[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/collineation-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/collineation-wikipedia\/","headline":"Collineation – Wikipedia","name":"Collineation – Wikipedia","description":"before-content-x4 In projective geometry, a bijection between projective spaces that preserves collinearity after-content-x4 In projective geometry, a collineation is a","datePublished":"2019-09-28","dateModified":"2019-09-28","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1e59f6f6df9cd31df7016afaf45faf7cd34cf77c","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1e59f6f6df9cd31df7016afaf45faf7cd34cf77c","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/collineation-wikipedia\/","wordCount":4111,"articleBody":" (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4In projective geometry, a bijection between projective spaces that preserves collinearity (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus an isomorphism between projective spaces, or an automorphism from a projective space to itself. Some authors restrict the definition of collineation to the case where it is an automorphism.[1] The set of all collineations of a space to itself form a group, called the collineation group.Table of Contents (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Definition[edit]Linear algebra[edit]Axiomatically[edit]Collineations of the projective line[edit]Projective linear transformations[edit]Automorphic collineations[edit]Fundamental theorem of projective geometry[edit]Linear structure[edit]History[edit]Anti-homography[edit]References[edit]External links[edit]Definition[edit]Simply, a collineation is a one-to-one map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. One may formalize this using various ways of presenting a projective space. Also, the case of the projective line is special, and hence generally treated differently.Linear algebra[edit]For a projective space defined in terms of linear algebra (as the projectivization of a vector space), a collineation is a map between the projective spaces that is order-preserving with respect to inclusion of subspaces.Formally, let V be a vector space over a field K and W a vector space over a field L. Consider the projective spaces PG(V) and PG(W), consisting of the vector lines of V and W.Call D(V) and D(W) the set of subspaces of V and W respectively. A collineation from PG(V) to PG(W) is a map \u03b1\u00a0: D(V) \u2192 D(W), such that:\u03b1 is a bijection.A \u2286 B \u21d4 \u03b1(A) \u2286 \u03b1(B) for all A, B in D(V).[2]Axiomatically[edit]Given a projective space defined axiomatically in terms of an incidence structure (a set of points P, lines L, and an incidence relation I specifying which points lie on which lines, satisfying certain axioms), a collineation between projective spaces thus defined then being a bijective function f between the sets of points and a bijective function g between the set of lines, preserving the incidence relation.[3] (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Every projective space of dimension greater than or equal to three is isomorphic to the projectivization of a linear space over a division ring, so in these dimensions this definition is no more general than the linear-algebraic one above, but in dimension two there are other projective planes, namely the non-Desarguesian planes, and this definition allows one to define collineations in such projective planes.For dimension one, the set of points lying on a single projective line defines a projective space, and the resulting notion of collineation is just any bijection of the set.Collineations of the projective line[edit]For a projective space of dimension one (a projective line; the projectivization of a vector space of dimension two), all points are collinear, so the collineation group is exactly the symmetric group of the points of the projective line. This is different from the behavior in higher dimensions, and thus one gives a more restrictive definition, specified so that the fundamental theorem of projective geometry holds.In this definition, when V has dimension two, a collineation from PG(V) to PG(W) is a map \u03b1\u00a0: D(V) \u2192 D(W), such that:The zero subspace of V is mapped to the zero subspace of W.V is mapped to W.There is a nonsingular semilinear map \u03b2 from V to W such that, for all v in V,\u03b1(\u27e8v\u27e9)=\u27e8\u03b2(v)\u27e9{displaystyle alpha (langle vrangle )=langle beta (v)rangle }This last requirement ensures that collineations are all semilinear maps.The main examples of collineations are projective linear transformations (also known as homographies) and automorphic collineations. For projective spaces coming from a linear space, the fundamental theorem of projective geometry states that all collineations are a combination of these, as described below.Projective linear transformations[edit]Projective linear transformations (homographies) are collineations (planes in a vector space correspond to lines in the associated projective space, and linear transformations map planes to planes, so projective linear transformations map lines to lines), but in general not all collineations are projective linear transformations. The group of projective linear transformations (PGL) is in general a proper subgroup of the collineation group.Automorphic collineations[edit]An automorphic collineation is a map that, in coordinates, is a field automorphism applied to the coordinates.Fundamental theorem of projective geometry[edit]If the geometric dimension of a pappian projective space is at least 2, then every collineation is the product of a homography (a projective linear transformation) and an automorphic collineation. More precisely, the collineation group is the projective semilinear group, which is the semidirect product of homographies by automorphic collineations.In particular, the collineations of PG(2, R) are exactly the homographies, as R has no non-trivial automorphisms (that is, Gal(R\/Q) is trivial).Suppose \u03c6 is a nonsingular semilinear map from V to W, with the dimension of V at least three. Define \u03b1\u00a0: D(V) \u2192 D(W) by saying that Z\u03b1 = {\u03c6(z)\u00a0: z \u2208 Z} for all Z in D(V). As \u03c6 is semilinear, one easily checks that this map is properly defined, and furthermore, as \u03c6 is not singular, it is bijective. It is obvious now that \u03b1 is a collineation. We say that \u03b1 is induced by \u03c6.The fundamental theorem of projective geometry states the converse:Suppose V is a vector space over a field K with dimension at least three, W is a vector space over a field L, and \u03b1 is a collineation from PG(V) to PG(W). This implies K and L are isomorphic fields, V and W have the same dimension, and there is a semilinear map \u03c6 such that \u03c6 induces \u03b1.For n \u2265 3, the collineation group is the projective semilinear group, P\u0393L \u2013 this is PGL, twisted by field automorphisms; formally, the semidirect product P\u0393L \u2245 PGL \u22ca Gal(K\/k), where k is the prime field for K.Linear structure[edit]Thus for K a prime field (Fp{displaystyle mathbb {F} _{p}} or Q{displaystyle mathbb {Q} }), we have PGL = P\u0393L, but for K not a prime field (such as C{displaystyle mathbb {C} } or Fpn{displaystyle mathbb {F} _{p^{n}}} for n \u2265 2), the projective linear group is in general a proper subgroup of the collineation group, which can be thought of as “transformations preserving a projective semi-linear structure”. Correspondingly, the quotient group P\u0393L \/ PGL \u2245 Gal(K\/k) corresponds to “choices of linear structure”, with the identity (base point) being the existing linear structure. Given a projective space without an identification as the projectivization of a linear space, there is no natural isomorphism between the collineation group and P\u0393L, and the choice of a linear structure (realization as projectivization of a linear space) corresponds to a choice of subgroup PGL < P\u0393L, these choices forming a torsor over Gal(K\/k).History[edit]The idea of a line was abstracted to a ternary relation determined by collinearity (points lying on a single line). According to Wilhelm Blaschke[4] it was August M\u00f6bius that first abstracted this essence of geometrical transformation:What do our geometric transformations mean now? M\u00f6bius threw out and fielded this question already in his Barycentric Calculus (1827). There he spoke not of transformations but of permutations [Verwandtschaften], when he said two elements drawn from a domain were permuted when they were interchanged by an arbitrary equation. In our particular case, linear equations between homogeneous point coordinates, M\u00f6bius called a permutation [Verwandtschaft] of both point spaces in particular a collineation. This signification would be changed later by Chasles to homography. M\u00f6bius\u2019 expression is immediately comprehended when we follow M\u00f6bius in calling points collinear when they lie on the same line. M\u00f6bius’ designation can be expressed by saying, collinear points are mapped by a permutation to collinear points, or in plain speech, straight lines stay straight.Contemporary mathematicians view geometry as an incidence structure with an automorphism group consisting of mappings of the underlying space that preserve incidence. Such a mapping permutes the lines of the incidence structure, and the notion of collineation persists.As mentioned by Blaschke and Klein, Michel Chasles preferred the term homography to collineation. A distinction between the terms arose when the distinction was clarified between the real projective plane and the complex projective line. Since there are no non-trivial field automorphisms of the real number field, all the collineations are homographies in the real projective plane,[5] however due to the field automorphism of complex conjugation, not all collineations of the complex projective line are homographies. In applications such as computer vision where the underlying field is the real number field, homography and collineation can be used interchangeably.Anti-homography[edit]The operation of taking the complex conjugate in the complex plane amounts to a reflection in the real line. With the notation z\u2217 for the conjugate of z, an anti-homography is given byf(z)=az\u2217+bcz\u2217+d.{displaystyle f(z)={frac {az^{*}+b}{cz^{*}+d}}.}Thus an anti-homography is the composition of conjugation with a homography, and so is an example of a collineation which is not an homography. For example, geometrically, the mapping f(z)=1\/z\u2217{displaystyle f(z)=1\/z^{*}} amounts to circle inversion.[6] The transformations of inversive geometry of the plane are frequently described as the collection of all homographies and anti-homographies of the complex plane.[7]^ For instance, Beutelspacher & Rosenbaum 1998, p.21, Casse 2006, p. 56 and Yale 2004, p. 226^ Geometers still commonly use an exponential type notation for functions and this condition will often appear as A \u2286 B \u21d4 A\u03b1 \u2286 B\u03b1 for all A, B in D(V).^ “Preserving the incidence relation” means that if point p is on line l then f(p) is in g(l); formally, if (p, l) \u2208 I then (f(p), g(l)) \u2208 I\u2032.^ Felix Klein (1926, 1949) Vorlesungen \u00fcber H\u00f6here Geometrie, edited by Blaschke, Seite 138^ Casse 2006, p. 64, Corollary 4.29^ Morley & Morley 1933, p. 38^ Blair 2000, p.\u00a043; Schwerdtfeger 2012, p.\u00a042.References[edit]Beutelspacher, Albrecht; Rosenbaum, Ute (1998), Projective Geometry \/ From Foundations to Applications, Cambridge University Press, ISBN\u00a00-521-48364-6Blair, David E. (2000), Inversion Theory and Conformal Mapping, Student mathematical library, vol.\u00a09, American Mathematical Society, ISBN\u00a09780821826362Blaschke, Wilhelm (1948), Projective Geometrie, Wolfenb\u00fctteler VerlagsanstaltCasse, Rey (2006), Projective Geometry \/ An Introduction, Oxford University Press, ISBN\u00a09780199298860Morley, Frank; Morley, F.V. (1933), Inversive Geometry, London: G. Bell and SonsSchwerdtfeger, Hans (2012), Geometry of Complex Numbers, Courier Dover Publications, ISBN\u00a09780486135861Yale, Paul B. (2004) [first published 1968], Geometry and Symmetry, Dover, ISBN\u00a00-486-43835-XExternal links[edit] (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/collineation-wikipedia\/#breadcrumbitem","name":"Collineation – Wikipedia"}}]}]