Finite Minkowski planes

In this paper we give second characterizations of a certain class of finite Minkowski planes.     

Two-transitive Minkowski planes

In this paper we determine all finite Minkowski planes with an automorphism group which satisfies the following transitivity property: any ordered pair of nonparallel points can be mapped onto any other ordered pair of nonparallel points.     

Arcs in Minkowski planes

The aim of this paper is to study some properties of k-arcs in Minkowski planes focalizing the attention on problems of existence and completness.Work done under the auspicies of G.N.S.A.G.A. supported by 40% grants of M.U.R.S.T.In memoriam Giuseppe Tallini     

Laguerre and Minkowski planes produced by dilatations

We show that each parabolic curve f in R² produces a Laguerre plane if f and all its images under dilatations are cycles. Likewise, two hyperbolic curves f₁,f₂ produce a Minkowski planeM(f₁,f₂). We determine for which curves is miquelian resp. ovoidal, and for which pairs f₁,f₂,M(f₁,f₂) is miquelian resp. satisfies the rectangle axiom, thus providing many examples of non-embeddable planes.     

Finite Minkowski planes and embedded inversive planes

We show that each known finite Minkowski plane of order even, contains embedded Miquelian inversive planes, (cf. Proposition 1). Received 2 July 1999.     

On embeddable Minkowski planes

Let (Q, R, R′) be a finite 2-net and F a non empty family of involutory permutations on Q. In this paper the properties of F are found for Q to be a hyperbolic quadric in a 3-dimensional Galois space PG(3,q) and F the family of the involutions of Q defined by the points in PG(3,q) not on Q.     

Toroidal circle planes that are not Minkowski planes

We construct first examples of circle planes on the torus that are no Minkowski planes, but satisfy the same axiom of joining as flat Minkowski planes. The circle planes constructed by us form a special class ofhyperbola structures (see [4]) or(B*)-Geometrien (see [2]).This research was supported by a Feodor Lynen Fellowship and an ARC International Research Fellowship.     

On resolvable finite Minkowski planes

Among the known finite Minkowski planes we determine an infinite family of examples admitting a partition of the set of blocks into equivalence classes, each of which in turn partitions the point set; in particular non-miquelian finite Minkowski planes with this property exist.work done within the activity of GNSAGA of CNR and supported by the Italian Ministry of Public Education     

On the Minkowski planes constructed by Artzy and Groh

The Minkowski planes constructed by R. Artzy and H. Groh [1] are characterized among the locally, connected and finite dimensional Minkowski planes as strongly semi-(p, w)-transitive Minkowski planes (see Theorem 2). The types of the Artzy-Groh planes in the typification of the Minkowski planes by M. Klein are determined (see Proposition 4). The second author was supported by a DAAD scholarship for a research visit at TU München. He sincerely thanks the Zentrum Mathematik der TU München for their hospitality.     

The symmetry axiom in Minkowski planes

We give a synthetic proof that in a symmetric Minkowski plane the rectangle axiom (G) holds. Dedicated to Professor Helmut Karzel     

Key distribution patterns using Minkowski planes

A key distribution pattern is a combinatorial structure which provides a secure method of distributing secret keys among a number of participants in a cryptographic scheme. Inversive and Laguerre planes have been used to construct key distribution patterns with storage requirements lower than the trivial distribution system. In this note we construct key distribution patterns from Minkowski planes, the third of the so-calledcircle geometries.The author acknowledges the support of the Australian Research Council     

Symmetric planes with non-classical tangent translation planes

We construct symmetric planes associated with an arbitrary locally compact connected nearfield . If is a proper nearfield, i.e. {;;}, then the tangent translation plane of this symmetric plane is not classical. All previously known examples of symmetric planes have classical tangent translation planes.Herrn Professor Dr. H. Salzmann zum 65. Geburtstag gewidmet