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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 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 = IdB (so that π has a right inverse). Conversely, if π has a right inverse i, then π ∘ i = IdB implies that i ∘ π ∘ i ∘ π = i ∘ IdB ∘ π = i ∘ π; that is, p = i ∘ π 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:

• In set theory:
  • An operation typified by the j-th projection map, written projj, that takes an element x = (x1, ..., xj, ..., xn) of the Cartesian product X1 × ⋯ × Xj × ⋯ × Xn to the value projj(x) = xj.¹ This map is always surjective and, when each space Xk 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 YX can be identified with the Cartesian product ∏ i ∈ X Y {\textstyle \prod _{i\in X}Y} , and the evaluation map is a projection map from the Cartesian product.
• For relational databases and query languages, the projection is a unary operation written as Π a 1 , … , a n ( R ) {\displaystyle \Pi _{a_{1},\ldots ,a_{n}}(R)} where a 1 , … , a n {\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 { a 1 , … , a n } {\displaystyle \{a_{1},\ldots ,a_{n}\}} .⁴⁵⁶[verification needed] R is a database-relation.
• 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.
• 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.⁹¹⁰[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.
• 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.
• The scalar projection (or resolute) of one vector onto another.
• 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), 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]

Further reading

• Craig, Thomas (1882) A Treatise on Projections from University of Michigan Historical Math Collection.
• Henrici, Olaus Magnus Friedrich (1911). "Projection" . Encyclopædia Britannica. Vol. 22 (11th ed.). pp. 427–434.

References

1. "Direct product - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-08-11. https://encyclopediaofmath.org/wiki/Direct_product ↩

2. 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 , … , X k {\displaystyle X_{1},\ldots ,X_{k}} are topological spaces. Show that each projection π i : X 1 × ⋯ × X k → X i {\displaystyle \pi _{i}:X_{1}\times \cdots \times X_{k}\to X_{i}} is an open map. 978-1-4419-9982-5 ↩

3. Brown, Arlen; Pearcy, Carl (1994-12-16). An Introduction to Analysis. Springer Science & Business Media. ISBN 978-0-387-94369-5. 978-0-387-94369-5 ↩

4. Alagic, Suad (2012-12-06). Relational Database Technology. Springer Science & Business Media. ISBN 978-1-4612-4922-1. 978-1-4612-4922-1 ↩

5. 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. 978-1-4493-9115-7 ↩

6. "Relational Algebra". www.cs.rochester.edu. Archived from the original on 30 January 2004. Retrieved 29 August 2021. https://web.archive.org/web/20040130014938/https://www.cs.rochester.edu/~nelson/courses/csc_173/relations/algebra.html ↩

7. 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. http://individual.utoronto.ca/acephalous/Sidoli_Berggren_2007.pdf ↩

8. "Stereographic projection - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-08-11. https://encyclopediaofmath.org/wiki/Stereographic_projection ↩

9. "Projection - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-08-11. https://encyclopediaofmath.org/wiki/Projection ↩

10. Roman, Steven (2007-09-20). Advanced Linear Algebra. Springer Science & Business Media. ISBN 978-0-387-72831-5. 978-0-387-72831-5 ↩

11. "Retraction - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-08-11. https://encyclopediaofmath.org/wiki/Retraction ↩

12. "Product of a family of objects in a category - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-08-11. https://encyclopediaofmath.org/wiki/Product_of_a_family_of_objects_in_a_category ↩