Collineation

In the mathematical fields of geometry and linear algebra, collineation denotes a bijective mapping of an affine or projective space onto itself, in which every straight line is mapped onto a straight line that is true to the line . The set of collineations in a room forms a group , in particular the reversals of collineations are always collineations.

The picture shows a collineation of the affine plane over the square number field . Although an affine point base (blue points ) of the plane is fixed by the collineation, an infinite number of points are not fixed, but mirrored at the origin : the points and all rational linear combinations ! ${\displaystyle K=\mathbb {Q} ({\sqrt {2}})}$ ${\displaystyle O,E_{1},E_{2}}$ ${\displaystyle O}$ ${\displaystyle X,Y}$ ${\displaystyle (a\cdot {\sqrt {2}},b\cdot {\sqrt {2}}),a,b\in \mathbb {Q} \setminus \{0\}}$

The term for one-dimensional spaces thus coincides with the term for the bijection of the straight line concerned. Therefore, mostly only collineations on at least two-dimensional spaces are studied.

Occasionally, the term collineation is also used for a bijective or even only injective, straight-line mapping of an affine or projective space into another space. The present article deals exclusively with collineations, which are straight-line, bijective self -images of a space.

→ In a more general sense, the automorphisms of finite incidence structures are also called collineations . See Finite Geometry # Automorphisms .

Collineations in synthetic geometry

In synthetic geometry , collineations on two-dimensional spaces (planes) are usually examined. Since the group of affinities or projectivities is often not rich enough for the non-Desargue planes to examine the structure of the plane, the group of collineations takes its place here. In an abstract incidence geometry , this group forms the characteristic automorphism group , because here the "position of points on a common straight line ( collinearity )" is the only structure in space and thus - in the sense of the Erlangen program - the only space, i.e. the plane here , is the characterizing invariant .

In-plane collineations and geometric automorphisms

A point D in a space with more than 2 points on each straight line lies in the plane defined by a triangle ABC if and only if the straight line DE connecting D with a point E on the straight line AB, which is neither equal to A nor equals B, at least one of the triangle sides (straight line!) BC or AC intersects at a point . This means that for affine and projective geometries with more than 2 points on each straight line, “level fidelity” can be reduced to “line fidelity”. ${\displaystyle F\neq E}$

• Every collineation of an affine plane is true to parallels , that is, for two straight lines in the plane . ${\displaystyle \kappa }$ ${\displaystyle g,h}$ ${\displaystyle g\parallel h\Leftrightarrow \kappa (g)\parallel \kappa (h)}$
• A collineation a minimum three-dimensional affine geometry is exactly then parallel loyal, if they give faithful , that is, when the images of any four coplanar points are always coplanar.
• A collineation of an affine geometry with more than 2 points on any straight line or any projective geometry is always true to plane. Compare the figure on the right and the 2nd order example below.
• A true-to-plane collineation is always a geometric automorphism of space, that is, it maps every subspace onto a subspace of the same dimension. Conversely, of course, every geometric automorphism is an in-plane collineation.
• A “bijection by changing the base with the same coordinates”, i. H. a mapping of the at least two-dimensional point space, in which each point is mapped to a point with the same coordinates (from a ternary body in the case of a plane, from a sloping body in the case of at least three-dimensional space), each subspace is mapped to a subspace with the same coordinate equations, but coordinates and equations are related to a different point base, is a true-to-plane collineation and thus a geometric automorphism
  • in the case of at least two-dimensional affine geometry,
  • in the case of at least three-dimensional projective geometry and
  • in the case of a Moufang level .
• Conversely, however, there are generally true-to-plane collineations that cannot be represented by changing the base for "coordinate identity".
• Every true-to-plane collineation of an at least two-dimensional affine geometry can be clearly continued to a collineation in its projective conclusion. There the remote hyperplane is then a fixed hyperplane of projective collineation.
• Conversely, a collineation in an at least two-dimensional projective geometry corresponds precisely to a true-to-plane collineation of the affine geometry, which is created by slitting the projective geometry when the collineation is slotted along a fixed hyperplane.
• The central or axial collineations, the plane perspectives , are important for synthetic geometry, especially for the study of non-Desarguean projective planes . These collineations create the subgroup of projectivities within the collineation group of a projective plane. The projectivities even form a normal divisor of this collineation group.

Collineations generalize geometric mappings

In both synthetic and analytical geometry , collineation generalizes mapping terms for which additional invariants are required:

1. A collineation of any affine space of finite dimension , in which every straight line has more than two points, is an affinity if and only if it is also partially true to the ratio . ${\displaystyle n>2}$
2. A collineation of a desargue affine plane is an affinity if and only if it is also partially true to the ratio.
3. A collineation of any affine plane is an affinity if and only if each of its restrictions on a straight line of the plane can be represented as a composition of finitely many bijective parallel projections.
4. A collineation of an at least three-dimensional projective space of finite dimension is precisely then a projectivity if it is also double-ratio true .
5. A collineation of a desargue projective plane is precisely then a projectivity if it is also double ratio true.
6. A collineation of any projective plane is projectivity if and only if it can be represented as a composition of finitely many projective perspectives .

Affinities and projectivities are always special collineations. In all cases they form a subgroup and even a normal divider of the group of all (true-to-plane) collineations of space, provided that this is at least two-dimensional.

Collineations in linear algebra, coordinate representation

Collineations on affine and projective spaces of finite dimensions over a body , more generally even over an inclined body , can be expressed by affinities or projectivities and a (inclined) body automorphism of the coordinate area. In linear algebra one usually restricts oneself to commutative oblique bodies, i.e. bodies as coordinate areas. Be a body or an oblique body, then: ${\displaystyle n>1}$ ${\displaystyle \sigma }$ ${\displaystyle K}$

1. Each collineation of a finite but at least 2-dimensional affine space has a unique representation as a composition with regard to a fixed affine coordinate system . First the automorphism is applied to the coordinates of a point and then the affinity to the new coordinate vector. ${\displaystyle \kappa }$ ${\displaystyle K}$ ${\displaystyle (|K|>2)}$ ${\displaystyle \kappa =\alpha \circ \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \alpha }$
2. Each collineation of a finite but at least 2-dimensional projective space has an unambiguous representation as a composition with regard to a fixed projective coordinate system . First the automorphism is applied to the coordinates of a point and then the projectivity is applied to the new coordinate vector. ${\displaystyle \kappa }$ ${\displaystyle K}$ ${\displaystyle \kappa =\pi \circ \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \pi }$
3. In particular, every non-identical (oblique) body automorphism of induces an affine or projective collineation of space , which depends on the chosen coordinate system and is not an affinity or projectivity. ${\displaystyle \sigma }$ ${\displaystyle K}$ ${\displaystyle \mathrm {id} _{R}\circ \sigma }$ ${\displaystyle R}$

In both representations, the automorphism is independent of the choice of the coordinate system. The split or double ratio of points, which is coordinate independent, becomes when the collineation is applied to the points . ${\displaystyle \sigma }$ ${\displaystyle t}$ ${\displaystyle \sigma (t)}$ ${\displaystyle \kappa }$

Inferences

• A collineation of a finite-dimensional Desarguean space is already an affinity or projectivity, ${\displaystyle \kappa }$
  • if the collineation leaves the partial or double ratios unchanged at all points on a straight line in space or
  • if the collineation has a straight line .
• Each collineation on an at least two-dimensional, Desargue's affine space A induces a bijective semilinear self-mapping of the space V of the connecting vectors, a finite-dimensional left vector space, which is uniquely determined . It then follows that the collineation with respect to a fixed point base of A can be represented uniquely as by a regular matrix T , the automorphism and the displacement component. ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa ({\overrightarrow {x}})=T\cdot \sigma ({\overrightarrow {x}})+{\overrightarrow {v_{0}}}}$ ${\displaystyle \sigma }$ ${\displaystyle {\overrightarrow {v_{0}}}=\kappa (O)}$
• Each collineation on an at least two-dimensional, Desargue's projective space P induces a bijective semilinear self-mapping of the coordinate vector space V , a finite-dimensional left vector space, which is uniquely determined . From this it follows that the collineation with respect to a fixed point base of P can be represented as a regular matrix T , which is unique except for scarar multiples, and the automorphism . ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa ({\overrightarrow {x}})=T\cdot \sigma ({\overrightarrow {x}})}$ ${\displaystyle \sigma }$

The affine spaces above the body must also be excluded for these conclusions : If the dimension of the space is greater than or equal to three, then these statements generally do not apply here ! ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$

Collineations in projective geometry

Every collineation of a projective space of dimension greater than or equal to 2 is a semi-linear mapping . So you have for ${\displaystyle \dim P(V)\geq 2}$

${\displaystyle \operatorname {Coll} (P(V))=\operatorname {P\Gamma L} (V)}$

for the group of collineations and the projective semilinear group . ${\displaystyle \operatorname {Coll} (P(V))}$ ${\displaystyle \operatorname {P\Gamma L} (V)}$

Examples

Spaces with at least 3 points on each straight line

The spaces considered in the following examples are always affine spaces over a body with more than two elements or projective spaces over any body, the dimension of the spaces is finite, but at least 2, ratio denotes the partial or double ratio :

• The composition of the conjugation and a projectivity of a complex projective space is called antiprojectivity . All collineations in the projective spaces are either projectivities or antiprojectivities. ${\displaystyle \mathbb {C} P^{n}}$
• Collineations on affine or projective spaces over a body , whose only automorphism is identity , are always affinities or projectivities. Such fields are all prime fields , i.e. the rational numbers and all residual class fields with prime numbers . ${\displaystyle K}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle p}$
  • The same applies to the collineations on spaces over the real numbers and more generally for spaces over any Euclidean solids , because like the prime solids these solids have no non-identical automorphisms. - Due to the equivalence of the statements “ ” and “ is solvable”, their natural arrangement is an algebraic invariant! ${\displaystyle a\geq 0}$ ${\displaystyle a=x^{2}}$
• Although collineations are generally not proportional, relationships that are in the prime body of a body are retained. If the characteristic of a body is not 2, then for example: ${\displaystyle K}$
  • In affine spaces , the center of a segment (in the sense of an ordered pair of points) is mapped to the center of the image segment for each collineation, ${\displaystyle K}$
  • in projective spaces the harmonious position of four collinear points is retained. ${\displaystyle K}$

Spaces with two points on each line

The three-dimensional affine space above the two-element body K. All 8 points, but only 14 of the 28 straight lines of this space are shown. The mapping, which swaps points C and F (green) and leaves all 6 other points fixed, is a collineation that is neither true to parallel nor true to plane. The straight lines CH and FH are swapped (red), while 2 of the 3 other parallels of their respective family remain fixed. The level is mapped to the set that is not a level. ${\displaystyle \{A,C,H,E\}}$ ${\displaystyle \{A,F,H,E\}}$

Every -dimensional affine geometry ( ) with exactly two points on each line is an affine space over the remainder class field . These are desargue affine geometries throughout, but the usual division ratio is degenerate, since there are no triples of different collinear points. In these special cases: ${\displaystyle n}$ ${\displaystyle n\geq 1}$ ${\displaystyle K=\mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle n\geq 2}$

• The group of straight line bijections of the point set (i.e. the collineations) is equal to the group of all bijections of the point set, i.e. isomorphic to the symmetrical group , because the line set consists exactly of all two-element point sets. ${\displaystyle S_{2^{n}}}$
• For this is true for the group of affinities. ${\displaystyle n=1;2}$
• For collineations, one also frequently demands level fidelity, i.e. that every two-dimensional subspace of the geometry is mapped onto a two-dimensional subspace. ${\displaystyle n\geq 3}$
• With this restricted concept of collineation:

Every true-to-plane collineation is an affinity in the sense of linear algebra and vice versa.

In contrast, the group of affinities (it has elements, compare linear group ) is for a real subgroup of . ${\displaystyle 2^{n}\cdot (2^{n}-2^{0})\cdot (2^{n}-2^{1})\cdots (2^{n}-2^{n-1})}$ ${\displaystyle n\geq 3}$ ${\displaystyle S_{2^{n}}}$

literature

• Walter Benz: A Century of Mathematics, 1890-1990 . Festschrift for the anniversary of the DMV . Ed .: Gerd Fischer (=  documents on the history of mathematics . Volume 6 ). Vieweg, Braunschweig 1990, ISBN 3-528-06326-2 (Contains many references to the history of the term “collineation” and related terms, including further references).
• Wendelin Degen, Lothar Profke: Fundamentals of affine and Euclidean geometry . 1st edition. Teubner, Stuttgart 1976, ISBN 3-519-02751-8 (On the meaning of the term collineation for “elementary” and school geometry).
• Gerd Fischer : Analytical Geometry . 6th, revised edition. Vieweg, Braunschweig a. a. 1992, ISBN 3-528-57235-3 (detailed description of the coordinate representation of arbitrary collineations of projective spaces over bodies).
• Günter Pickert : Projective levels . 2nd Edition. Springer, Berlin / Heidelberg / New York 1975, ISBN 3-540-07280-2 (on the structure of the collineation group).
• Hermann Schaal: Linear Algebra and Analytical Geometry . 2nd, revised edition. tape 2 . Vieweg, Braunschweig 1980, ISBN 3-528-13057-1 (connection between collineations and correlations , mainly for the case of a two- or three-dimensional real geometry).
• Günter Scheja, Uwe Storch: Textbook of Algebra: including linear algebra . 2., revised. and exp. Edition. Teubner, Stuttgart 1994, ISBN 3-519-12203-0 (In this textbook, the special cases that occur with bodies of characteristic 2 are discussed in more detail).

Individual evidence

1. ↑ G. Fischer: Analytical Geometry . 1992, p. 163 .
2. ↑ ^{a b c d} G. Scheja, U. Storch: Textbook of Algebra: including linear algebra. 1994.
3. ↑ ^{a b c d} The statements remain valid even in the special case of the body , if one also demands a "collineation" in this case, see the sections # In-plane collineations and geometric automorphisms and # Spaces with two points on each straight line . ${\displaystyle K=\mathbb {Z} /2\mathbb {Z} }$
4. ↑ ^{a b} H. Schaal: Linear Algebra and Analytical Geometry. Volume II, 1980, p. 198.