Collineation

The term referred collineation in the fields of mathematics and geometry, linear algebra a bijective mapping of an affine or projective space to itself, in which each is mapped to a line Line that is just so true. The amount of collineations of a space forms a group, especially the reversals of collineations are always collineations.

Thus, the term falls for one-dimensional spaces with the notion of bijection of the respective straight lines together. Therefore, usually only collineations be studied on at least two-dimensional spaces.

Occasionally, the term collineation is also used for a bijective or even just injective faithful picture of an affine or projective space to another room. The present article deals exclusively with collineations that are just loyal, bijective self-maps of a room.

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

• 3.1 Consequences

• 4.1 rooms with at least three points on each line
• 4.2 rooms with two points on each line

Collineations in synthetic geometry

In the synthetic geometry ( planes) are examined in typically collineations on two-dimensional spaces. As for the nichtdesargueschen levels the group of affinities or projectivities often is not rich enough to investigate the structure of the plane, here the group of collineations in place occurs. In an abstract incidence geometry, this group forms the characteristic automorphism, since the " position of points on a common line ( collinear ) " the only structure on the space, and thus - in the sense of the Erlangen program - the only space, in this case the level, characterizing invariant.

Eben loyalty collineations and geometric automorphisms

• Every collineation of an affine plane is parallel faithful, that is, valid for two straight lines of the plane.
• A collineation of at least three-dimensional affine geometry is then exactly parallel faithful if it is just true, 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 each line or an arbitrary projective geometry is always precisely true. Compare the figure to the right, and the example of order 2 below.
• A layer is always a faithful collineation geometric automorphism of the space that is to form each sub- area on a sub-space of the same dimension from. Conversely, of course, any geometric automorphism a newly loyal collineation.
• A " bijection by base exchange with the same coordinate ," ie, an image of the at least two- point space, in which each point to a point with the same coordinates ( a Ternärkörper in the case of a plane of a swash body in the case of an at least three-dimensional space ), each subspace is mapped onto a sub-space with the same coordinates equations, however, coordinates and equations are based on a different point of the base is a flat, and thus a geometrical fidelity collineation 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 Moufangebene.

• Conversely, however, there is generally planar loyal collineations which can not be represented by a change of basis in "Coordinate identity ".
• Each just a loyal collineation at least two-dimensional affine geometry can be uniquely continued to a collineation in its projective completion. There, the remote hyperplane is a Fixhyperebene of projective collineation.
• Conversely, to a collineation in a least two-dimensional projective geometry if and only just a loyal collineation of the affine geometry that arises by slitting the projective geometry when is slit along a Fixhyperebene the collineation.
• Important for the synthetic geometry, in particular for the study of nichtdesarguesschen projective planes, are the central or axial collineations, the flat Perspektivitäten. These collineations generate the subgroup of projectivities within the collineation group of a projective plane. The projectivities even form a normal subgroup of the collineation.

Collineations generalize geometric figures

In the synthetic and in analytic geometry collineation generalized Figure items in which additional invariants are required:

Affinities and projectivities are always special collineations. They form in all cases, a sub-group, and even a normal subgroup of the group of all (ground true ) collineations of space if it is at least two-dimensional.

Collineations in linear algebra, coordinate representation

Collineations on affine and projective spaces of finite dimension over a field, generally even over a skew field can be expressed by affinities or projectivities and a (skew - ) Körperautomorphismus of the coordinate area. In linear algebra, limited usually to commutative skew field, ie body as coordinate ranges. Be a body or skew field, then:

In two representations of the automorphism is independent of the choice of the coordinate system. The part or double ratio of dots which is independent coordinates, will, when the collineation is applied to the points.

Conclusions

• A collineation of a finite-dimensional space is already Desargues then an affinity or projectivity

• If the collineation the partial or cross ratios on a line of space can be the same for all points or
• If the collineation has a fixed point straight.

• Each collineation on a minimum two-dimensional Desargues affine space A induces a uniquely determined by bijective semilinear self- mapping of the space V of the connection vectors of a finite dimensional vector space links. It follows then that the collineation with respect to a fixed point selected based on A clearly than by a regular matrix T, the automorphism and the offset component can be represented.
• Each collineation on a minimum two-dimensional Desargues projective space P induces a uniquely determined by semilinear bijective self-map of the coordinate vector space V, a finite-dimensional left vector space. It follows then that the collineation can be represented by a regular, unique up to multiples skarare matrix T and the automorphism with respect to a fixed point selected as the basis of P.

Also for these conclusions the affine spaces must be excluded on the body: If the dimension of the space is greater than or equal to three, then these statements are true in general here not to!

Examples

Rooms with at least three points on each line

The consideration in the following examples are always areas affine spaces over a field having more than two elements or projective space to a desired body, the dimension of the space is finite, and at least 2, the ratio referred to Part or double ratio:

• The composition of conjugation and a projectivity of a complex projective space is called Antiprojektivität. All collineations in projective spaces are either projectivities or Antiprojektivitäten.
• Collineations on affine or projective spaces over a field whose only automorphism is the identity, are always affinities or projectivities. Such bodies are all prime field, ie the rational numbers and all the residue field with prime.

• The same applies to the collineations in spaces over the real numbers and more generally for spaces over arbitrary euclidean bodies, because these bodies have as the prime field no non-identical automorphisms. - Due to the equivalence of the statements, " " and " is releasable" is their natural arrangement of an algebraic invariant!

• Although collineations in general are not relatively loyal, proportions must be maintained, which lie in the prime field of a body. If the characteristics of a body not 2, then applies, for example:

• In affine space about the center of a track is mapped (in the sense of an ordered pair of points ) at each collineation to the center of the image line,
• In projective spaces over the harmonic situation of four collinear points remain.

Rooms with two points on each line

Each -dimensional affine geometry () with exactly two points on each line is an affine space over the residue field. These are for consistently desarguesche affine geometries, but the usual split ratio is degenerate, then there are no triples of different collinear points. In these special cases:

• The group of even loyal bijections of the set of points (ie the collineations ) is equal to the group of all bijections of the set of points, ie, isomorphic to the symmetric group, because the set of lines consists exactly of all two-element sets of points.
• For this is also true for the group of affinities.
• For one often calls for collineations in addition Eben loyalty, so that every two-dimensional subspace of the geometry 'll mapped onto a two-dimensional subspace.
• With this restricted Kollineationsbegriff then:

In contrast, the group of affinities is (it has elements, Linear compare group ) for a proper subgroup of.