As an affine translation plane or short translational level an affine plane is then called in synthetic geometry, when their translation group sharply simply transitively on her and she therefore largely by this group of their translations ( parallel shifts ) can be described by any point of the plane assigned to a translational will. The endomorphism ring of the translation group, which is always commutative at a translational level, contains a skew field, the division ring of endomorphisms putts. The group of translations is a module over this skew field.
Pure geometric an affine plane if and only a translational level when in her small affine set of Desargues (see the illustration at the end of the introduction) is universal, ie, a lock set, which is used in synthetic geometry as an axiom.
In addition, the term translational projective plane is used in the synthetic geometry rare. These special projective planes are closely related to the affine translation planes. This relationship is explained in the section in this article projective translation plane. The terms affine translation plane or projective translation plane are generalizations of the terms desarguesian affine or projective plane desarguesian.
The investigation of translations and their putts endomorphisms is next to the description by a Koordinatenternärkörper a common method to algebraisieren nichtdesarguesche levels. For desarguesche and certainly for pappusssche levels of the skew field of putts endomorphisms coincides with the coordinate Chief body in translation planes it is contained in the coordinate quasi body as the core.
→ The algebraization an affine plane using coordinates on a straight line of the plane algebraic links of these coordinates as well as the terms and quasi Ternärkörper body that are used in this article are shown in more detail in the respective main articles.
- 2.1 core of the coordinate quasi body
- 2.2 commensurable points, stretch factor, part ratio
- 2.3 Strahlensatz and dilations
Definitions and characteristics
Translations in affine incidence levels
A bijective self-map of an affine plane is called translation if and only if
- The image of each line is a straight line, that is, is a collineation
- Is for every line of the plane
- Is no fixed point or the identity mapping of the plane.
Each translation is uniquely determined by a point pixel pair.
For non-identical translations is the connecting line of a track and straight. Just the parallels of these lines form the set of all traces of. The parallel class of the tracks is called the direction of translation and one calls a shift in direction.
Translation group and directional stability endomorphisms
The set of translations of an affine incidence plane forms with respect to the composition of a group. This group is commutative if there is ( non-identical ), translations in the plane (at least) two different directions. A Gruppenendomorphismus track called faithful if for each non-identical translation is the traces of the footsteps of match or the 0 - endomorphism. Equivalently: changes its direction at no translation.
If the translation group commutative and non-trivial, then the amount of putts endomorphisms is the links
Into a ring with zeros element and one element, a subring of Endomorhismenringes. The order in which the homomorphisms are used in the definition of the multiplication on translations, determines whether the translation group to a left or right module over. With the chosen definition and here for the " scalar " is a links module.
Affine translation plane
An affine incidence plane is called affine translation plane if one of the following equivalent conditions is true:
- Thus there are in an affine translation plane if one selects a point as the origin determines a natural bijection between the points of the plane and the translation group. A translation plane can be identified with its translation group.
- On the other hand, can be identified with an equivalence class of shift same parent pairs of points ( " arrows " ) any translation, there are two arrows are equivalent, if and applies the same translation. We call these equivalence classes of arrows also " vectors".
For uniquely determined translation, which maps a point on a point is written for short. This notation refers to both the equivalence class of the same shift "arrows".
Since a translational level, each track is endomorphism faithful even a Gruppenautomorphismus, the ring is here even a skew field. The group of translations ( " vectors" in the sense described above ) form a links module. If we let Skalarkörper as a vector space is also a skew field to how the occasionally happens in the literature, the group of translations thus forms a real vector space links.
As a result, the order of every non-identical translation is determined by the characteristics of: Is this characteristic is a prime number, then all non-identical translations have this order, it is 0, then all non-identical translations have infinite order. Just when the characteristic is different from 2, the translational level satisfies the ( affine ) Fano axiom.
Coordinate quasi body and directional stability endomorphisms
Core of the coordinate quasi body
A ( left ) quasi- body differs from a skew field in that no Rechtsdistributivgesetz and no associative law of multiplication is required. One defines a quasi body
As its core, then this core is a skew field and this is isomorphic to the division ring of endomorphisms of putts for translation layer above. Isomorphism on this will also be the coordinates of the body to a quasi -left module in the direction of the first coordinate axis in the translation group is isomorphic to the sub- module of the translations.
If one has the multiplication as defined and the " scalar multiplication " of right and then you have to isomorphism the multiplication not be reversed, since the elements of the core after construction operate distributive and associative and right- and then becomes a law module. But it is in the literature usually only right quasi body - in which the definition of the core are adjusted accordingly must - be provided with such a right module structure, as a casual gives a geometric interpretation of a group of geometric transformations in " equilateral " structure.
Commensurable points, stretch factor, part ratio
Three collinear points the translational level are called commensurable if a track faithful endomorphism exists that transforms the translation that moves on in the translation that moves on. Vectorially written. In this case, it is called the stretch factor for the Punktetripel. From the stretching factor can reversibly clearly a part ratio can be obtained ( for three different collinear and commensurate points):
The fraction notation here is a problem, because all occurring elements of mutually commute.
Strahlensatz and dilations
Are five points of an affine translation plane with the properties (see the figure on the right ):
- Are not collinear,
- Are collinear and commensurable
- Are collinear,
This first set of beams for translation planes justifies it, to denote the truer endomorphisms as " Centric stretching " the translational level and motivates the term " stretch factor ": If you choose an origin and assigns, as explained above each point of the translation, then operated on any " stretch factor " on the points of the plane as a collineation and even as dilatation. This dilation of the origin fixed point and all lines is through the origin are Fixgeraden. Conversely operates each dilation, which has exactly the origin as a fixed point, by conjugation on the translations and this operation is a faithful trace endomorphism of the translation group. Therefore, the subgroup of the generalized dilations with center and the subgroup of dilations described here with a stretch factor of identical sub- groups of the affinity group at a translational level.
It further follows: in the configuration shown above, and triangles, and each co-linear, and is then either both of the collinear triples commensurable or both incommensurable. If they are incommensurable, then there is no dilation, which has a fixed point and on, on maps. Nor can no affinity exist with this property!
Since the stretching factor acts as a figure on the parallel shifts, results under the conditions of the first beam set and the additional condition a second set of beams corresponding statement: - this formula remains true also in the trivial case. So the first two rays sentences shall apply mutatis mutandis in any desargueschen level, in which case can be omitted, the condition of commensurability, in general, while the third set of beams, which is called in synthetic geometry, three-jet rate, can generally be proved only for pappussche levels.
(→ Compare the main article Centric stretching and Strahlensatz )
A translation plane with associated skew field of putts Endomomorphismen of exactly then a desarguesche level when a following equivalent conditions is true:
Since the coordinates are unambiguously determined by the affine plane, up to isomorphism, can the statements about these areas " A coordinate quasi body ... " here equivalent with " Each coordinate quasi body ... " are formulated.
On the other hand, each "real ", ie nichtdesarguesche translational level contains a desarguesche level as a proper subset: If we choose a coordinate system and consider only points with coordinates that gets to and are commensurable, and only those lines whose coefficients have this property, then one a to desargueschen level isomorphic affine incidence structure.
If you can in a translational level, define a orthogonality and the characteristic of the skew field is not 2, that is, the ( affine ) Fano axiom is true, then the generality of the orthocenter set and the means solders rate equivalent and - if these are universal - is in the plane the set of Pappus generally valid and the coordinates quasi body even a body. (See Präeuklidische level).
General performs a translational level the set of Pappus iff
- If it is desarguessch and the multiplication in the division ring of endomorphisms of the truer translation group is commutative, so is a body or equivalent
- If its coordinate quasi body is a body.
Is the order of the translational level finite, then the skew field is always a body. Then the translational level is exactly then pappussch if it is desarguessch.
A affine or projective plane is called finite if it is their order and thus also the number of points in the plane. The order is in an affinity level the number of points on a straight line in a projective plane, the order of the affinity level of the slots formed by the projective plane. From the fact that the coordinate quasi body of an affine translation plane is a left vector space over the skew field of putts endomorphisms, arise together with the set of Wedderburn, which states that a finite division ring is always commutative, that is a finite field, implications for the finite Translations - and Moufangebenen:
- The skew field of a finite translation plane is a finite field, so it has elements of a prime and.
- The coordinate quasi body is a vector space over endlichdimensionarer and therefore has elements. Thus, the order of the level of translation, these prime power, is, then the translation level is pappussche layer over the body.
- There are a number of finite affine translation planes that are not desarguesch, for example, 4 different (non- isomorphic ) of order 9, (→ See the examples in the article Ternärkörper. )
- The formal analogue of affine translation planes under the projective planes are the Moufangebenen, in which small set of projective Desargues is generally valid. Ruth Moufang has shown that real, that is nichtdesarguesche Moufangebenen are always infinite. It follows that in a finite affine translation plane projective extension iff is a Moufangebene when both levels to levels desarguesch and therefore equivalent over a finite field.
→ For more general statements about the possible orders of finite levels are found in Articles Projective plane and projective geometry.
Projective translation plane
A projective plane is called translation plane with respect to one of their lines when they met in respect of this line as axis of the small set of projective Desargues. An equivalent description of such a projective translation plane: It belongs to one of the classes IVa, V or VII in the classification of projective planes by Hanfried Lenz.
The projective completion of an affine translation plane is always a projective translation plane. On the other hand, a projective plane of translation is slit along a projective line, creates an affine plane in which this straight line is the straight line distance. The affine plane so produced is then precisely an affine translation plane, when the projective plane satisfies the small set of projective Desargues in relation to the axis. Equivalently: The straight line an axis in the Lenz - figure of the projective plane has to be.
Examples and counter-examples
- Each desarguesche plane is a translation plane, ie in particular the affine plane over a skew field. Just Right skew field of putts endomorphisms ( up to isomorphism ) the same as the coordinates chief body.
- The real octonions form a quasi body which is not a division ring: Although both distributive laws are valid, but the multiplication is not associative. Thus the affine translation plane is a nichtdesarguesche translational level.
- The real Moulton plane is an affine plane, which is not a translation plane: If the ( "normal" and Moulton -level ) Just to have on the some Moultongeraden their "kink ", then the translation group consists exactly of the "normal" shifts the real plane in the direction of the straight line, it is thus for commutative group isomorphic. Each of Gruppenautomorphismus is directionally stable, but since not simply the translation group operates transitively on the Moulton plane, it is of little to describe this geometry.
- In contrast, the finite Moultonebenen are always affine translation planes. There exist infinitely many nichtdesarguesche finite translation planes of this type, see the section body finite quasi - Moulton planes in the article Quasi body.
→ The article Ternärkörper contains further examples of affine translation planes, especially for verbose examples of finite translation planes nichtdesarguesche (→ subsection examples of order 9 ).