Mapping equal to its square under mapping composition From Wikipedia, the free encyclopedia
In mathematics, a projection is an idempotentmapping 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 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 lineCP 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 of the plane with the line parallel to D passing through P. See Affine space §Projection for an accurate definition, generalized to any dimension.^{[citation needed]}
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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.
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 = Id_{B} (so that π has a right inverse). Conversely, if π has a right inverse i, then π ∘ i = Id_{B} implies that i ∘ π ∘ i ∘ π = i ∘ Id_{B} ∘ 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 the j-^{th}projection map, written proj_{j}, that takes an element x = (x_{1}, ..., x_{j}, ..., x_{n}) of the Cartesian productX_{1} × ⋯ × X_{j} × ⋯ × X_{n} to the value proj_{j}(x) = x_{j}.^{[1]} This map is always surjective and, when each space X_{k} has a topology, this map is also continuous and open.^{[2]}
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 $\Pi _{a_{1},\ldots ,a_{n}}(R)$ where $a_{1},\ldots ,a_{n}$ 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 $\{a_{1},\ldots ,a_{n}\}$.^{[4]}^{[5]}^{[6]}^{[verification needed]}R is a database-relation.^{[citation needed]}
In spherical geometry, projection of a sphere upon a plane was used by Ptolemy (~150) in his Planisphaerium.^{[7]} The method is called stereographic projection and uses a plane tangent to a sphere and a pole C diametrically opposite the point of tangency. Any point P on the sphere besides C determines a line CP intersecting the plane at the projected point for P.^{[8]} The correspondence makes the sphere a 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 linear algebra, a linear transformation that remains unchanged if applied twice: p(u) = p(p(u)). In other words, an idempotent 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 dimensions n for the domain and k ≤ n 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.^{[9]}^{[10]}^{[verification needed]}
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.^{[citation needed]}
In topology, a retraction is a continuous mapr: X → X which restricts to the identity map on its image.^{[11]} This satisfies a similar idempotency condition r^{2} = 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.^{[citation needed]}
Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol.218 (Seconded.). p.606. doi:10.1007/978-1-4419-9982-5. ISBN978-1-4419-9982-5. Exercise A.32. Suppose $X_{1},\ldots ,X_{k}$ are topological spaces. Show that each projection $\pi _{i}:X_{1}\times \cdots \times X_{k}\to X_{i}$ is an open map.