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A geometrical characterization of Minkowski planes of order 3 and 4

100%

Jakóbowski J.

1999

tom Vol. 32, nr 2

441-456

In [5] H. A. Wilbrink proved that a certain class of Minkowski planes induce nearaffine planes. Of course, this class contains all Minkowski planes over fields. But only Minkowski planes of order 3 and 4 induce nearaffine planes which are also Minkowski planes (moreover they are affine planes, too.

Multicentral automorphisms in geometries of circles

63%

Jakóbowski J. , Matraś A.

2001

tom Vol. 49, no 4

417-432

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.

Extending nearaffine planes to hyperbola structures

63%

Cudna-Salmanowicz K. , Jakóbowski J.

2007

tom Vol. 55, no 1

71-80

H. A. Wilbrink [Geom. Dedicata 12 (1982)] considered a class of Minkowski planes whose restrictions, called residual planes, are nearaffine planes. Our study goes in the opposite direction: what conditions on a nearaffine plane are necessary and sufficient to get an extension which is a hyperbola structure.