Affine translation plane

In synthetic geometry, an affine plane of translation or, for short , plane of translation , is an affine plane if its translation group operates on it simply and transitively and it can therefore be largely described by this group of its translations ( parallel displacements ) by assigning a translation to each point on the plane becomes. The endomorphism ring of the translation group, which is always commutative in a translation plane , contains an oblique body , the oblique body of the track-true endomorphisms . The group of translations is a module above this oblique body.

From a purely geometric point of view, an affine plane is a translation plane if and only if Desargues' small affine theorem (compare the figure at the end of the introduction) is generally valid in it, i.e. a closure theorem that is used as an axiom in synthetic geometry.

In addition, the term projective translation plane is used less often in synthetic geometry . These special projective planes are closely related to the affine translation planes . This relationship is explained in this article in the section Projective Translation Plane. The terms affine translation level and  projective translation level are generalizations of the terms desargue affine and  desargue projective level .

The investigation of the translations and their true-to-track endomorphisms is a common method of algebraising non-Desargue planes in addition to the description by means of a coordinate ternary body. For desargue and even more so for pappus planes, the oblique body of the true-track endomorphisms coincides with the coordinate oblique body; in the case of translation planes, it is contained in the coordinate quasi-body as a core .

The algebraization of an affine plane with the help of coordinates on a straight line of the plane, algebraic connections of these coordinates as well as the terms ternary bodies and quasi-bodies , which are used in this article, are presented in more detail in the corresponding main articles.

Desargues' little affine theorem says: Are and triangles in which the "straight lines of association" are parallel: then it follows from the parallelism of two pairs of triangle sides (e.g. and ) that the third pair of sides is also parallel (in the example ) . ${\displaystyle A_{1}A_{2}A_{3}}$ ${\displaystyle B_{1}B_{2}B_{3}}$ ${\displaystyle A_{1}B_{1}\parallel A_{2}B_{2}\parallel A_{3}B_{3}}$ ${\displaystyle A_{1}A_{2}\parallel B_{1}B_{2}}$ ${\displaystyle A_{2}A_{3}\parallel B_{2}B_{3}}$ ${\displaystyle A_{3}A_{1}\parallel B_{3}B_{1}}$

Definitions and characteristics

Translations in affine incidence planes

A bijective self-mapping of an affine plane is called translation if it holds ${\displaystyle \tau :{\mathcal {A}}\rightarrow {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}}$

• the image of each straight line is a straight line, i.e. H. is a collineation ${\displaystyle \tau }$
• for each line is the level , ${\displaystyle g}$ ${\displaystyle \tau (g)\parallel g}$
• ${\displaystyle \tau }$is fixed points or identical mapping of the plane , . ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \;\operatorname {Id} _{\mathcal {A}}}$

Each translation is uniquely determined by a point-image point pair . ${\displaystyle \tau }$ ${\displaystyle (P,\tau (P))\in {\mathcal {A}}^{2}}$

For non-identical translations is connecting straight of and a trace line . Exactly the parallels of these straight lines form the set of all traces of . The parallel set of tracks is called the direction of translation and is then also called a shift in direction . ${\displaystyle P}$ ${\displaystyle \tau (P)}$ ${\displaystyle \tau }$ ${\displaystyle R}$ ${\displaystyle \tau }$ ${\displaystyle \tau }$ ${\displaystyle R}$

Translation group and endomorphisms that are true to track

The set of translations of an affine incidence level forms a group with regard to the composition . This group is commutative if there are (non-identical) translations of the plane in (at least) two different directions. A group endomorphism is called true to track if for each non- identical translation the traces of coincide with the traces of or if the endomorphism is 0 . Equivalent: does not change its direction with any translation. ${\displaystyle ({\mathcal {T}},\circ )}$ ${\displaystyle \alpha :{\mathcal {T}}\rightarrow {\mathcal {T}}}$ ${\displaystyle \tau \in {\mathcal {T}}\setminus \lbrace \operatorname {Id} _{\mathcal {A}}\rbrace }$ ${\displaystyle \tau }$ ${\displaystyle \alpha (\tau )}$ ${\displaystyle \alpha }$ ${\displaystyle 0_{S}:\tau \mapsto \operatorname {Id} _{\mathcal {A}}}$ ${\displaystyle \alpha }$

If the translation group is commutative and nontrivial, then the set of true-track endomorphisms is determined by the links ${\displaystyle S}$

${\displaystyle \alpha +\beta :\tau \mapsto \alpha (\tau )\circ \beta (\tau )\quad }$ and

${\displaystyle \alpha \cdot \beta :\tau \mapsto \alpha (\beta (\tau ))\quad }$

to a ring with zero element and one element , a sub-ring of the endomorphism ring. The order in which the homomorphisms in the definition of multiplication are applied to translations determines whether the translation group becomes a left or right module . With the definition chosen here and for the "scalar multiplication", it is a left module. ${\displaystyle 0_{S}:\tau \mapsto \operatorname {Id} _{\mathcal {A}}}$ ${\displaystyle 1_{S}=\operatorname {Id} _{\mathcal {T}}}$ ${\displaystyle S}$ ${\displaystyle \alpha \cdot \beta =\alpha \circ \beta }$ ${\displaystyle \alpha \cdot \tau =\alpha (\tau )}$ ${\displaystyle S}$

Affine translation plane

An affine incidence plane is called an affine translation plane if one of the following equivalent conditions applies:

1. The small affine set of Desargues applies . ${\displaystyle {\mathcal {A}}}$
2. The coordinate ternary body, which can be assigned by choosing any coordinate system, is a quasi-body . ${\displaystyle {\mathcal {A}}}$
3. There is always a translation with for two points . ${\displaystyle P,Q\in A}$ ${\displaystyle \tau \in {\mathcal {T}}}$ ${\displaystyle \tau (P)=Q}$
4. The translational group operates on sharply simply transitive . ${\displaystyle {\mathcal {A}}}$

• Thus, in an affine translation plane , if one firmly chooses a point as the origin, there is a natural bijection between the points of the plane and the translation group. A translation level can thus be identified with its translation group. ${\displaystyle {\mathcal {A}}}$ ${\displaystyle O\in {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}\ni P\leftrightarrow \tau _{P}\in {\mathcal {T}}:\tau _{P}(O)=P}$

• On the other hand, every translation can be identified with an equivalence class of ordered pairs of points (“arrows”) with the same displacement, two arrows being equivalent if and with the same translation . These equivalence classes of arrows are also called “vectors”. ${\displaystyle (P_{1},Q_{1}),(P_{2},Q_{2})\in {\mathcal {A}}^{2}}$ ${\displaystyle \tau (P_{1})=Q_{1}}$ ${\displaystyle \tau (P_{2})=Q_{2}}$ ${\displaystyle \tau \in {\mathcal {T}}}$

An abbreviation is used for the uniquely determined translation, which maps a point onto a point . This notation also denotes the equivalence class of the "arrows" which are to be displaced. ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle {\overrightarrow {PQ}}}$ ${\displaystyle (P,Q)}$

Since every true-track endomorphism in a translation plane is even a group auto morphism, the ring here is even an inclined body. The group of translations (“vectors” in the sense described above) form a link module. If a skewed body is also permitted as the scalar field of a vector space , as happens occasionally in the literature, the group of translations actually forms a -left vector space. ${\displaystyle \alpha \in S\setminus \lbrace 0\rbrace }$ ${\displaystyle (S,+,\cdot )}$ ${\displaystyle S}$ ${\displaystyle S}$

As a consequence, the order of every nonidentical translation is determined by the characteristic of : If this characteristic is a prime number , then all nonidentical translations have this order , if it is 0, then all nonidentical translations have infinite order. Exactly when the characteristic is different from 2, the translation plane fulfills the (affine) Fano axiom . ${\displaystyle \tau \in {\mathcal {T}}}$ ${\displaystyle S}$ ${\displaystyle p}$ ${\displaystyle p}$

Coordinate quasi-bodies and true-track endomorphisms

Core of the coordinate quasi-body

A (left) quasi-body differs from an inclined body in that no law of distribution of law and no law of associative multiplication is required. Defined for a quasi-body ${\displaystyle K}$

${\displaystyle \operatorname {Kern} (K)=\lbrace x\in K:\;\forall a,b\in K\left((a+b)x=ax+bx\land (ab)x=a(bx)\right)\rbrace }$

as its core , then this core forms a skew field and this is isomorphic to the skew of the truer endomorphism of the translational level over . Via this isomorphism, the coordinate quasi- body also becomes a left module, which is isomorphic to the sub-module of the translations in the direction of the first coordinate axis in the translation group . ${\displaystyle S}$ ${\displaystyle {\mathcal {T}}({\mathcal {A}})}$ ${\displaystyle {\mathcal {A}}=K^{2}}$ ${\displaystyle K}$ ${\displaystyle S}$ ${\displaystyle {\mathcal {T}}({\mathcal {A}})}$

If you have defined the multiplication in as and the “scalar multiplication” from the right as, then the multiplication does not have to be reversed for the isomorphism , since the elements of the kernel also operate distributively and associatively from the right and then become a right module. However, it is customary in the literature to provide only right quasibodies - for which the definition of the core has to be adapted accordingly - with such a right module structure, since an "equilateral" structure results in a more informal geometric interpretation of as a group of geometric images. ${\displaystyle S}$ ${\displaystyle \alpha \cdot \beta =\beta \circ \alpha }$ ${\displaystyle \tau \cdot \alpha =\alpha (\tau ),}$ ${\displaystyle S\cong \operatorname {Kern} (K)}$ ${\displaystyle K}$ ${\displaystyle S}$ ${\displaystyle S^{*}}$

Commensurable points, aspect ratio, division ratio

Three collinear points the translational level is called commensurable if a trace faithful Endomorphismus exists, the translation that on that turns moves in translation on shifts. Vectorially written . In this case the stretching factor is called the point triple . A partial ratio can be obtained from the stretch factor (for three different collinear and commensurable points) in a reversible manner: ${\displaystyle A,B,C}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \alpha \in S}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle C}$ ${\displaystyle \alpha ({\overrightarrow {AB}})={\overrightarrow {AC}}}$ ${\displaystyle \alpha \in S}$ ${\displaystyle \alpha =\operatorname {SF} (A,B,C)}$ ${\displaystyle (A,B,C)}$ ${\displaystyle \lambda =\operatorname {TV} (ABC)\in S}$

${\displaystyle \alpha ={\frac {\lambda }{1+\lambda }},\quad \lambda ={\frac {\alpha }{1-\alpha }}.}$

The fraction notation is not a problem here, because all elements of occurring commute with each other. ${\displaystyle S}$

Theorem of rays and stretching

To the theorem of rays for translation planes.

Are five points of an affine translation plane with the properties (see the figure on the right): ${\displaystyle O,A_{1},A_{2},B_{1},B_{2}}$

• ${\displaystyle O,A_{1},B_{1}}$ are not collinear,
• ${\displaystyle O,A_{1},A_{2}}$ are collinear and commensurate,
• ${\displaystyle O,B_{1},B_{2}}$ are collinear,

then applies: ${\displaystyle A_{1}B_{1}\parallel A_{2}B_{2}\;\Leftrightarrow \;{\begin{cases}(O,B_{1},B_{2})\quad {\mbox{sind kommensurabel und}}\\\operatorname {SF} (O,A_{1},A_{2})=\operatorname {SF} (O,B_{1},B_{2})\end{cases}}}$

This first set of beams for translation planes justifies to call the truer Endomorphisms as "Centric stretching" the translational level and motivates the term "stretch factor": If you select an origin and assigns as stated above each point the translation to, then operates each "stretch factor" on the points of the plane as collineation and even as dilation . In this dilatation, the origin is a fixed point and all straight lines through the origin are fixed lines. Conversely, every dilation that has the exact origin as a fixed point operates by conjugation on the translations and this operation is a true-to-track endomorphism of the translation group. Therefore, in the case of a translation plane, the subgroup of the generalized stretches with a center and the subgroup of the stretches described here around with a stretching factor are identical subgroups of the affinity group. ${\displaystyle O\in {\mathcal {A}}}$ ${\displaystyle P}$ ${\displaystyle {\overrightarrow {OP}}}$ ${\displaystyle \alpha \in S\setminus \lbrace 0,1\rbrace }$ ${\displaystyle \Delta _{O}({\mathcal {A}})}$ ${\displaystyle O}$ ${\displaystyle O}$ ${\displaystyle S^{\ast }}$

It also follows: If, in the configuration shown above, and are triangles, and are both collinear and then, of the collinear triples, either both are commensurable or both are incommensurable. They are incommensurable, then there is no dilation, which has a fixed point and on , on maps. So there can be no affinity with this property! ${\displaystyle OA_{1}B_{1}}$ ${\displaystyle OA_{2}B_{2}}$ ${\displaystyle O,A_{1},A_{2}}$ ${\displaystyle O,B_{1},B_{2}}$ ${\displaystyle A_{1}A_{2}\parallel B_{1}B_{2}}$ ${\displaystyle (O,A_{1},A_{2})}$ ${\displaystyle (O,B_{1},B_{2})}$ ${\displaystyle O}$ ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$ ${\displaystyle B_{1}}$ ${\displaystyle B_{2}}$

Since the stretching factor acts as a mapping on the parallel displacements, under the conditions of the first theorem of rays and the additional condition, a statement corresponding to the second theorem of rays results : - This formula remains correct even in the trivial case . The first two ray theorems apply accordingly in every desargue plane, whereby the condition of commensurability can then be omitted, quite generally, while the third ray theorem, which is also called three- ray theorem in synthetic geometry , can only be proven generally for pappus planes. ${\displaystyle A_{1}B_{1}\parallel A_{2}B_{2}}$ ${\displaystyle \operatorname {SF} (O,A_{1},A_{2})\cdot {\overrightarrow {A_{1}B_{1}}}={\overrightarrow {A_{2}B_{2}}}}$ ${\displaystyle \operatorname {SF} (O,A_{1},A_{2})=1}$

(Compare the main articles centric stretching and the theorem of rays )

Desargue planes

A translation plane with the corresponding oblique body of the true-to-track endomorphisms of is a Desargue plane if and only if one of the following equivalent conditions applies: ${\displaystyle {\mathcal {A}}}$ ${\displaystyle S}$ ${\displaystyle {\mathcal {T}}({\mathcal {A}})}$

1. The great affinity set of Desargues applies . ${\displaystyle {\mathcal {A}}}$
2. A coordinate quasi-body of coincides with its core. ${\displaystyle {\mathcal {A}}}$
3. A coordinate quasi-body of is an oblique body. ${\displaystyle {\mathcal {A}}}$
4. A coordinate quasi-body of is isomorphic to . ${\displaystyle {\mathcal {A}}}$ ${\displaystyle S}$
5. If three points of the plane lie on a straight line, they are always commensurable.
6. The translations form a two- dimensional left vector space . ${\displaystyle S}$

Since the coordinate areas are uniquely determined by the affine plane up to isomorphism, the statements about these areas "A coordinate quasi-body ..." can also be formulated here with "Every coordinate quasi-body ...".

On the other hand, every "real", i.e. non-Desargue translation plane, contains a Desargue plane as a real subset: If one chooses a coordinate system and only considers points with coordinates that are to and commensurable, and only those straight lines whose coefficients have this property, one obtains one for desargue level isomorphic affine incidence structure. ${\displaystyle (O,E_{1},E_{2})}$ ${\displaystyle x_{1},x_{2}\in OE_{1}=K}$ ${\displaystyle 0=O}$ ${\displaystyle 1=E_{1}}$ ${\displaystyle \operatorname {Kern} (K)^{2}}$

Pappus planes

If an orthogonality relation can be defined in a translation plane and the characteristic of the inclined body is not 2, that is, the (affine) Fano axiom holds, then the general validity of the vertical intersection theorem and the central perpendicular theorem is equivalent and - if these are generally valid - is in the Level, the Pappos theorem is generally applicable and the coordinate quasibody is even a body . (See Pre-Euclidean Plane ). ${\displaystyle S}$

In general, a translation plane satisfies Pappos' theorem if and only if

• if it is desarguean and the multiplication in the oblique body of the true-to-track endomorphisms of the translation group is commutative, i.e. a body or equivalent ${\displaystyle S}$ ${\displaystyle S}$
• if their coordinate quasi-body is a body.

If the order of the translation plane is finite, then the oblique body is always a body. Then the plane of translation is pappusian if and only if it is desarguean. ${\displaystyle S}$

Finite levels

An affine or projective plane is called finite if it is its order and therefore also the number of points on the plane. In the case of an affine plane, the order is the number of points on a straight line; in the case of a projective plane, the order of the affine plane, which is created by slitting the projective plane. From the fact that the coordinate quasi-body of an affine translation plane is a left vector space over the oblique body of the true-track endomorphisms, together with Wedderburn's theorem , which says that a finite oblique body is always commutative, i.e. a finite body , conclusions for finite translations result - and Moufang levels : ${\displaystyle n}$ ${\displaystyle S}$

• The oblique body of a finite translation plane is a finite body , so it has elements with a prime number and . ${\displaystyle S}$ ${\displaystyle \mathbb {F} _{q}}$ ${\displaystyle q=p^{m}}$ ${\displaystyle p}$ ${\displaystyle m\in \mathbb {N} ,m\geq 1}$
• The coordinate quasi-body is a finite-dimensional vector space over and therefore has elements. So the order of the translation plane is this prime power, if there is, then the translation plane is the Pappus plane above the body . ${\displaystyle S=\mathbb {F} _{q}}$ ${\displaystyle q^{r}=p^{mr}\!\,\,(r\in \mathbb {N} ,r\geq 1)}$ ${\displaystyle r=1}$ ${\displaystyle S^{2}}$ ${\displaystyle S=\mathbb {F} _{q}}$
• There are numerous finite affine translation planes that are not desarguean, for example 4 different (non-isomorphic) of order 9 (see the examples in the article Ternary Bodies .)
• The formal analogue of affine translation planes among the projective planes are the Moufan planes , in which Desargues' little projective theorem is generally valid. Ruth Moufang has shown that real , that is, non-desargue, Moufang levels are always infinite. It follows from this that with a finite affine translation plane, the projective expansion is a Moufang plane if and only if both planes are desarguean and therefore equivalent to planes over a finite field.

More general statements about the possible orders of finite planes can be found in the articles Projective Plane and Projective Geometry .

Projective translational plane

A projective plane is called a translation plane with respect to one of its straight lines if it satisfies Desargues' little projective theorem with respect to this straight line as an axis . An equivalent description of such a projective translation level : It belongs to one of the classes IVa, V or VII in the classification of projective levels according to Hanfried Lenz .

The projective closure of an affine translation plane is always a projective translation plane. If, on the other hand, a projective translation plane is slotted along a projective straight line , an affine plane is created in which this straight line represents the distance line. The affine plane generated in this way is an affine translation plane if and only if the projective plane satisfies Desargues' little projective theorem with respect to as an axis . Equivalent: The straight line must be an axis in the Lenz figure of the projective plane. ${\displaystyle u}$ ${\displaystyle u}$ ${\displaystyle u}$

Examples and counterexamples

• Every Desargue plane is a translation plane, in particular the affine plane over an oblique body . Here the oblique body of the true-to-track endomorphisms (except for isomorphism) corresponds to the coordinate oblique body . ${\displaystyle K^{2}}$ ${\displaystyle K}$ ${\displaystyle S}$
• The real octonions form a quasi-body, which is not a skewed body: Both distributive laws apply, but the multiplication is not associative. The affine translation plane is thus a non-Desargue translation plane. ${\displaystyle \mathbb {O} }$ ${\displaystyle \mathbb {O} ^{2}}$
• The real Moulton plane is an affine plane that is not a translation plane: If the (“normal” and Moulton plane) straight line on which some Moulton straight lines have their “kink”, then the translation group consists exactly of the “normal” displacements the real plane in the direction of the straight line , it is isomorphic to the commutative group . Every group automorphism of is on track, but since the translation group does not simply operate transitively on the Moulton plane, this is of little use in describing this geometry. ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle (\mathbb {R} ,+)}$ ${\displaystyle (\mathbb {R} ,+)}$
• In contrast, the finite moulton planes are always affine translation planes. There are an infinite number of non-Desargue finite translation planes of this type, see the section Quasi-bodies of finite Moulton planes in the article Quasi-bodies .

The article Ternary Body contains further examples for affine translation planes, in particular also detailed examples for finite, non-Desargue translation planes (see subsection Examples of order 9 ).

literature

• Wendelin Degen, Lothar Profke: Fundamentals of affine and Euclidean geometry . 1st edition. Teubner, Stuttgart 1976, ISBN 3-519-02751-8 .
• Heinz Lüneburg : Translation planes . 1st edition. Springer, Berlin / Heidelberg / New York 1980, ISBN 3-540-09614-0 .
• Günter Pickert : Axiomatic justification of the plane Euclidean geometry in vector representation . In: Mathematical-physical semester reports . tape 10 , 1963, ISSN  0025-5823 , pp. 65-85 .
• Charles Weibel: Survey of Non-Desarguesian Planes . In: Notices of the American Mathematical Society . tape 54 . American Mathematical Society, November 2007, pp. 1294–1303 (English, ams.org [PDF; 702 kB ; accessed on January 24, 2012]).

References and comments

1. ↑ ^{a b c d e f} sword (1976)
2. ↑ ^{a b} Weibel (2007)
3. ↑ The translation group is trivial if and only if there is no translation apart from the identical mapping. In this case the translation group is commutative, but its only endomorphism is identity, so the ring of true-track endomorphisms defined here would be a null ring .