# Collineation

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{{short description|In projective geometry, a bijection between projective spaces that preserves collinearity}}
In [projective geometry](/source/projective_geometry), a '''collineation''' is a [one-to-one](/source/injective_function) and [onto](/source/surjection) map (a [bijection](/source/bijection)) from one [projective space](/source/projective_space) to another, or from a projective space to itself, such that the [images](/source/Image_(mathematics)) of [collinear](/source/collinear) points are themselves collinear. A collineation is thus an ''[isomorphism](/source/isomorphism)'' between projective spaces, or an [automorphism](/source/automorphism) from a projective space to itself. Some authors restrict the definition of collineation to the case where it is an automorphism.<ref>For instance, {{harvnb|Beutelspacher|Rosenbaum|1998|loc=p.21}}, {{harvnb|Casse|2006|loc=p. 56}} and {{harvnb|Yale|2004|loc=p. 226}}</ref> The [set](/source/set_(mathematics)) of all collineations of a space to itself form a [group](/source/group_(mathematics)), called the '''collineation group'''.

==Definition==
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===
For a projective space defined in terms of [linear algebra](/source/linear_algebra) (as the projectivization of a [vector space](/source/vector_space)), a collineation is a map between the projective spaces that is [order-preserving](/source/order-preserving) with respect to [inclusion](/source/inclusion_(set_theory)) of subspaces.

Formally, let ''V'' be a vector space over a [field](/source/field_(mathematics)) ''K'' and ''W'' a vector space over a field ''L''.  Consider the projective spaces ''PG''(''V'') and ''PG''(''W''), consisting of the [vector line](/source/vector_line)s 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 α : ''D''(''V'') → ''D''(''W''), such that:
* α is a bijection.
* ''A'' ⊆ ''B'' ⇔ α(''A'') ⊆ α(''B'') for all ''A'', ''B'' in ''D''(''V'').<ref>Geometers still commonly use an exponential type notation for functions and this condition will often appear as ''A'' ⊆ ''B'' ⇔ ''A''<sup>α</sup> ⊆ ''B''<sup>α</sup> for all ''A'', ''B'' in ''D''(''V'').</ref>

===Axiomatically===
Given a [projective space defined axiomatically](/source/Projective_space) in terms of an [incidence structure](/source/incidence_structure) (a set of points ''P,'' lines ''L,'' and an [incidence relation](/source/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.<ref>"Preserving the incidence relation" means that if point {{math|''p''}} is on line {{math|''l''}} then {{math|''f''(''p'')}} is in {{math|''g''(''l'')}}; formally, if {{math|(''p'', ''l'') ∈ ''I''}} then {{math|(''f''(''p''), ''g''(''l'')) ∈ ''I''′}}.</ref>

Every projective space of dimension greater than or equal to three is isomorphic to the [projectivization](/source/projectivization) of a linear space over a [division ring](/source/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 plane](/source/non-Desarguesian_plane)s, 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===
For a projective space of dimension one (a projective line; the projectivization of a vector space of [dimension](/source/dimension_(vector_space)) two), all points are collinear, so the collineation group is exactly the [symmetric group](/source/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 {{nowrap|''α'' : ''D''(''V'') → ''D''(''W'')}}, such that:
* The [zero subspace](/source/Examples_of_vector_spaces) of ''V'' is mapped to the zero subspace of ''W''.
* ''V'' is mapped to ''W''.
* There is a nonsingular [semilinear map](/source/semilinear_map) ''β'' from ''V'' to ''W'' such that, for all ''v'' in ''V'', <math display="block">\alpha(\langle v\rangle)=\langle \beta(v)\rangle</math>
This last requirement ensures that collineations are all semilinear maps.

==Types==
The main examples of collineations are  projective linear transformations (also known as [homographies](/source/homography)) 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.
<!-- This illustrates the pitfall of not restricting the definition of collineation. Correlations are not automorphisms of the projective space.

A [duality](/source/duality_(projective_geometry)) is a collineation from a projective space onto its dual space, taking points to hyperplanes (and vice versa) and preserving incidence. A [correlation](/source/correlation_(projective_geometry)) is a duality from a projective space onto itself (this implies that the space is self-dual). A [polarity](/source/pole_and_polar) is an [involutory](/source/involution_(mathematics)) correlation.
-->

===Projective linear transformations===

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](/source/Projective_linear_group)) is in general a proper [subgroup](/source/subgroup) of the collineation group.

===Automorphic collineations===
An '''{{Visible anchor|automorphic collineation}}''' is a map that, in coordinates, is a [field automorphism](/source/field_automorphism) applied to the coordinates.

==Fundamental theorem of projective geometry==
{{see also|Homography#Fundamental theorem of projective geometry}}
If the geometric dimension of a [pappian](/source/Pappus's_hexagon_theorem) 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](/source/projective_semilinear_group), which is the [semidirect product](/source/semidirect_product) of homographies by automorphic collineations.

In particular, the collineations of the [real projective plane](/source/Projective_plane) {{nowrap|PG(2, '''R''')}} are exactly the homographies, as '''R''' has no non-trivial automorphisms (see [Automorphism#Examples](/source/Automorphism) and footnote ''d'' in [Real number](/source/Real_number)).

Suppose ''φ'' is a nonsingular semilinear map from ''V'' to ''W'', with the dimension of ''V'' at least three.  Define {{nowrap|''α'' : ''D''(''V'') → ''D''(''W'')}} by saying that {{nowrap|1=''Z''<sup>''α''</sup> = {''φ''(''z'') : ''z'' ∈ ''Z''}{{null}}}} for all ''Z'' in ''D''(''V''). As ''φ'' is semilinear, one easily checks that this map is properly defined, and furthermore, as ''φ'' is not singular, it is bijective.  It is obvious now that ''α'' is a collineation.  We say that ''α'' is induced by ''φ''.

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 ''α'' 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 ''φ'' such that ''φ'' induces ''α''.

For {{nowrap|''n'' ≥ 3}}, the collineation group is the [projective semilinear group](/source/projective_semilinear_group), P&Gamma;L – this is PGL, twisted by [field automorphism](/source/field_automorphism)s; formally, the [semidirect product](/source/semidirect_product) {{nowrap|P&Gamma;L ≅ PGL ⋊ Gal(''K''/''k'')}}, where ''k'' is the [prime field](/source/prime_field) for ''K''.

===Linear structure===
Thus for ''K'' a prime field (<math>\mathbb{F}_p</math> or <math>\mathbb{Q}</math>), we have {{nowrap|1=PGL = P&Gamma;L}}, but for ''K'' not a prime field (such as <math>\mathbb{C}</math> or <math>\mathbb{F}_{p^n}</math> for {{nowrap|''n'' ≥ 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 {{nowrap|P&Gamma;L / PGL ≅ 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 PΓL, and the choice of a linear structure (realization as projectivization of a linear space) corresponds to a choice of subgroup {{nowrap|PGL < P&Gamma;L}}, these choices forming a [torsor](/source/torsor) over Gal(''K''/''k'').

==History==
The idea of a [line](/source/line_(geometry)) was abstracted to a [ternary relation](/source/ternary_relation) determined by [collinearity](/source/collinearity) (points lying on a single line).   According to [Wilhelm Blaschke](/source/Wilhelm_Blaschke)<ref>[Felix Klein](/source/Felix_Klein) (1926, 1949) ''Vorlesungen über Höhere Geometrie'', edited by Blaschke, Seite 138</ref> it was [August Möbius](/source/August_M%C3%B6bius) that first abstracted this essence of geometrical transformation:
:What do our geometric transformations mean now? Möbius 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, Möbius called a permutation [Verwandtschaft] of both point spaces in particular a ''collineation''. This signification would be changed later by [Chasles](/source/Michel_Chasles) to ''homography''. Möbius’ expression is immediately comprehended when we follow Möbius in calling points [collinear](/source/collinear) when they lie on the same line. Möbius' 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](/source/incidence_structure) with an [automorphism group](/source/automorphism_group) consisting of mappings of the underlying space that preserve [incidence](/source/incidence_(geometry)). Such a mapping permutes the lines of the incidence structure, and the notion of collineation persists.

As mentioned by Blaschke and Klein, [Michel Chasles](/source/Michel_Chasles) preferred the term ''homography'' to ''collineation''. A distinction between the terms arose when the distinction was clarified between the [real projective plane](/source/real_projective_plane) and the [complex projective line](/source/complex_projective_line). Since there are no non-trivial field automorphisms of the [real number](/source/real_number) field, all the collineations are homographies in the real projective plane,<ref>{{harvnb|Casse|2006|loc=p. 64, Corollary 4.29}}</ref> however due to the field automorphism of [complex conjugation](/source/complex_conjugation), not all collineations of the complex projective line are homographies. In applications such as [computer vision](/source/computer_vision) where the underlying field is the real number field, ''homography'' and ''collineation'' can be used interchangeably.

==Anti-homography==
The operation of taking the [complex conjugate](/source/complex_conjugate) in the [complex plane](/source/complex_plane) amounts to a [reflection](/source/reflection_(mathematics)) in the [real line](/source/real_line).  With the notation ''z''<sup>∗</sup> for the conjugate of ''z'', an '''anti-homography''' is given by
:<math>f(z) = \frac {a z^* + b}{c z^* + d}.</math>
Thus an anti-homography is the [composition](/source/function_composition) of conjugation with a [homography](/source/homography), and so is an example of a collineation which is not an homography. For example, geometrically, the mapping <math>f(z) = 1/z^*</math> amounts to [circle inversion](/source/inversive_geometry).<ref>{{harvnb|Morley|Morley|1933|loc=p. 38}}</ref> The transformations of [inversive geometry](/source/inversive_geometry) of the plane are frequently described as the collection of all homographies and anti-homographies of the complex plane.<ref>{{harvnb|Blair|2000}}, [https://books.google.com/books?id=sxHKpaAuclgC&pg=PA43 p.&nbsp;43]; {{harvnb|Schwerdtfeger|2012}}, [https://books.google.com/books?id=4XE2_AqxYVkC&pg=PA42 p.&nbsp;42].</ref>

==Notes==
{{Reflist}}

==References==
* {{citation|first1=Albrecht|last1=Beutelspacher|first2=Ute|last2=Rosenbaum|author-link1=Albrecht Beutelspacher|title=Projective Geometry / From Foundations to Applications|publisher=Cambridge University Press|year=1998|isbn=0-521-48364-6}} 
* {{citation|title=Inversion Theory and Conformal Mapping|volume=9|series=Student mathematical library|first=David E.|last=Blair|publisher=American Mathematical Society|year=2000|isbn=9780821826362}}
* {{citation|first=Wilhelm|last=Blaschke|author-link=Wilhelm Blaschke|title=Projective Geometrie|year=1948|publisher=Wolfenbütteler Verlagsanstalt}}
* {{citation|first=Rey|last=Casse|title=Projective Geometry / An Introduction|publisher=Oxford University Press|year=2006|isbn=9780199298860}}
* {{citation|first1=Frank|last1=Morley|author-link1=Frank Morley|first2=F.V.|last2=Morley|year=1933|title=Inversive Geometry|place=London|publisher=G. Bell and Sons}}
* {{citation|title=Geometry of Complex Numbers|title-link=Geometry of Complex Numbers|first=Hans|last=Schwerdtfeger|author-link=Hans Schwerdtfeger|publisher=Courier Dover Publications|year=2012|isbn=9780486135861}}
* {{citation|first=Paul B.|last=Yale|title=Geometry and Symmetry|publisher=Dover|year=2004|orig-year=first published 1968|isbn=0-486-43835-X}}

==External links==
* {{PlanetMath|urlname=Collineation|title=projectivity}}

{{Authority control}}

Category:Projective geometry

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