The symmetry axiom in Minkowski planes

Tangent Segments in Minkowski Planes

AMinkowski plane is Euclidean iff the two tangent segments from any exterior point to the unit circle have the same length. MSC 2000: 52A10, 46B20, 46C99, 51M04

Minkowski Planes with Miquelian Pairs

On Reuleaux Triangles in Minkowski Planes

In this paper we prove some results on Reuleaux triangles in (Minkowski or) normed planes. For example, we reprove Wernicke’s result (see [21]) that the unit disc and Reuleaux triangles in a normed plane are homothets if and only if the unit circle is either an affine regular hexagon or a parallelogram. Also we show that the ratio of the area of the unit ball of a Minkowski plane to that of a R...

The Symmetry Axioms in Laguerre Planes

We introduce two axioms in Laguerre geometry and prove that they provide a characterization of miquelian planes over fields of the characteristic different from 2. They allow to describe an involutory automorphism that sheds some new light on a Laguerre inversion as well as on a symmetry with respect to a pair of generators.

On the Klein–Kroll types of flat Minkowski planes

Klein and Kroll classified Minkowski planes with respect to subgroups of central automorphisms and obtained a multitude of types. Some of these types are known to exist only in finite Minkowski planes or as the type of a proper subgroup in the automorphism group. In this paper we investigate the Klein– Kroll types of flat Minkowski planes with respect to the full automorphism groups of these pl...