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2 Bulletin of the Polish Academy of Sciences. Mathematics

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Central automorphisms of Veblenian nearaffine planes

Cudna-Lasecka K., Jakóbowski J.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 3

337-347

The paper deals with nearaffine planes described by H. A. Wilbrink. We consider their central automorphisms, i.e. automorphisms satisfying the Veblen condition, which become central collineations in connected projective planes. Moreover, a concept of central pseudo-automorphism is considered, i.e. some bijections in a nearaffine plane are not automorphisms but they become central collineations in the related projective planes.

Multicentral automorphisms in geometries of circles

Jakóbowski J., Matraś A.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

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.