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1

A residual skewaffine plane of a Mobius or Minkowski plane

Matraś A.

Demonstratio Mathematica

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2007

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Vol. 40, nr 4

913-924

EN

For Mobius and Minkowski planes of characteristic different from 2 a residual skewaffine piane associated with any point p is constructed. Following the construction given by Andre (cf. [1]) we obtain the residual piane as the group space of some normally transitive group of automorphisms fixing p. This is a skewaffine piane without straight lines in the Mobius case and with two families of straight lines in the Minkowski case.

2

The double tangency symmetries in Laguerre planes

Kosiorek J., Matraś A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2007

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Vol. 55, no 2

123-137

EN

The group generated by double tangency symmetries in a Laguerre plane is investigated. The geometric classification of involutions of a symmetric Laguerre plane is given. We introduce the notion of projective automorphisms using the double tangency and parallel perspectivities. We give the description of the groups of projective automorphisms and automorphisms generated by double tangency symmetries as subgroups of the group M(F, R) of automorphisms of a chain geometry Σ(Y, R) following Benz.

3

A construction of a skewaffine structure in Laguerre geometry

Matraś A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2006

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Vol. 54, no 3-4

277-289

EN

J. Andre constructed a skewaffine structure as a group space of a normally transitive group. In this paper his construction is used to describe the structure of the set of circles not passing through a point of a Laguerre plane. Sufficient conditions to ensure that this structure is a skewaffine plane are given.

4

Some connections between Minkowski and hyperbolic planes

Kosiorek J., Matraś A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2004

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Vol. 52, nr 2

133-139

EN

The model of the Minkowski plane in the projective plane with a fixed conic sheds a new light on the connection between the Minkowski and hyperbolic geometries. The construction of the Minkowski plane in a hyperbolic plane over a Euclidean field is given. It is also proved that the geometry in an orthogonal bundle of circles is hyperbolic in a natural way.

5

Multicentral automorphisms in geometries of circles

Jakóbowski J., Matraś A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2001

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Vol. 49, no 4

417-432

EN

We consider three types of geometries of circles (Moebius plane, Laguerre plane and Minkowski plane, cf. [4) with respect to so-called multicentral automorphisms. An automorphism [phi] of any geometry of circles is central if it has a fix point P and [phi] becomes a central collineation in the derived projective plane M(P). For any central automorphism [phi] we try to establish the whole set of points R such that [phi] becomes a central collineation in M(R.). Than [phi] is called multicentral if this set contains at least two points. Moreover, [phi] is proper if existing of a point [R is not equal to P], is not caused by the fact that [phi] is central in M(P). There is no proper multicentral automorphism in a Moebius plane. The most interesting proper multicentral automorphisms are involutorial mappings: double homotheties in Minkowski planes, and (sigma, tau)homologies in Laguerre planes. We give some examples.

6

Central automorphisms of Laguerre planes

Makowiecka H., Matraś A.

Demonstratio Mathematica

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1998

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Vol. 31, nr 4

753-763

EN

Any automorphism of a Benz plane having at least one fixed point induces a collineation on the projective extension of the residual affine plane with reference to this point. When this collineation is a central automorphism , the initial automorphism is called the central automorphism (or central-axial automorphism, cf.([3]). In this paper we present an analytical description of central automorphisms of a miguelian Laguerre planes with the characteristic different from two. This description is applied to find transitive groups of homotheties and translations of types occuring in the classification theorems of R. Kleinewillinghofer ([2]). Some examples over an arbitrary commutative field are constructed, the other over the finite field Z3 and z5. It is interesting that two types of the Kleinewillinghofer classification ( [2] ) appear only as automorphism subgroups of finite plane of order three or five. This will give a clear characterization of these planes. Throughaut we assume that the characteristic of a plane is not equal to two.