Projection (mathematics): Difference between revisions

In [[mathematics]], a '''projection''' is aan [[Idempotence|idempotent]] [[function (mathematics)|mapping]] of a [[Setset (mathematics)|set]] (or other [[mathematical structure]]) into a [[subset]] (or sub-structure),. whichIn isthis equalcase, toidempotent itsmeans squarethat forprojecting [[functiontwice composition|mappingis composition]]the (or,same in other words,as whichprojecting isonce. [[idempotence|idempotent]]). The [[restriction (mathematics)|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 (paper sheet). of Thepaper): the projection of a point is its shadow on the sheet of paper, sheet.and Thethe projection (shadow) of a point on the paper sheet of paper is thisthat 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]] of three dimensions onto a plane in it, like the shadow example. The two main projections of this kind are:

* {{anchor|Central projection}}The '''projection from a point onto a plane''' or '''central projection''': If ''{{mvar|C''}} is a point, called the '''center of projection''', then the projection of a point ''{{mvar|P''}} different from ''{{mvar|C''}} onto a plane that does not contain ''{{mvar|C''}} is the intersection of the [[line ''(geometry)|line]] {{mvar|CP''}} with the plane. The points ''{{mvar|P''}} such that the line ''{{mvar|CP''}} is [[parallel (geometry)|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 [[projectiveProjective geometry]] for a formalization of this terminology). The projection of the point ''{{mvar|C''}} itself is not defined.

* The '''projection parallel to a direction {{mvar|D}}, onto a plane''' or '''[[parallel projection]]''': The image of a point ''{{mvar|P''}} is the intersection withof the plane ofwith the line parallel to ''{{mvar|D''}} passing through ''{{mvar|P''}}. See {{slink|Affine space|Projection}} for an accurate definition, generalized to any dimension.{{Citation needed|date=August 2021}}

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 [[geometry|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|date=August 2021}}

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|date=August 2021}}

[[File:Proj-map.svg|thumb|191x191px|The commutativity of this diagram is the universality of the projection {{mvar|π}}, for any map ''{{mvar|f''}} and set {{mvar|X}}.]]

InGenerally, ana abstractmapping settingwhere wethe can[[Domain generally say thatof a ''projection''function|domain]] isand a[[codomain]] mappingare ofthe asame [[set (mathematics)|set]] (or of a [[mathematical structure]]) whichis a projection if the mapping is [[idempotent]], which means that a projection is equal to its [[Function composition|composition]] with itself. A '''projection''' may also refer to a mapping which has a [[Inverse_function#Right_inverses|right inverse]]. Both notions are strongly related, as follows. Let ''{{mvar|p''}} be an idempotent [[map (mathematics)|map]]mapping from a set ''{{mvar|A''}} into itself (thus {{math|1=''p''  ∘ ''p''  = ''p''}}) and {{math|1=''B''  = ''p''(''A'')}} be the image of ''{{mvar|p''}}. If we denote by ''{{mvar|π''}} the map ''{{mvar|p''}} viewed as a map from ''{{mvar|A''}} onto ''{{mvar|B''}} and by ''{{mvar|i''}} the [[injective function|injection]] of ''{{mvar|B''}} into ''{{mvar|A''}} (so that {{math|1=''p''  = ''i''  ∘ ''π''}}), then we have {{math|1=''π''  ∘ ''i''  = Id_''B''}} (so that ''{{mvar|π''}} has a right inverse). Conversely, if ''{{mvar|π''}} has a right inverse {{mvar|i}}, then {{math|1=''π''  ∘  ''i'' = Id_''B''}} implies that {{math|1=''i''&nbsp ∘ ''π'' ∘ ''i'' ∘ ''π'' = ''i'' ∘ Id_''B'' ∘ ''π'' = ''i'' ∘ ''π''}}; that is, {{math|1=''p'' = ''i'' ∘ ''π''}} is idempotent.{{Citation needed|date=August 2021}}

** An operation typified by the ''{{mvar|j''&nbsp;}}-^th [[projection (set theory)|projection map]], written {{math|proj_''j''&nbsp;}}, that takes an element {{nowrapmath|1='''x''' = (''x''₁, …..., ''x''_''j'', …..., ''x''_''kn'')}} of the [[Cartesian product]] {{nowrapmath|''X''₁ × ⋯ × ''X''_''j'' × ⋯ × ''X''_''kn''}} to the value {{nowrapmath|1=proj_''j''&nbsp;('''x''') = ''x''_''j''.}}<ref>{{Cite web|title=Direct product - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Direct_product|access-date=2021-08-11|website=encyclopediaofmath.org}}</ref> This map is always [[surjective]] and, when each space {{math|''X''_''k''}} has a [[Topological space|topology]], this map is also [[Continuity (topology)|continuous]] and [[open map|open]].<ref>{{cite book|last=Lee|first=John M.|date=2012|title=Introduction to Smooth Manifolds|edition=Second|series=Graduate Texts in Mathematics|volume=218|isbn=978-1-4419-9982-5|doi=10.1007/978-1-4419-9982-5|page=606|url=https://zenodo.org/record/4461500|quote='''Exercise A.32.''' Suppose <math>X_1, \ldots, X_k</math> are topological spaces. Show that each projection <math>\pi_i : X_1 \times \cdots \times X_k \to X_i</math> is an open map.}}</ref>

** A mapping that takes an element to its [[equivalence class]] under a given [[equivalence relation]] is known as the [[canonical projection]].<ref>{{Cite book|lastlast1=Brown|firstfirst1=Arlen|url=https://books.google.com/books?id=Y2Mwck8Q9A4C&newbks=0&printsec=frontcover&pg=PA8&hl=en|title=An Introduction to Analysis|last2=Pearcy|first2=Carl|date=1994-12-16|publisher=Springer Science & Business Media|isbn=978-0-387-94369-5|language=en}}</ref>

** The evaluation map sends a function ''{{mvar|f''}} to the value {{math|''f''(''x'')}} for a fixed ''{{mvar|x''}}. The space of functions {{math|''Y''^''X''}} can be identified with the Cartesian product <math display="inline">\prod_{i\in X}Y</math>, and the evaluation map is a projection map from the Cartesian product.{{Citation needed|date=August 2021}}

* For [[relational database]]s and [[query language]]s, the [[Projection (relational algebra)|projection]] is a [[unary operation]] written as <math>\Pi_{a_1, \ldots,a_n}( R )</math> where <math>a_1,\ldots,a_n</math> is a set of attribute names. The result of such projection is defined as the [[Set (mathematics)|set]] that is obtained when all [[tuple]]s in ''{{mvar|R''}} are restricted to the set <math>\{a_1,\ldots,a_n\}</math>.<ref>{{Cite book|last=Alagic|first=Suad|url=https://books.google.com/books?id=1SLvBwAAQBAJ&q=projection+relational+database&pg=PA25|title=Relational Database Technology|date=2012-12-06|publisher=Springer Science & Business Media|isbn=978-1-4612-4922-1|language=en}}</ref><ref>{{Cite book|last=Date|first=C. J.|url=https://books.google.com/books?id=vWKClUCN2HYC&q=projection+relational+database&pg=PA72|title=The Relational Database Dictionary: A Comprehensive Glossary of Relational Terms and Concepts, with Illustrative Examples|date=2006-08-28|publisher="O'Reilly Media, Inc."|isbn=978-1-4493-9115-7|language=en}}</ref><ref>{{Cite web|title=Relational Algebra|url=https://www.cs.rochester.edu/~nelson/courses/csc_173/relations/algebra.html|url-status=dead|archive-url=https://web.archive.org/web/20040130014938/https://www.cs.rochester.edu/~nelson/courses/csc_173/relations/algebra.html|archive-date=30 January 2004|access-date=29 August 2021|website=www.cs.rochester.edu}}</ref>{{Verify source|date=August 2021}} ''{{mvar|R''}} is a [[Relation (database)|database-relation]].{{Citation needed|date=August 2021}}

* In [[spherical geometry]], projection of a sphere upon a plane was used by [[Ptolemy]] (~150) in his [[Planisphaerium]].<ref>{{Cite journal|last1=Sidoli|first1=Nathan|last2=Berggren|first2=J. L.|date=2007|title=The Arabic version of Ptolemy's Planisphere or Flattening the Surface of the Sphere: Text, Translation, Commentary|url=http://individual.utoronto.ca/acephalous/Sidoli_Berggren_2007.pdf|journal=Sciamvs|volume=8|access-date=11 August 2021}}</ref> 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 ''{{mvar|P''}} on the sphere besides ''{{mvar|C''}} determines a line ''{{mvar|CP''}} intersecting the plane at the projected point for ''{{mvar|P''}}.<ref>{{Cite web|title=Stereographic projection - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Stereographic_projection|access-date=2021-08-11|website=encyclopediaofmath.org}}</ref> The correspondence makes the sphere a [[one-point compactification]] for the plane when a [[point at infinity]] is included to correspond to ''{{mvar|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 [[Sphere#Hemisphere|hemisphere]] is frequently projected onto a plane using the [[gnomonic projection]].{{Citation needed|date=August 2021}}

* In [[linear algebra]], a [[linear transformation]] that remains unchanged if applied twice: ({{math|1=''p''(''u'') = ''p''(''p''(''u''))),}}. inIn other words, an [[idempotent]] operator. For example, the mapping that takes a point {{math|(''x'', ''y'', ''z'')}} in three dimensions to the point {{math|(''x'', ''y'', 0) in the plane}} is a projection. This type of projection naturally generalizes to any number of dimensions ''{{mvar|n''}} for the sourcedomain and {{math|''k'' ≤ ''n''}} for the targetcodomain of the mapping. See [[orthogonalOrthogonal projection]], [[projectionProjection (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.<ref>{{Cite web|title=Projection - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Projection|access-date=2021-08-11|website=encyclopediaofmath.org}}</ref><ref>{{Cite book|last=Roman|first=Steven|url=https://books.google.com/books?id=bSyQr-wUys8C&newbks=0&printsec=frontcover&pg=PA231&dq=%22idempotent%22+AND+%22projection%22+AND+%22linear+algebra%22&hlpg=enPA231|title=Advanced Linear Algebra|date=2007-09-20|publisher=Springer Science & Business Media|isbn=978-0-387-72831-5|language=en}}</ref>{{Verify source|date=August 2021}}

* 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 map|open]] and surjective.{{Citation needed|date=August 2021}}

* In [[topology]], a [[retraction (topology)|retraction]] is a [[Continuous map (topology)|continuous map]] {{math|''r'': ''X'' → ''X''}} which restricts to the [[identity map]] on its image.<ref>{{Cite web|title=Retraction - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Retraction|access-date=2021-08-11|website=encyclopediaofmath.org}}</ref><ref>{{Cite web|title=retract|url=https://planetmath.org/Retract|access-date=2021-08-11|website=planetmath.org}}</ref> This satisfies a similar idempotency condition {{math|1=''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|date=August 2021}}