Projection (mathematics)

Last updated March 19, 2023

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 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 ]}

Definition

The commutativity of this diagram is the universality of the projection p, for any map f and set X. Proj-map.svg — The commutativity of this diagram is the universality of the projection π, for any map f and set X.

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, then π ∘ i = Id_B implies that i ∘ π is idempotent.^{[ citation needed ]}

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 j^th projection map, written proj_j, that takes an element x = (x₁, ..., x_j, ..., x_n) of the Cartesian product X₁ × ⋯ × 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]
- A mapping that takes an element to its equivalence class under a given equivalence relation is known as the 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 $\Pi _{a_{1},\ldots ,a_{n}}(R)$ 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 map r: X → X which restricts to the identity map on its image.^[11] 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.^{[ 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 ]}

Related Research Articles

In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions.

In mathematics, a surjective function (also known as surjection, or onto function ) is a function $f$ such that every element $y$ can be mapped from element $x$ so that $f (x) = y$ . In other words, every element of the function's codomain is the image of at least one element of its domain. It is not required that $x$ be unique; the function $f$ may map one or more elements of $X$ to the same element of $Y$ .

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 mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere, onto a plane perpendicular to the diameter through the point. It is a smooth, bijective function from the entire sphere except the center of projection to the entire plane. It maps circles on the sphere to circles or lines on the plane, and is conformal, meaning that it preserves angles at which curves meet and thus locally approximately preserves shapes. It is neither isometric nor equiareal.

In database theory, relational algebra is a theory that uses algebraic structures for modeling data, and defining queries on it with a well founded semantics. The theory was introduced by Edgar F. Codd.

<span class="mw-page-title-main">Quotient space (topology)</span> Topological space construction

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.

<span class="mw-page-title-main">Fiber bundle</span> Continuous surjection satisfying a local triviality condition

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 mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion.

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.

<span class="mw-page-title-main">Manifold</span> Topological space that locally resembles Euclidean space

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

In topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of (based) loops in X, i.e. continuous pointed maps from the pointed circle S¹ 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.

<span class="mw-page-title-main">Clifford torus</span> Geometrical object in four-dimensional space

In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles S¹
_a and S¹
_b. It is named after William Kingdon Clifford. It resides in R⁴, as opposed to in R³. To see why R⁴ is necessary, note that if S¹
_a and S¹
_b each exists in its own independent embedding space R²
_a and R²
_b, the resulting product space will be R⁴ rather than R³. The historically popular view that the cartesian product of two circles is an R³ torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis z available to it after the first circle consumes x and y.

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.

<span class="mw-page-title-main">Euclidean plane</span> Geometric model of the planar projection of the physical universe

In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted $E 2$ . It is a geometric space in which two real numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of parallel lines. It has also metrical properties induced by a distance, which allows to define circles, and angle measurement.

In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the n-dimensional torus

References

↑ "Direct product - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-08-11.
↑ Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). p. 606. doi:10.1007/978-1-4419-9982-5. ISBN 978-1-4419-9982-5. 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.
↑ Brown, Arlen; Pearcy, Carl (1994-12-16). An Introduction to Analysis. Springer Science & Business Media. ISBN 978-0-387-94369-5.
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↑ "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.
↑ "Product of a family of objects in a category - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-08-11.

Projection (mathematics)

Contents

Definition

Applications

Related Research Articles

References

Further reading