Although these morphisms are often epimorphisms and even surjective, they do not have to be.^{[12][verification needed}
  ]