Correlation (projective geometry)

A correlation is in projective geometry a (incidence structure ) isomorphism between a projective plane and its dual plane. From the plane is additionally required in the most important cases that it meets the set of Pappus, can be so koordinatisiert by a commutative field. The presentation and classification of correlations are similar to the collineations of a projective plane. Important differences are Kollineatieonen: A correlation of the level forms of points on straight lines and straight lines to points from. While collineations of a projective plane always exist, correlations do not exist, if the projective plane (or more generally the projective space ) is not pappossch.

An important application have projective polarities, which are double- ratio -abiding, self-reciprocal correlations in the absolute geometry, because such a correlation as absolute polarity of the " metric " of a projective- metric space indicates there and defines its motion group. They are a generalization of the assignment described in the article pole and polar ( a hyperbolic projective polarity), which is determined by a conic section. This can also be a projective polarity of a certain projective line within a broader projective space be interesting: it can be through a (not necessarily positive - definite, but a formal ) describe scalar product, which on a line of the projective space, an elliptical, projective Polarinvolution, that is, induces a fixed-point- free, projective polarity on a straight line. This Polarinvolution on an excellent line at supplies in the projective description of absolute geometry for the " Euclidean special case " the invariant that provides the projective polarity in non-Euclidean case. Here, a relationship to the shows (initially projective two -dimensional ) Minkowski space, which is not itself a model of absolute geometry: The Minkowski metric induced on an excellent line at the level of a hyperbolic projective Polarinvolution.

The term correlation is more commonly used in the obvious sense in projective spaces of higher dimension and for nichtdesarguessche levels.

2.1 Coordinate representation
2.2 Projective polarities and conic sections
2.3 Projective correlations and bilinear forms

3.1 A non - projective, elliptic polarity
3.2 Hyperbolic polarities
3.3 An elliptic polarity

4.1 polarities over finite spaces

5.1 Special Polarinvolutionen

6.1 Desargues spaces of arbitrary, finite-dimensional
6.2 Nichtdesarguessche levels

Definitions

Correlation

A correlation of a pappusschen projective plane is an incidence -preserving bijective mapping this space to the dual plane, with bijective, and is bijective on mapped. Point set and set of lines are reversed in so in the dual plane.

Projective correlation

A correlation is called projective if each one-dimensional basic structure is projective, ie mapped double ratio faithful. This means in practice:

Polarity, pole, polar and conjugated elements

A self-reciprocal correlation ( it need not be necessary projective ) is called polarity. It assigns to each point of a well-defined straight line ( its polar ) and each line a well-defined point ( their pole ), the pole of the polar of a point again is the original point and the polar of the pole of a straight line back to the original line.

Two points are called conjugate to each other (in terms of polarity) when everyone is on the polar of the other: two lines are called conjugate to each other (in terms of polarity), where each passes through the pole of the other. A point is called self-conjugate if it is on its polars, a straight line if it contains its pole.

Hyperbolic and elliptic polarities

A polarity is called hyperbolic if it has self-conjugate points (and thus equivalent to self-conjugate line), otherwise it is called elliptic.

Preparation and Properties

Are correlations on the same projective plane, then the concatenation is a collineation this level (and also a collineation of the dual plane).

The correlations projective, the concatenation is a projectivity both the plane ( a point set ) and the dual- layer (as the figure on the set of lines ).
Also can be a projectivity within the meaning of the previous statement, when none of the two correlations is projective.

A correlation is exactly one plane then a polarity if the identical mapping of the plane is ( its point set and its set of lines ).

Coordinate representation

Be a body. The vector space provides the Standard Model of the projective plane. After selecting a projective point basis, ie an ordered complete quadrilateral can also be an abstract projective plane then identify with the standard model. It is agreed: column vectors represent the homogeneous coordinates of points, lines vectors for the homogeneous coordinates of lines. A point and a line are incident if and only if the formal matrix product has value.

For a projective correlation mapping must map the coordinates of each point linearly, so be ready with a regular matrix A. It must also apply to the line coordinates. Thus, the " incidence form " merges in itself, must apply between the regular matrices of the connection. The correlation is then exactly in involution when.

At any correlation, the assignments must be semilinear, then for the coordinate vectors of points and for the coordinate vectors of lines. Here is a Körperautomorphismus of K. The Körperautomorphismus on the selected coordinate system independent, compare this: # collineation coordinate representation. Here, too, must apply between the regular matrices of the connection. The correlation is then exactly in involution if and.

Projective polarities and conic sections

If a hyperbolic polarity projective, so the self-conjugate points and lines form a conic section k is referred to by Karl von Staudt as the fundamental curve of polarity. The pole of an arbitrary straight line then is also called " her pole with respect to k" and the polars of any point " its polar with respect to k", as explained in the article pole and polar.

For elliptic polarities exists no single defining fundamental curve.

Projective correlations and bilinear forms

One can also look at in isolation from the geometrical interpretation given to the hyperplanes of points by assigning illustration. The terms and attributes radical nullteilig isotropic and which are defined in the abstract linear algebra, are also found in the geometric literature. They overlap with partially labeled the same, but not quite equivalent terms from the classification of quadrics. The explanations here are governed by Bachmann ( 1973).

It is first A be an arbitrary matrix with entries from a field K, the m-dimensional vector space over K with its standard vector space basis. Then by

A bilinear form F defined.

The radical left is the core of the linear mapping, ie, the solution space of the equation, formally the is a picture of Vekorraums in its ( algebraic ) dual space, because acts as a linear form on vectors.
The right radical is the core of the linear mapping.
For a subspace.
For a subspace.
If the bilinear form F symmetric, then left and right radical are the same, you call this quantity then radical of V with respect to the form F. It is however sufficient to the fact that a symmetric matrix. This is always given for a projective polarity.

For the concept of isotropy it just depends on the values of the bilinear form. A vector is called isotropic if is. From the definition it follows that any vector that belongs to the right or left radical is isotropic.

Conversely, in a symmetric bilinear form any isotropic vector contained in the radical, then the bilinear form is called nullteilig

For the cases described in this article applies the following dictionary ( all of the above figures are projective in the first column and a linear or bilinear in the second and third ):

Examples

A non - projective, elliptic polarity

Be the field of complex numbers. Then, by and the complex conjugate ( the unit matrix is ) defined a correlation to the projective plane that is involutory, but not projective, ie a polarity. This is elliptical, since the equation for self-conjugate vectors has no solution other than the zero vector.

Hyperbolic polarities

The unit circle of the affine plane over the real numbers is the projective completion of this level to the fundamental curve of a polarity. If you choose the line at infinity, then the circle equation is projective. The " shape matrix " of this quadric the diagonal matrix is also the point spread matrix of the corresponding polarity. So it is the affine point in the projective projective polar, affine over. This is a straight line which is different from the origin affine points P perpendicular to line OP and passes through the point P on the unit circle is a mirror image with respect.

Polar of the origin is the line at infinity,
The points on the projective conic are exactly the self-conjugate points of the polarity ( exactly they are incident with their polars ). The polarity is so because there is self-conjugate points, hyperbolic.

The hyperbola of the affine plane above, in the projective degree to the shape of the matrix, is equivalent to the form of matrix, which has the advantage similar to the form of matrix in the previous example, to be ( projective is the conic section, in this example is equivalent to the unit circle ). The permutation matrix forms (as projectivity ) the unit circle k on h from, it is the shape matrix A of the unit circle. Thus, if a pole - polar pair with respect to the unit circle, then a pole - polar pair with respect to the hyperbola.

The by certain polarity is hyperbolic and projective.
The far point of the axis, the polar, that is, that the affine -axis.
The far point of the axis, the polar, that is, that the affine -axis.
The self-conjugate points lie on the conic h, the self-conjugate lines are the tangents.
For example, touch the two bisectors of the coordinate system as the asymptotes of the hyperbola affine projective conic h in their respective distance point, this far point is each pole of the asymptote. ( Mathematically, for the first bisector: ).
As with the circle ( and in any conic section with a central point) is the affine center of the conic section, here the origin, polar to the line at infinity.

An elliptical polarity

Be. We consider the three-dimensional vector space mapping which assigns to each vector to him (in the sense of the usual scalar product ) vertical two-dimensional subspace. In projective space, this corresponds to the correlation. This is a projective polarity. There are no self-conjugate point (one-dimensional sub-spaces of V) or straight lines ( two-dimensional sub-spaces of V ), thus the polarity is elliptical.

The real projective plane may be considered as model of the real elliptic geometry by cutting the subspaces of V with a ball S around the zero point of V: the pair of points is then From the projective point where the "straight line" the sphere S is true ( antipodes of the sphere are thus becoming an elliptic point " glued " ), from the projective line is the great circle in which the vector space plane intersects the sphere.

So behave Polar and pole as the Earth's equator to the geographic poles. The polar to a ( elliptical ) point (that is, a pair consisting of a point and its counter- point) then the great circle that is farthest away from this. The pole to a great circle p ( the polars ) is characterized in that all great circles which are perpendicular to p, intersect there.

Is defined in the projective plane by a vertical relation

Then were introduced with the described elliptical polarity projective a " metric " in which these projective plane to form an elliptical plane, more precisely to the elliptical is ( up to isomorphism clear ) layer over the field of real numbers. Each elliptical polarity of the real projective plane can in fact bring on the form of this elliptical polarity by a suitable choice of the coordinate system.

Projective polarity in projective spaces of arbitrary, finite-dimensional

In at least two-dimensional, pappusschen projective space over a field, one has. Fixed by a projective polarity a certain one-to -one mapping between points and hyperplanes of the space This is particularly uniformly elliptical case, the fact that there is no self-conjugate point, geometric means that no point is situated on the polar to him hyperplane.

Polarities over finite spaces

Through a drawer argument could also be refined to an enumeration of self-conjugate elements in a finite polarity can be proved: Is there at a projective, elliptic polarity and is the characteristic of K is not 2, then K must be infinite.

Equivalently: If K is finite with elements and, and is a regular matrix, then has the equation for self-conjugate points

A nontrivial solution.

It suffices to consider the case: one can under the conditions stated, the matrix with the projective quadric methods in the article shown, notably by completing the square on the diagonal form bring, geometrically speaking to Choose an orthogonal basis of. The equation to be solved is then equivalent

Substituting and considers all elements that arise on the left side of the equation when used for all q -state elements, then these are different numbers, because each of exactly two different numbers gives the same value, the setting provides an additional one. If 0 is below the values as shown, then you set and has a non-trivial solution 0 is not among them, so are all displayable by the term on the left-hand side of equation ( ) numbers in included, then it must include also a square number, for decays into exactly two square classes, the class of square numbers, which is a proper subgroup of, and their real coset, both classes each contain elements, ie less than those given during insertion into the left side of ( ). Thus, there must be for self-conjugate points again a nontrivial solution of the equation ( ).

→ The exact numbers selbstkonjugierter points for polarities over finite spaces resulting in the most important cases from the theorems about quadratic quantities.

Polarinvolution as polarity on a straight line

It should be - the following considerations apply but over arbitrary fields with. We consider the " geometry " in which, ie the one-dimensional subspaces consists only of the line through the origin. Each sub- area is characterized by a " direction ". It is. On the other hand, applies exactly to the points of a line, the homogeneous equation. The coefficient vector is the normal vector of the line. Since both the direction and the normal vectors "homogeneous" are (only determined up to a multiplication by ) the considered geometry is a one-dimensional projective geometry and the assignment with is a projective involution correlation of these projective line, ie a one-dimensional projective polarity. Describing the affine plane over with " orthogonality " as the actual level in the projective plane over, then you have by this one-dimensional projective polarity on the line at infinity, ie, the straight line with the coordinates of a projective invariant, which (in the case described, usual ) orthogonality projective describes: the projective geometry itself assigns each parallel class a remote point as the direction to which Polarinvolution assigns each direction to their " polar " direction, which in turn is leading to the parallel class, of which one is out, vertical band.

Generally called a polarity on a projective line, the part of ( at least two-dimensional ) projective space is Polarinvolution. Since a projective line is the set of points dual to itself, any correlation of the line is also a collineation, projective each correlation is a projectivity and usually only the case of a projective correlation on a straight line in a larger projective space is geometrically interesting.

Special Polarinvolutionen

A Polarinvolution is called projective if it is projective as a collineation, ie a (one-dimensional ) is projectivity.
A Polarinvolution is called elliptic if it has no fixed points. This definition transfers the corresponding property of the two-dimensional polarity, with the aggravation that here means incidence of a tie.
A Polarinvolution is called hyperbolic if it has at least one fixed point.

The one-dimensional projective group operates sharply triply transitive on the line, therefore, a non-identical projective collineation not only have one or exactly two fixed points. This demonstrates an analogy to the two-dimensional case: The Fixelementmengen that can occur in a hyperbolic, projective Polarinvolution consist of a " ( double -counting ) point" or a pair of points. These are precisely the " conic sections " that can occur next to the empty set and the whole line in one-dimensional space.

In the case of finite straight lines, the total number of points on the line because the general assumption is straight, since the order of the lines is an odd number and the case excluded exactly a fixed point for involution.

A hyperbolic, projective Polarinvolution but is generally characterized by the set of its fixed points is not uniquely determined, unlike in the two-dimensional case, a hyperbolic, projective polarity by the amount of their self - conjugate points.

Generalizations

Desargues any spaces, finite-dimensional

On a projective line is the set of points dual to itself and the correlation term coincides with the term collineation. Each bijection of the set of points ( that is, the points on the single straight line ) is a correlation. Interesting here is the study of the self-reciprocal, projective collineations. Projective "spaces" of dimension 0 (points) and -1 ( empty set) supply obviously nothing interesting.

Each of at least three-dimensional projective geometry is desarguesch, ie as represented - dimensional space over a skew field K. Here, the term correlation can be transmitted with almost no restrictions when K is isomorphic to its opposite ring: as incidence structure with the basic structures point, line, ... hyperplane isomorphic to the dual structure (incidence versa thereby possibly to ). Every bijective mapping which each point is a hyperplane, each line is one -dimensional subspace, etc. assigns true incidence of a correlation. As in the flat case:

The complete correlation is determined by the images of the points. If a coordinate system fixed chosen then determines each semilinear map (regular matrix Schiefkörperautomorphismus K ) the point coordinate vectors on the hyperplane coordinate vectors correlation clearly, each correlation is displayed.
A correlation is exactly projective if the dot image with respect to a coordinate system (and then in each coordinate system ) is linear so the Körperautomorphismus is identical.
Such a correlation is in involution as in the two-dimensional case under the same conditions.
For the composition of two correlations and the square of a correlation, the same relationships to collineations and the identity as given above for the two-dimensional case apply.
The self-conjugate points of a projective involution correlation form a ( possibly empty ) hypersurface of second order self-conjugate and the hyperplanes are exactly the Tangentialhyperebenen this hypersurface, if K is commutative and its characteristic is not 2. Without this, this need not apply! Therefore, it is usually a pappussche geometry that satisfies the Fano axiom ahead when one speaks of polarities.

Nichtdesarguessche levels

For an arbitrary projective plane the dual plane is always another projective plane. In general, the level is not isomorphic to its dual plane. Only when it is, therefore, a correlation exists at all, then it is labeled as such. A correlation always exists under the following condition:

Then on the representation of the correlation as a semilinear point the figure in the previous section said.

Involution (mathematics) Spacetime Identity matrix Antipodal point Characteristic (algebra) Affine plane (incidence geometry)

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