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In mathematics, a **projection** is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition, i.e., which is idempotent. 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**or*D*, onto a plane**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 projective geometry. However, a projective transformation is a bijection of a projective space, a property *not* shared with the *projections* of this article.^{[ citation needed ]}

In an abstract setting we can generally say that a *projection* is a mapping of a set (or of a mathematical structure) 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 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, then *π* ∘ *i* = Id_{B} implies that *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:

- In set theory:
- 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 product*X*_{1}× ⋯ ×*X*_{j}× ⋯ ×*X*_{n}to the value proj_{j}(**x**) =*x*_{j}.^{ [1] }This map is always surjective. - A mapping that takes an element to its equivalence class under a given equivalence relation is known as the canonical projection.
^{ [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 , and the evaluation map is a projection map from the Cartesian product.^{[ citation needed ]}

- An operation typified by the
- 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 .^{ [3] }^{ [4] }^{ [5] }^{[ 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.
^{ [6] }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*.^{ [7] }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.^{ [8] }^{ [9] }^{[ 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 map
*r*:*X*→*X*which restricts to the identity map on its image.^{ [10] }^{ [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 ]} - The scalar projection (or resolute) of one vector onto another.
^{[ citation needed ]} - 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. This projection will take many forms in different categories. The projection from the Cartesian product of sets, the product topology of topological spaces (which is always surjective and 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 ]}

**Euclidean space** is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the *Euclidean plane*. It was introduced by the Ancient Greek mathematician Euclid of Alexandria, and the qualifier *Euclidean* is used to distinguish it from other spaces that were later discovered in physics and modern mathematics.

In geometry, a **torus** is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.

In geometry, a **coordinate system** is a system that uses one or more numbers, or **coordinates**, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the *x*-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and *vice versa*; this is the basis of analytic geometry.

In geometry, the **stereographic projection** is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles at which curves meet. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.

In topology and related areas of mathematics, the **quotient space** of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the **quotient topology**, that is, with the finest topology that makes continuous the canonical projection map. In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space.

In mathematics, **hyperbolic geometry** is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

In mathematics, the concept of a **projective space** originated from the visual effect of perspective, where parallel lines seem to meet *at infinity*. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines.

In mathematics, specifically algebraic topology, a **covering map** is a continuous function from a topological space to a topological space such that each point in has an open neighborhood **evenly covered** by . In this case, is called a **covering space** and the **base space** of the covering projection. The definition implies that every covering map is a local homeomorphism.

In mathematics, the **open unit disk** around *P*, is the set of points whose distance from *P* is less than 1:

In mathematics, and particularly topology, a **fiber bundle** is a space that is *locally* a product space, but *globally* may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map, that in small regions of behaves just like a projection from corresponding regions of to The map called the **projection** or **submersion** of the bundle, is regarded as part of the structure of the bundle. The space is known as the **total space** of the fiber bundle, as the **base space**, and the **fiber**.

In mathematics, a **principal bundle** is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group . In the same way as with the Cartesian product, a principal bundle is equipped with

- An action of on , analogous to for a product space.
- A projection onto . For a product space, this is just the projection onto the first factor, .

In topology, a branch of mathematics, a **fibration** is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space. A fibration is like a fiber bundle, except that the fibers need not be the same space, nor even homeomorphic; rather, they are just homotopy equivalent. Weak fibrations discard even this equivalence for a more technical property.

In the mathematical field of differential topology, the **Hopf fibration** describes a 3-sphere in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle. Technically, Hopf found a many-to-one continuous function from the 3-sphere onto the 2-sphere such that each distinct *point* of the 2-sphere is mapped from a distinct great circle of the 3-sphere. Thus the 3-sphere is composed of fibers, where each fiber is a circle — one for each point of the 2-sphere.

In the mathematical field of algebraic topology, the **homotopy groups of spheres** describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or *n-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space.

In topology, a branch of mathematics, the **loop space** Ω*X* of a pointed topological space *X* is the space of (based) loops in *X*, i.e. continuous pointed maps from the pointed circle *S*^{1} to *X*, equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an *A*_{∞}-space. That is, the multiplication is homotopy-coherently associative.

In differential geometry, in the category of differentiable manifolds, a **fibered manifold** is a surjective submersion

The **Lambert azimuthal equal-area projection** is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772. "Zenithal" being synonymous with "azimuthal", the projection is also known as the **Lambert zenithal equal-area projection**.

In category theory, a branch of mathematics, given a morphism *f*: *X* → *Y* and a morphism *g*: *Z* → *Y*, a **lift** or **lifting** of *f* to *Z* is a morphism *h*: *X* → *Z* such that *f* = *g*∘*h*. We say that *f* factors through *h*.

**Two-dimensional space** is a geometric setting in which two values are required to determine the position of an element. The set of pairs of real numbers with appropriate structure often serves as the canonical example of a two-dimensional Euclidean space. For a generalization of the concept, see dimension.

- ↑ "Direct product - Encyclopedia of Mathematics".
*encyclopediaofmath.org*. Retrieved 2021-08-11. - ↑ Brown, Arlen; Pearcy, Carl (1994-12-16).
*An Introduction to Analysis*. Springer Science & Business Media. ISBN 978-0-387-94369-5. - ↑ Alagic, Suad (2012-12-06).
*Relational Database Technology*. Springer Science & Business Media. ISBN 978-1-4612-4922-1. - ↑ Date, C. J. (2006-08-28).
*The Relational Database Dictionary: A Comprehensive Glossary of Relational Terms and Concepts, with Illustrative Examples*. "O'Reilly Media, Inc.". ISBN 978-1-4493-9115-7. - ↑ "Relational Algebra".
*www.cs.rochester.edu*. Archived from the original on 30 January 2004. Retrieved 29 August 2021. - ↑ Sidoli, Nathan; Berggren, J. L. (2007). "The Arabic version of Ptolemy's Planisphere or Flattening the Surface of the Sphere: Text, Translation, Commentary" (PDF).
*Sciamvs*.**8**. Retrieved 11 August 2021. - ↑ "Stereographic projection - Encyclopedia of Mathematics".
*encyclopediaofmath.org*. Retrieved 2021-08-11. - ↑ "Projection - Encyclopedia of Mathematics".
*encyclopediaofmath.org*. Retrieved 2021-08-11. - ↑ Roman, Steven (2007-09-20).
*Advanced Linear Algebra*. Springer Science & Business Media. ISBN 978-0-387-72831-5. - ↑ "Retraction - Encyclopedia of Mathematics".
*encyclopediaofmath.org*. Retrieved 2021-08-11. - ↑ "retract".
*planetmath.org*. Retrieved 2021-08-11. - ↑ "Product of a family of objects in a category - Encyclopedia of Mathematics".
*encyclopediaofmath.org*. Retrieved 2021-08-11.

- Thomas Craig (1882) A Treatise on Projections from University of Michigan Historical Math Collection.

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