Central collineation

Central collineation: For each point are collinear ${\displaystyle P}$ ${\displaystyle Z,P,\pi (P)}$

In geometry, a central collineation (short: perspectivity ) is a collineation that has a center and a fixed point hyperplane. The center is a point of the projective space with the property that every straight line through this point is a fixed straight line of perspectivity.

The concept of the perspective position of one-dimensional structures relative to one another is older than the term perspectivity in the sense of a bijective self- image of an at least two-dimensional projective space , compare the figure below on the right. More modern one speaks here of a central perspective assignment or dual of an axial perspective assignment . These mappings, which are already important for Pascal's theorem , for example , can generally only be extended to a perspective of the total space if this space is papposian and fulfills the Fano axiom . To put it algebraically: if this more extensive space is one over a commutative body with a characteristic . Since up to the second half of the nineteenth century (implicitly, because an axiomatic of real numbers was only developed at that time) real , at most three- dimensional projective geometry (as the geometry of the position ) was practiced, perspective assignment and perspectivity are not sharp in the older literature differentiated and often referred to identically. ${\displaystyle \mathbb {P} ^{n}(K),n\geq 2}$ ${\displaystyle K}$ ${\displaystyle \operatorname {char} (K)\neq 2}$

The initial configuration of Desargues' theorem was a typical case of "perspectivity" in the geometry of the position : the colored triangles and are in perspective position to one another when viewed from the point . There is therefore a ( central ) perspective assignment that assigns the unlined points to the deleted ones. If Desargue's theorem applies, then the triangles (as three sides ) are also ( axially ) perspective to one another, seen from the axis . Then there is exactly one perspective (in the sense of the more recent projective geometry) of the entire plane, which maps the points assigned to perspective onto one another. ${\displaystyle \Delta (ABC)}$ ${\displaystyle \Delta (A'B'C')}$ ${\displaystyle Z}$ ${\displaystyle a}$

In synthetic geometry , the term “plane perspectivity” is defined for projective planes independently of the term “projectivity”: There a perspectivity is a (projective) collineation with a center and a straight line (axis). For projective planes, the term is synonymous with the term central-axial collineation .

The definition of synthetic geometry for Desargue's projective planes - these are precisely the planes that can be understood as two-dimensional projective spaces in the sense of analytical geometry - is equivalent to the definition as projectivities with center and axis. It allows the concept of “projectivity” to be generalized to non-Desarguean levels.

→ The plane perspectives have an important application in the classification of projective planes .

Definitions

Perspectivity in a Desargue space

Let be an oblique body , and the - dimensional projective space above . Then a projectivity is called projective perspectivity if one of the following equivalent conditions is met: ${\displaystyle K}$ ${\displaystyle n\in \mathbb {N} ,\;n\geq 2}$ ${\displaystyle \mathbb {P} ^{n}(K)}$ ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle \pi :\mathbb {P} ^{n}(K)\rightarrow \mathbb {P} ^{n}(K)}$

1. There is a point such that every line is through a fixed line of , so it holds. ${\displaystyle Z\in \mathbb {P} ^{n}(K)}$ ${\displaystyle g}$ ${\displaystyle Z}$ ${\displaystyle \pi }$ ${\displaystyle \pi (g)=g}$
2. There is a fixed point hyperplane, the axis of , that is, a -dimensional projective subspace , so that the restriction is the identical mapping of . ${\displaystyle H}$ ${\displaystyle \pi }$ ${\displaystyle n-1}$ ${\displaystyle H<\mathbb {P} ^{n}(K)}$ ${\displaystyle \left.\pi \right|_{H}}$ ${\displaystyle H}$

Perspectivity on a projective level

Be a projective plane. Then a collineation is called projective perspectivity if one of the following equivalent conditions is met: ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle \kappa :{\mathfrak {P}}\rightarrow {\mathfrak {P}}}$

1. There is a point such that every line is through a fixed line of , so it holds. ${\displaystyle Z\in {\mathfrak {P}}}$ ${\displaystyle g}$ ${\displaystyle Z}$ ${\displaystyle \kappa }$ ${\displaystyle \kappa (g)=g}$
2. There is a fixed point line of , that is, a straight line of the plane , so that the restriction is the identical mapping of . ${\displaystyle h}$ ${\displaystyle \kappa }$ ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle \left.\kappa \right|_{h}}$ ${\displaystyle h}$

Connection of definitions

A desargue projective plane is always isomorphic to a two-dimensional projective space over a sloping body clearly determined by the plane up to isomorphism . A collineation of such a space is already true to the double ratio if it does not change the double ratios for the points on a projective straight line (→ compare the article collineation ). Since a perspectivity is a collineation with a fixed point straight line, it is initially true to the double ratio for this straight line and thus in general and is therefore a projectivity. ${\displaystyle \mathbb {P} ^{2}(K)}$ ${\displaystyle K}$

Projectivity on a non-Desarguese level

In synthetic geometry one defines: Let be any projective plane. Then an image is called projectivity if it can be represented as a composition from a finite number of perspectives. ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle \kappa :{\mathfrak {P}}\rightarrow {\mathfrak {P}}}$

As a composition of special collineations, such an image is of course also a collineation, in particular bijective . At a Desarguese level, like the perspectives, it is true to the double ratio. One can show that a collineation true to the double ratio can always be represented by a chain of perspectives and that for this composition representation never more than three perspectives have to be connected. Thus the definitions of linear algebra and synthetic geometry are equivalent for Desarguean planes. ${\displaystyle \kappa }$

Note, however, that the concatenation of two perspectives is generally not a perspective.

Level perspectives

• Every collineation of an affine plane can clearly be continued to a collineation in its projective conclusion. There the distance line is then a fixed line of the projective collineation. Conversely, a collineation in a projective plane corresponds to a collineation of the affine plane that is created by slitting the projective plane when slitting is made along a fixed line of the collineation.
• The generalized terms “ affinity ” and “projectivity” (see above) of synthetic geometry are compatible: A collineation of a projective plane with (at least) one fixed line is a projectivity if and only if its restriction to one (equivalent: to each) Its affine plane, which emerged by slitting along a fixed line, is an affinity, a collineation on an affine plane is an affinity if and only if its continuation on the projective end of the plane is a projectivity. However, there are also projectivities without a fixed line.
• A collineation of a projective plane is called axial collineation if a straight line exists that is a fixed point straight line of the collineation, that is, the restriction of the collineation in question to is the identical mapping of the straight line. In this case it is called the axis of axial collineation. ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle a}$
• A collineation of a projective plane is called central collineation , if a point exists so that every straight line is through a fixed line of the collineation. This automatically becomes a fixed point of the collineation and is referred to as the center of the collineation. ${\displaystyle Z}$ ${\displaystyle Z}$ ${\displaystyle Z}$

Properties and names

• The terms axial collineation and central collineation are dual to one another.
• A non-identical collineation has at most one center and at most one axis.
• A collineation is central if and only if it is axial.
  • A collineation that is central or axial (and thus both) is also referred to as central-axial collineation or plane perspectivity .
• For a non-identical perspective, the following applies:

1. The set of fixed points consists exactly of the set of points on the axis together with the center,
2. the set of fixed lines consists exactly of the axis together with all straight lines through the center,
3. it is uniquely determined by its axis, its center and a point, image point pair (neither on the axis nor the center).

• The set of central collineations with a fixed center forms a subgroup of the projective group,
• the set of axial collineations with a fixed axis forms a subgroup of the projective group and ${\displaystyle a}$
  • the set of central-axial collineations centered on the fixed axis forms a subgroup of the last-mentioned group. ${\displaystyle a}$

Image construction, existence and uniqueness

Image construction with a flat perspective from its axis and the center (blue) with the help of a given point-image point pair . ${\displaystyle a}$ ${\displaystyle Z}$ ${\displaystyle (A_{1},A_{2})}$

The axis and the center are given to us from a plane perspective . Compare the figure on the right: the axis and center are blue. In addition , the image point of a point that is not on the axis and does not coincide with the center is known. This must lie on the connecting line, as it is a fixed line. ${\displaystyle a}$ ${\displaystyle Z}$ ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$ ${\displaystyle A_{1}+Z}$

1. We draw the connecting line to another point , it intersects the axis at a fixed point . ${\displaystyle B_{1}}$ ${\displaystyle B_{1}+A_{1}}$ ${\displaystyle a}$ ${\displaystyle F}$
2. The image of is the straight line . ${\displaystyle B_{1}+A_{1}=A_{1}+F}$ ${\displaystyle A_{2}+F}$
3. The connecting line is a fixed line. ${\displaystyle B_{1}+Z}$
4. The picture from below the perspective is . This is the intersection of the fixed line from 3 and the straight line from 2. ${\displaystyle B_{1}}$ ${\displaystyle B_{2}}$ ${\displaystyle B_{1}+Z}$ ${\displaystyle A_{2}+F}$

Special cases:

• If the point lies on the fixed line , then the image of an auxiliary point outside of the fixed line and the axis must first be constructed according to the specified construction text . This pair of auxiliary points can then be used for construction. ${\displaystyle B_{1}}$ ${\displaystyle A_{1}+A_{2}=A_{1}+Z}$ ${\displaystyle H_{2}}$ ${\displaystyle H_{1}}$
• The construction description can also be used when the center is on the axis . ${\displaystyle Z}$ ${\displaystyle a}$

Uniqueness and Existence:
The specifications are as indicated above: When there is a unique fixed point collineation with straight and fixed point that the point on maps? We assume , but not initially . ${\displaystyle a}$ ${\displaystyle Z}$ ${\displaystyle A_{1}}$ ${\displaystyle A_{2}\in (A_{1}+Z)}$ ${\displaystyle A_{1},A_{2}\not \in a;\;Z\not \in \{A_{1},A_{2}\}}$ ${\displaystyle A_{1}\neq A_{2}}$

• If such a collineation exists, it is axial because it has a fixed point line, so it is a perspectivity. It must therefore also have a center and this can only be (or the collineation is the identical mapping), since it is a fixed line. The design text makes it clear: There can be no further collineation that meets the requirements! ${\displaystyle Z}$ ${\displaystyle A_{1}+Z}$
• In particular, the collineation exists for and is then the identical mapping. ${\displaystyle A_{1}=A_{2}}$
• It is sufficient for the existence in the case that the couple is contained in the Lenz-Barlotti figure of the plane. ${\displaystyle A_{1}\neq A_{2}}$ ${\displaystyle (Z,a)}$
• Then there is a collineation for any pair and any pair of different points with if the projective plane is a Moufang plane, i.e. belongs to Lenz class VII. ${\displaystyle (Z,a);Z\in a}$ ${\displaystyle (A_{1},A_{2})}$ ${\displaystyle Z\in (A_{1}+A_{2}),\,Z\not \in \{A_{1},A_{2}\},\,A_{1},A_{2}\not \in a}$
• Then there is a collineation for every arbitrary pair and every pair of different points with if the projective plane is desarguessic, i.e. belongs to the Lenz-Barlotti class VII.2. ${\displaystyle (Z,a)}$ ${\displaystyle (A_{1},A_{2})}$ ${\displaystyle Z\in (A_{1}+A_{2}),\,Z\not \in \{A_{1},A_{2}\},\,A_{1},A_{2}\not \in a}$
• A special case is the Fano plane , the minimal model of a projective plane that has exactly three points on each straight line. It is a Desarguessian and even Pappusian plane and the aforementioned condition is empty here: Every collineation with an axis and a fixed point outside the axis is the identical mapping, since no different image point exists for a point . ${\displaystyle a}$ ${\displaystyle Z\not \in a}$ ${\displaystyle A_{1}\not \in \{Z\}\cup a}$ ${\displaystyle A_{1}}$ ${\displaystyle (Z+A_{1})\setminus (\{Z\}\cup a)=\{A_{1}\}}$

Ways of speaking

If you hold a certain straight line as a distance line in a projective plane, which is implicitly already done by selecting a projective coordinate system, then one usually calls a plane perspective

• axial collineation if its center is on the distant line but its axis is not the distant line,
• central collineation , if its axis is the distant line but its center is not a distant point,
• (projective) translation if its axis is the distant line and its center is a distant point.

The motivation for this language regulation becomes clear in the related examples below. For non-identical perspectives, in which neither the center nor the axis are improper, there is no language regulation in the situation described; the distance line cannot be a fixed line, so they do not operate on the affine section of the projective plane.

Examples

When specifying the axis and the center, it is always assumed in the following examples that the observed collineation is not the identity of the plane.

• In any affine incidence plane, the projective continuation of a translation is a perspectivity (a “projective translation”): the axis is the distant line and the center is the distant point of the straight line of the displacement.
• In any affine incidence plane, the projective continuation of a dilatation is a perspective: the axis is the distant line, the center is the affine fixed point, if such a point exists, otherwise the dilatation is a translation.
• In a Desarguessian plane, the projective continuation of a centric stretching is a perspectivity (a “central collineation”). The center is the midpoint of the extension, the axis is again the long-distance line. Since the concept of centric stretching can be generalized to affine translation planes, this also applies to these planes.
• In a Desargue plane, the projective continuation of a shear is central-axial (an "axial collineation"): the axis is the affine fixed point line together with its far point, the center is this far point.
• In a Desargue plane that satisfies Fano's axiom , the continuation of an oblique mirroring is a central-axial collineation (an "axial collineation"): the axis is the mirror axis together with its far point, the center is the direction in which the mirror is mirrored.
• In contrast, the projective continuation of a rotation of the Euclidean plane is only a perspective if the rotation takes place by a multiple of 180 °, i.e. the rotation is a point reflection or the identity. Since every rotation of the Euclidean plane is a composition of two perpendicular axis reflections, i.e. special oblique reflections (see congruence mapping ), projective continuations of rotations provide examples of projectivities that are not perspectives.

literature

• Albrecht Beutelspacher , Ute Rosenbaum: Projective geometry . From the basics to the applications (=  Vieweg Studium: advanced course in mathematics ). 2nd, revised and expanded edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X .
• Arrigo Bonisoli: On collineation groups of finite planes . Socrates Intensive Program, Dipartimento di Matematic a Università della Basilicata, Potenza, Italy ( full text [PDF; accessed on January 8, 2012] As the title suggests: group structure of the collineation group).
• Harold Scott MacDonald Coxeter : Real Projective Geometry of the Plane . After the 2nd engl. Edition translated by W. Burau. Ed .: Wilhelm Blaschke (=  Mathematical Individual Writings . Volume 3 ). 1st German edition. R. Oldenbourg, Munich 1955 (The textbook brings the classic, real "geometry of the situation" of the 19th century in a relatively modern formulation. Above all, the author or translator explains in detail who certain ideas and ways of speaking are based on and the translator explains differences between German and American usage).
• Erich Hartmann: Projective Geometry . Technische Universität, Darmstadt 2006 ( full text [PDF; accessed on January 8, 2012] short script).
• Lars Kadison, Matthias T. Kromann: Projective Geometry and Modern Algebra . Birkhäuser, Boston / Basel / Berlin 1996, ISBN 3-7643-3900-4 (Consequences of the Fano axiom and the theorems of Desargues and Pappos for the transitivity properties of projective groups).
• Günter Pickert : Projective levels . 2nd Edition. Springer, Berlin / Heidelberg / New York 1975, ISBN 3-540-07280-2 (application of perspectives especially in non-Desarguese levels).
• Hans Walser: Projective images, graphic approach . Eidgenössische Technische Hochschule, Zurich ( full text [PDF; accessed on January 8, 2012] lecture notes; numerous illustrations, most of which, however, belong to exercises and must therefore be completed (following instructions in the text)).

References and comments

1. ↑ This can be, for example, a series of points, i.e. the set of points on a fixed straight line, a plane bundle of lines, i.e. the set of straight lines through a fixed point, or a non-degenerate conic section .
2. ↑ ^{a b} Beutelspacher & Rosenbaum (2004)
3. ↑ ^{a b c} Bonisoli, Prop. 2.3
4. ↑ Hartmann 2.4
5. ↑ Walser chap. 4th
6. ↑ Bonisoli, Prop. 2.1 and 2.2
7. ^ Hauke ​​Klein: Collineations. Geometry. University of Kiel, accessed on January 8, 2012 (English, definition of central-axial collineations and description of some important groups of such collineations). 
8. ↑ According to Pickert (1975), the plus sign between points is understood to mean that the sum of the points represents the straight line connecting them. In Desargue's case, the standard model is actually the sum of two subspaces of a (left) vector space.