Correlation (Projective Geometry)

In projective geometry, a correlation is an ( incidence structure ) isomorphism between a projective plane and its dual plane. In the most important cases, the level is additionally required to fulfill Pappos' theorem , i.e. to be able to be coordinated by a commutative body . The representation and classification of correlations largely correspond to those of collineations of a projective plane. Important differences to collineations are: A correlation of the plane maps points onto straight lines and straight lines onto points. While collineations of a projective plane always exist, correlations need not exist if the projective plane (or more generally the projective space) is not Papposian.

Projective polarities have an important application , that is , involutional correlations true to the double ratio in absolute geometry , because such a correlation there, as absolute polarity, characterizes the “metric” of a projective-metric space and defines its movement group. They are a generalization of the mapping described in the article Pol and Polare (a hyperbolic projective polarity) determined by a conic section . A projective polarity of a certain projective straight line within a more comprehensive projective space can also be of interest here: It can be described by a (not necessarily positive-definite , but rather a formal ) scalar product which, on a straight line in projective space, has an elliptical, projective polar involution , that is, a fixed point-free, projective polarity is induced on a straight line . In the projective description of the absolute geometry for the “Euclidean special case”, this polar involution on an excellent long-distance straight line supplies the invariant which the projective polarity supplies in the non-Euclidean case. This shows a relationship to the (initially projective two- dimensional) Minkowski space , which is not itself a model of an absolute geometry: The Minkowski metric induces a hyperbolic projective polar involution on an excellent long-distance line of the plane .

The term correlation is used more generally in the obvious sense for projective spaces of higher dimensions and for non-Desargue planes.

Definitions

correlation

A correlation of a Pappus projective plane is an incidence- preserving bijective mapping of this space on the dual plane , with bijective mapping on and bijective mapping on . The set of points and set of lines are thus interchanged in the dual plane. ${\displaystyle ({\mathfrak {P}},{\mathfrak {G}},I)}$ ${\displaystyle ({\mathfrak {G}},{\mathfrak {P}},I^{-1})}$ ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle {\mathfrak {G}}}$ ${\displaystyle {\mathfrak {G}}}$ ${\displaystyle {\mathfrak {P}}}$

Projective Correlation

A correlation is called projective if every one-dimensional basic structure is mapped projectively, i.e. true to the double ratio. Specifically, this means: ${\displaystyle \kappa }$

1. If four points are on a straight line, then their double ratio is equal to the double ratio of the four straight lines . ${\displaystyle A,B,C,D}$ ${\displaystyle \kappa (A),\kappa (B),\kappa (C),\kappa (D)}$
2. If there are four straight lines that go through a common point, then their double ratio is equal to the double ratio of the four points . ${\displaystyle a,b,c,d}$ ${\displaystyle \kappa (a),\kappa (b),\kappa (c),\kappa (d)}$

Polarity, pole, polar and conjugate elements

An involutorial correlation (it does not need to be projective) is called polarity . It assigns a well-defined straight line (its polar ) to every point and a well-defined point (its pole ) to every straight line , the pole of the polar of a point being the original point again and the polar of the pole of a straight line being the original straight line. ${\displaystyle \kappa }$

Two points are called conjugate to each other (in polarity), when each is on the polar of the other: two straight hot conjugated to one another (in polarity) when each passes through the pole of the other: . A point is called self-conjugated if it lies on its polar, a straight line if it contains its pole. ${\displaystyle A,B}$ ${\displaystyle AI\kappa (B),BI\kappa (A)}$ ${\displaystyle a,b}$ ${\displaystyle \kappa (a)Ib,\kappa (b)Ia}$

Hyperbolic and elliptical polarities

A polarity is called hyperbolic if it has self-conjugate points (and thus equivalently self-conjugate straight lines), otherwise it is called elliptic.

Appearance and properties

• If correlations are on the same projective plane, the concatenation is a collineation of this plane (and also a collineation of the dual plane). ${\displaystyle \kappa _{1},\kappa _{2}}$ ${\displaystyle \kappa _{2}\circ \kappa _{1}}$

• If the correlations are projective, then the concatenation is a projectivity of both the plane (as a set of points) and of the dual plane (as a mapping on the set of lines). ${\displaystyle \kappa _{1},\kappa _{2}}$ ${\displaystyle \kappa _{2}\circ \kappa _{1}}$
• ${\displaystyle \kappa _{2}\circ \kappa _{1}}$can also be projectivity in the sense of the previous statement if neither of the two correlations is projective.

• A correlation of a plane is a polarity if and only if the plane (its point set and its line set) is identical . ${\displaystyle \kappa }$ ${\displaystyle \kappa ^{2}}$

Coordinate representation

Be a body . The vector space delivers the standard model of the projective plane . After selecting a projective point base , i.e. an ordered complete quadrilateral , an abstract projective plane can then also be identified with the standard model. It is agreed: Column vectors stand for the homogeneous coordinates of points, row vectors for the homogeneous coordinates of straight lines. A point and a straight line intersect if and only if the formal matrix product has the value . ${\displaystyle K}$ ${\displaystyle K^{3}}$ ${\displaystyle K}$ ${\displaystyle x\in K^{3}\setminus \{0\}}$ ${\displaystyle [u],u=[u]^{T}\in K^{3}\setminus \{0\}}$ ${\displaystyle x}$ ${\displaystyle [u]}$ ${\displaystyle u^{T}\cdot x}$ ${\displaystyle (0)=0}$

For a projective correlation , the assignment map the coordinates of each point linear, is thus with a regular matrix A . The same must apply to the straight line coordinates . In order for the "incidence form" to pass into itself, the relationship between the regular matrices must apply. The correlation is involutor if and only if is. ${\displaystyle \kappa }$ ${\displaystyle \kappa (x)=(A\cdot x)^{T}}$ ${\displaystyle 3\times 3}$ ${\displaystyle \kappa ([u])=B\cdot u}$ ${\displaystyle u^{T}\cdot x}$ ${\displaystyle A,B}$ ${\displaystyle B^{T}=r\cdot A^{-1},r\neq 0,r\in K}$ ${\displaystyle A^{T}=A}$

For any correlation , the assignments must be semilinear , then is for the coordinate vectors of points and for the coordinate vectors of straight lines. It is one of Körperautomorphismus K . The body automorphism is independent of the selected coordinate system, compare: Collineation # Coordinate representation . Here, too, must be between the regular matrices of connection apply. The correlation is involutive if and only if and is. ${\displaystyle \kappa }$ ${\displaystyle \kappa (x)=(A\cdot \alpha (x))^{T}}$ ${\displaystyle \kappa ([u])=B\cdot \alpha (u)}$ ${\displaystyle \alpha }$ ${\displaystyle A,B}$ ${\displaystyle B^{T}=r\cdot A^{-1},r\neq 0,r\in K}$ ${\displaystyle A^{T}=\alpha (A)}$ ${\displaystyle \alpha ^{-1}=\alpha }$

Projective polarities and conic sections

If a hyperbolic polarity is projective, the self-conjugate points and straight lines form a conic section which, according to Karl von Staudt, is called the fundamental curve of polarity. The pole of any straight line is then also called “its pole in relation to ” and the polar of any point “its polar in relation to ”, as explained in the article Polars and Polars . ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle k}$

There is no defining fundamental curve for elliptical polarities.

Projective correlations and bilinear forms

→ Main article : bilinear form

The mapping given by the assignment for points on hyperplanes can also be viewed separately from the geometrical interpretation. The terms radical and the isotropic and zero-part attributes , which are defined in abstract linear algebra , also appear in geometric literature. They overlap with terms from the classification of quadrics, some of which have the same name but are not entirely equivalent. The explanations given here are based on Bachmann (1973). ${\displaystyle x\mapsto A\cdot x}$

First let any - matrix with entries from a body , the dimensional vector space over with its standard vector space base . Then it's through ${\displaystyle A}$ ${\displaystyle m\times m}$ ${\displaystyle K}$ ${\displaystyle V=K^{m}}$ ${\displaystyle m}$ ${\displaystyle K}$

${\displaystyle F:V\times V\rightarrow K;\quad (x,y)\mapsto F(x,y):=x^{T}\cdot A\cdot x}$

defines a bilinear shape. ${\displaystyle F}$

radical

• The left radical is the core of the linear mapping , i.e. the solution space of the equation , formally this is a mapping of the vector space into its (algebraic) dual space , because it acts as a linear form on vectors. ${\displaystyle V\rightarrow V^{*}:x\mapsto x^{T}\cdot A}$ ${\displaystyle x^{T}\cdot A=0}$ ${\displaystyle V^{*}}$ ${\displaystyle x^{T}\cdot A}$
• The right radical is the core of linear mapping . ${\displaystyle V\rightarrow V:y\mapsto A\cdot y}$
• For a subspace is . ${\displaystyle U\leq V}$ ${\displaystyle ^{\perp }U=\{v\in v|\forall u\in U:F(v,u)=0\}}$
• For a subspace is . ${\displaystyle U\leq V}$ ${\displaystyle U^{\perp }=\{v\in v|\forall u\in U:F(u,v)=0\}}$
• If the bilinear form is symmetrical, then left and right radicals are identical; this set is then called radical of with regard to the form . For this way, it is sufficient that a symmetric matrix is . This is always the case for a projective polarity . ${\displaystyle F}$ ${\displaystyle V}$ ${\displaystyle F}$ ${\displaystyle A}$ ${\displaystyle (A^{T}=A)}$

Isotropic vectors, zero division

For the concept of isotropy , only the shape values ​​of the bilinear form are important. A vector is called isotropic if is. From the definition it follows that every vector belonging to the right or left radical is isotropic. ${\displaystyle F(x,x)=x^{T}\cdot A\cdot x}$ ${\displaystyle x\in V}$ ${\displaystyle F(x,x)=0}$

Is inverted at a symmetric bilinear form, each vector included in the isotropic radical, then that means the bilinear nullteilig

The following dictionary applies to the cases described in this article (all figures mentioned are projective in the first column and linear or bilinear in the second and third):

Projective geometry	Matrix display	Vector space
Point mapping of a correlation	${\displaystyle x\mapsto x^{T}A}$, Matrix is regular ${\displaystyle A}$	Right and left radicals are the zero vector space
Correlation is a polarity	Matrix A is regular and symmetric	The bilinear form is symmetrical, its radical is the zero vector space ${\displaystyle F}$
from here a projective polarity:	from here a regular, symmetrical matrix:	from here a non-degenerate , symmetrical bilinear form:
Point is self-conjugated. ${\displaystyle <x>}$	${\displaystyle x^{T}\cdot A\cdot x=0}$	${\displaystyle x}$ is isotropic
Hyperplane is self-conjugated ${\displaystyle [y]}$	${\displaystyle y^{T}\cdot A\cdot y=0}$	${\displaystyle y}$ is isotropic
Point and hyperplane are polar ${\displaystyle x}$ ${\displaystyle [y]}$	${\displaystyle y^{T}\cdot A\cdot x=0}$	${\displaystyle ^{\perp }<x>=[y]}$ equivalent ${\displaystyle [y]^{\perp }=x}$
Polarity is elliptical	${\displaystyle x^{T}Ax=0\Rightarrow x=0}$	Every isotropic vector is in the radical, so it is here . is zero-part ${\displaystyle 0}$ ${\displaystyle F}$
Polarity is hyperbolic	${\displaystyle \exists x\in V\setminus \{0\}:x^{T}Ax=0}$	There are isotropic vectors that are not in the radical. is not zero-part. ${\displaystyle F}$

Examples

A non-projective, elliptical polarity

Be the body of complex numbers . Then ( be the - identity matrix ) and , the complex conjugation defines a correlation on the projective level , which is involutor but not projective, i.e. a polarity. This is elliptical because the equation for self-conjugate vectors has no solution other than the zero vector. ${\displaystyle K=\mathbb {C} }$ ${\displaystyle A=B=E_{3}}$ ${\displaystyle E_{3}}$ ${\displaystyle 3\times 3}$ ${\displaystyle \alpha (t)={\overline {t}}}$ ${\displaystyle \mathbb {P} ^{2}(\mathbb {C} )}$ ${\displaystyle {\overline {x}}^{T}\cdot A\cdot x={\overline {x}}^{T}\cdot x=0}$ ${\displaystyle x\in \mathbb {C} ^{3}}$

Hyperbolic polarities

• The unit circle of the affine plane over the real numbers becomes the fundamental curve of a polarity in the projective closure of this plane. If you choose the distance line, the circular equation reads projective . The “shape matrix” of this quadric, the diagonal matrix, is also the point mapping matrix of the associated polarity. So the affine point , projectively, goes over into the projective polar , affine . This is a straight line that stands for affine points different from the origin perpendicular to the straight line OP and passes through the point that is a mirror image of the unit circle line. ${\displaystyle x_{1}^{2}+x_{2}^{2}=1}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {P} ^{2}(\mathbb {R} )}$ ${\displaystyle x_{0}=0}$ ${\displaystyle k:-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=0}$ ${\displaystyle A=\operatorname {diag} (-1,1,1)}$ ${\displaystyle (p_{1}|p_{2})}$ ${\displaystyle (1,p_{1},p_{2})}$ ${\displaystyle -x_{0}+p_{1}\cdot x_{1}+p_{2}\cdot x_{2}=0}$ ${\displaystyle p_{1}\cdot x_{1}+p_{2}\cdot x_{2}=1}$ ${\displaystyle P}$ ${\displaystyle P'}$ ${\displaystyle P}$

• The polar of origin is the distant straight line,
• Polars of a far point are the straight lines through the origin, the perpendicular direction of which is affine, the straight lines , ${\displaystyle R(0,r_{1},r_{2})}$ ${\displaystyle R}$ ${\displaystyle r_{1}\cdot x_{1}+r_{2}\cdot x_{2}=0}$
• the points on the projective conic section are exactly the self-conjugate points of the polarity (exactly they intersect with their polar). So since there are self-conjugate points, the polarity is hyperbolic. ${\displaystyle -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=0}$

• In the projective conclusion, the hyperbola of the affine plane becomes with the shape matrix , which is equivalent to the shape matrix , which has the advantage of being similar to the shape matrix from the previous example (projectively, the conic in this example is equivalent to the unit circle). The permutation matrix maps (as projectivity) the unit circle on , it is with the shape matrix of the unit circle. So if a pole-polar pair is with respect to the unit circle, then there is a pole-polar pair with respect to the hyperbola. ${\displaystyle x_{1}^{2}-x_{2}^{2}=1}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle -x_{0}^{2}+x_{1}^{2}-x_{2}^{2}=0}$ ${\displaystyle \operatorname {diag} (-1,1,-1)}$ ${\displaystyle h:x_{0}^{2}-x_{1}^{2}+x_{2}^{2}=0}$ ${\displaystyle H=\operatorname {diag} (1,-1,1)}$ ${\displaystyle S={\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}}}$ ${\displaystyle k}$ ${\displaystyle h}$ ${\displaystyle SAS^{-1}=H}$ ${\displaystyle A}$ ${\displaystyle (x,u)}$ ${\displaystyle (Sx,Su)}$

• The polarity determined by it is hyperbolic and projective. ${\displaystyle H,\alpha =1}$
• The far point of the -axis has the polar , that is , the affine -axis. ${\displaystyle x_{1}}$ ${\displaystyle x=(0,1,0)^{T}}$ ${\displaystyle (H\cdot x^{T})^{T}=[0,-1,0]}$ ${\displaystyle x_{1}=0}$ ${\displaystyle x_{2}}$
• The far point of the -axis has the polar , that is , the affine -axis. ${\displaystyle x_{2}}$ ${\displaystyle y=(0,0,1)^{T}}$ ${\displaystyle (H\cdot y^{T})^{T}=[0,0,1]}$ ${\displaystyle x_{2}=0}$ ${\displaystyle x_{1}}$
• The self-conjugate points lie on the conic section h , the self-conjugate straight lines are its tangents .
• For example, the two bisectors of the coordinate system as asymptotes of the affine hyperbola touch the projective conic section h in their respective far point, this far point is each pole of the asymptote. (Mathematically for the first bisector:) . ${\displaystyle w:x_{1}-x_{2}=0,w=[0,1,-1]\mapsto Hw^{T}=(0,-1,-1)^{T}}$
• As with the circle (and with every conic section with a center point) the affine center point of the conic section, here the origin, is polar to the distance line.

An elliptical polarity

Elliptical polarity on the sphere: An elliptical point (pair of antipodes) is assigned the great circle as a polar , which is cut out of the sphere by the plane that is too perpendicular . ${\displaystyle (A,A')}$ ${\displaystyle a}$ ${\displaystyle {\vec {AA'}}}$ ${\displaystyle M}$ ${\displaystyle S}$

Be . In three-dimensional vector space we consider the assignment that assigns each vector to the two-dimensional subspace that is perpendicular to it (in the sense of the usual scalar product ) . In projective space this corresponds to the correlation with . This is a projective polarity. There are no self-conjugate points (one-dimensional subspaces of ) or straight lines (two-dimensional subspaces of ), so the polarity is elliptical. ${\displaystyle K=\mathbb {R} }$ ${\displaystyle V=\mathbb {R} ^{3}}$ ${\displaystyle x\in V\setminus \{0\}}$ ${\displaystyle <x>^{\perp }}$ ${\displaystyle \mathbb {P} ^{2}(\mathbb {R} )}$ ${\displaystyle A=B=E_{3},\alpha =1}$ ${\displaystyle V}$ ${\displaystyle V}$

The real projective plane can be understood as a model of real elliptical geometry by intersecting the subspaces of with a sphere around the zero point of : The projective point then becomes the pair of points in which the "straight line" meets the sphere ( antipodes of the sphere thus "glued" to an elliptical point ), the projective straight line becomes the great circle in which the vector space plane intersects the sphere. ${\displaystyle V}$ ${\displaystyle S}$ ${\displaystyle V}$ ${\displaystyle <x>,x\in V\setminus \{0\}}$ ${\displaystyle x}$ ${\displaystyle S}$ ${\displaystyle <x>^{\perp },x\in V\setminus \{0\}}$

So the polar and pole behave like the earth equator to the geographic poles . The polar to an (elliptical) point (i.e. to a pair of a point and its counterpoint) is then the great circle that is furthest away from it. The pole to a great circle (the polar ) is characterized in that all great circles that are perpendicular to each other intersect there. ${\displaystyle \textstyle \mathbf {p} }$ ${\displaystyle \textstyle \mathbf {p} }$

Define a vertical relation in the projective plane ${\displaystyle \mathbb {P} ^{2}(\mathbb {R} )}$

" Shall apply if and only if the pole (in the sense of the polarity defined above) contains", ${\displaystyle g\perp h}$ ${\displaystyle h}$ ${\displaystyle g}$

then one has with the described elliptical polarity projective a "metric" in introduced with this projective plane to a elliptic plane , more precisely to the (up to isomorphism unique) elliptic plane over the field of real numbers. Each elliptical polarity of the real projective plane can namely be brought to the shape of this elliptical polarity by a suitable choice of the coordinate system . ${\displaystyle \mathbb {P} ^{2}(\mathbb {R} )}$

Projective polarity in projective spaces of any finite dimension

In an at least two-dimensional, pappusian projective space over a body , one has a certain one-to-one assignment between the points and hyperplanes of the space through a fixed projective polarity. This is particularly uniform in the elliptical case: The fact that there are no self-conjugate points means geometrically that no point lies on the hyperplane that is polar to it. ${\displaystyle \mathbb {P} ^{n}(K)}$

Polarities over finite spaces

By a drawer argument , which could also be refined to a counting of the self-conjugate elements in a finite polarity can be demonstrated: Exist on , a projective elliptical polarity and is the characteristic of not , then it must be infinite. ${\displaystyle \mathbb {P} ^{n}(K)}$ ${\displaystyle n\geq 2}$ ${\displaystyle K}$ ${\displaystyle 2}$ ${\displaystyle K}$

Equivalent: If K is finite with elements and , and is a regular matrix, then has the equation for self-conjugate points ${\displaystyle q=\#K}$ ${\displaystyle \operatorname {char} (K)\neq 2}$ ${\displaystyle n\geq 2}$ ${\displaystyle A\in \mathrm {GL} _{n+1}(K)}$

${\displaystyle x^{T}\cdot A\cdot x=0}$

a nontrivial solution . ${\displaystyle x\in K^{n+1}\setminus \{0\}}$

It suffices to consider the case : Under the given conditions , one can bring the matrix to the diagonal form with the methods presented in the article projective quadric , in particular by adding a square ; geometrically speaking, one chooses an orthogonal base of . The equation to be solved is then equivalent ${\displaystyle n=2}$ ${\displaystyle (\operatorname {char} (K)\neq 2)}$ ${\displaystyle A=\operatorname {diag} (a_{0},a_{1},a_{2}),a_{i}\in K^{*}=K\setminus \{0\}}$ ${\displaystyle K^{3}}$ ${\displaystyle a_{0}\cdot x_{0}^{2}+a_{1}\cdot x_{1}^{2}+a_{2}\cdot x_{2}^{2}=0}$

${\displaystyle \mathrm {(+)} \quad -a_{2}^{-1}a_{0}\cdot x_{0}^{2}-a_{2}^{-1}a_{1}\cdot x_{1}^{2}=x_{2}^{2}}$.

If one sets and considers all the elements that result on the left side of the equation, if body elements are substituted for all q , then these are different numbers, because the same value results for exactly two different numbers , the substitution yields an additional one. If 0 is below the values ​​shown in this way, then one sets and has a nontrivial solution, if 0 is not below, so if all the numbers represented by the term on the left side of the equation (+) are contained in, then there must also be a square number below it because it breaks down into exactly two square classes, the class of the square numbers, which is a real subgroup of , and its real subsidiary class , both classes each contain elements, i.e. fewer than would result from inserting into the left side of (+). So there has to be a nontrivial solution of the equation (+) for self-conjugated points. ${\displaystyle x_{0}:=1}$ ${\displaystyle x_{1}}$ ${\displaystyle {\frac {q-1}{2}}+1}$ ${\displaystyle x_{1},x_{1}'\in K^{*}}$ ${\displaystyle x_{1}=0}$ ${\displaystyle x_{2}=0}$ ${\displaystyle K^{*}}$ ${\displaystyle K^{*}}$ ${\displaystyle Q_{1}=(K^{*})^{2}}$ ${\displaystyle K^{*}}$ ${\displaystyle K^{*}\setminus Q_{1}}$ ${\displaystyle {\frac {q-1}{2}}}$

→ The exact numbers of self-conjugate points for polarities over finite spaces result in the most important cases from the theorems about quadratic sets .

Polar involution as polarity on a straight line

Leading example

Let it be - the following considerations apply to arbitrary bodies . We consider the "geometry" in the , which consists only of the straight lines through the origin , ie the one-dimensional subspaces. Each subspace is identified by a “direction” . It is . On the other hand, the homogeneous equation applies precisely to the points of a straight line . The coefficient vector is the normal vector of the straight line. Since both the directional as well as the normal vectors are "homogeneous" (only determined except for a multiplication by ), the geometry under consideration is a one-dimensional projective geometry and the assignment with is a projective, involutorial correlation of these projective straight lines, i.e. a one-dimensional projective polarity . If one describes the affine plane over with "orthogonality" as the actual plane within the projective plane over , then one has a projective invariant through this one-dimensional projective polarity on the distant straight line , i.e. the straight line with the coordinates , which is the orthogonality that is usual in the case described projectively describes: The projective geometry itself assigns a far point as a direction to each set of parallels , the polar involution assigns the “polar” direction to each direction, which in turn is the set perpendicular to the set of parallels from which one started. ${\displaystyle K=\mathbb {R} }$ ${\displaystyle \operatorname {char} (K)\neq 2}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle g_{\vec {v}}}$ ${\displaystyle {\vec {v}}=(v_{1},v_{2})^{T}\in \mathbb {R} ^{2}\setminus \{(0,0)\}}$ ${\displaystyle {\vec {x}}\in g\Leftrightarrow \exists s\in K,{\vec {x}}=s\cdot {\vec {v}}}$ ${\displaystyle (x_{1},x_{2})^{T}}$ ${\displaystyle g_{\vec {v}}}$ ${\displaystyle -v_{2}\cdot x_{1}+v_{1}\cdot x_{2}=0}$ ${\displaystyle (-v_{2},v_{1})}$ ${\displaystyle s\in \mathbb {R} ^{*}}$ ${\displaystyle {\vec {v}}\mapsto {\vec {n}}=A\cdot {\vec {v}}}$ ${\displaystyle A={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle x_{0}=0}$ ${\displaystyle [1,0,0]}$

In general, a polarity on a projective straight line that is part of a (at least two-dimensional) projective space is called polar involution . Since the set of points is dual to itself in a projective straight line, every correlation of the straight lines is also a collineation, every projective correlation a projectivity and usually only this case of a projective correlation on a straight line in a larger projective space is geometrically interesting .

Special polar involutions

• A polar involution is called projective if it is projective as collineation, i.e. a (one-dimensional) projectivity.
• A polar involution is called elliptical if it has no fixed points. This definition transmits the corresponding property of two-dimensional polarity, with the tightening of the fact that here incidence for points means equality.
• A polar involution is called hyperbolic if it has at least one fixed point.

The one-dimensional projective group operates sharply threefold transitive on the straight line , therefore a non-identical projective collineation can only have no, one or exactly two fixed points here. This shows an analogy to the two-dimensional case: The sets of fixed elements that can occur in a hyperbolic, projective polar involution consist of a "(double-counting) point" or a pair of points. These are exactly the "conic sections" that can appear in one-dimensional space next to the empty set and the whole straight line. ${\displaystyle \operatorname {PGL} (2,K),\operatorname {char} (K)\neq 2}$ ${\displaystyle \mathbb {P} ^{1}(K)}$

In the case of a finite line, the total number of points on the line is even because of the general assumption that the order of the line is odd and the case of exactly one fixed point for an involution is excluded. ${\displaystyle q+1}$ ${\displaystyle \operatorname {char} (K)\neq 2}$ ${\displaystyle q}$

A hyperbolic, projective polar involution is generally not uniquely determined by the set of its fixed points , unlike in the two-dimensional case a hyperbolic, projective polarity by the set of its self-conjugate points.

Generalizations

Desargue spaces of arbitrary, finite dimensions

Straight lines, points

On a projective straight line, the set of points is dual to itself and the term correlation coincides with the term collineation. Every bijection of the set of points (i.e. the points on the single straight line) is a correlation. Only the investigation of the involutive, projective collineations is of interest here. Projective “spaces” of the dimensions (points) and (empty set) obviously do not provide anything interesting. ${\displaystyle 0}$ ${\displaystyle -1}$

At least three-dimensional spaces

Every at least three-dimensional projective geometry can be represented as a -dimensional space over a sloping body . Here, the term correlation can be applied almost without restrictions if is isomorphic to its counter-ring : is the incidence structure with the basic structures point, straight line, ..., hyperplane isomorphic to the dual structure (incidence is reversed if necessary). Every bijective mapping that assigns a hyperplane to every point, a -dimensional subspace etc. to every straight line , is a correlation. As in the case above: ${\displaystyle n}$ ${\displaystyle \mathbb {P} ^{n}(K),n\geq 3}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle \mathbb {P} ^{n}(K),n\geq 3}$ ${\displaystyle n-2}$

• The complete correlation is determined by the images of the points. If a coordinate system is fixed, then every semi-linear mapping ( regular matrix, inclined body automorphism of K ) of the point coordinate vectors on hyperplane coordinate vectors determines the correlation uniquely, every correlation can be represented in this way. ${\displaystyle A\circ \alpha }$ ${\displaystyle A}$ ${\displaystyle (n+1)\times (n+1)}$ ${\displaystyle \alpha }$
• A correlation is projective if the point image of a coordinate system (and then in each coordinate system) with respect to linear so the Körperautomorphismus is identical. ${\displaystyle \alpha }$
• Such a correlation is involutive under the same conditions as in the two-dimensional case.
• For the composition of two correlations and the square of a correlation, the same relationships to collineations or to identity apply as given above for the two-dimensional case.
• The self-conjugate points of a projective involutive correlation form a (possibly empty) hypersurface of the second order and the self-conjugate hyperplanes are precisely the tangential hyperplanes of this hypersurface, if is commutative and its characteristic is not . Without these requirements, this does not have to apply! Therefore, when speaking of polarities, one usually assumes a Pappusian geometry that satisfies the Fano axiom. ${\displaystyle K}$ ${\displaystyle 2}$

Non-Desargue plains

For any projective level , the dual level is always a projective level again. In general, however, the plane is not isomorphic to its dual plane. Only if is does a correlation exist at all, then it is also called that. A correlation always exists under the following condition: ${\displaystyle E=({\mathfrak {P}},{\mathfrak {G}},I)}$ ${\displaystyle E^{D}=({\mathfrak {G}},{\mathfrak {P}},I^{-1})}$ ${\displaystyle E\cong E^{D}}$

E is a Desargue's plane over an oblique body that is isomorphic to its counter ring.

Then what was said in the previous section applies to the representation of the correlation as a semilinear point map.

literature

Projective geometry (in the sense of the usual linear algebra)

• Günther Eisenreich: Linear Algebra and Analytical Geometry . 3rd, exp. and corrected edition. Akademie-Verlag, Berlin 1991, ISBN 3-05-501301-8 , pp. 286-290 .
• Benno Klotzek: Analytical Geometry and Linear Algebra . Harri Deutsch, Thun / Frankfurt am Main 1997, ISBN 3-8171-1532-6 , p. 218 f .
• Uwe Storch, Hartmut Wiebe: Textbook of Mathematics, Volume II: Linear Algebra . In: Textbook of mathematics for mathematicians, computer scientists and physicists: in 4 volumes . BI-Wissenschafts-Verlag, Mannheim / Leipzig / Vienna / Zurich 1990, ISBN 3-411-14101-8 .

Application in absolute geometry

• Friedrich Bachmann : Structure of geometry from the concept of reflection . 2nd supplemented edition. Springer, Berlin / Heidelberg / New York 1973, ISBN 3-540-06136-3 .

References and comments

1. ↑ In this context an involution is a mapping with , i.e. never the identity, according to construction a correlation cannot be identical anyway, since it maps points on straight lines. Bachmann (1973) ${\displaystyle \alpha }$ ${\displaystyle \alpha ^{2}=1,\alpha \neq 1}$
2. ↑ If the incidence relation is not viewed as symmetrical, as is sometimes done in the literature, this relation is reversed. ${\displaystyle I}$
3. ↑ ^{a b c d} Bachmann (1973), p. 88f.
4. ↑ Often it is still required that the characteristic of this body is not 2, geometrically that means: the projective plane fulfills the Fano axiom .
5. ↑ So for the coefficient triple of a homogeneous plane equation im ${\displaystyle K^{3}}$
6. ↑ there §8.2: Metric vector spaces and orthogonal groups.
7. ↑ A basis of 3 vectors that are orthogonal in pairs with respect to the given non-degenerate bilinear form.
8. ↑ ^{a b} If K is commutative, i.e. a field, then this condition is trivially fulfilled.
9. ↑ In the case of a symmetrical incidence relation.