Andreas E. Schroth

Three-dimensional quadrangles and flat Laguerre planes

Every three-dimensional generalized quadrangle can be constructed from flat Laguerre planes.

Generalized quadrangles constructed from topological Laguerre planes

The Lie geometry of a finite-dimensional locally compact connected Laguerre plane is a topological generalized quadrangle.

How to draw a hexagon

Abstract Pictures of the G 2 (2) hexagon and its dual are presented. A way to obtain these pictures is discussed.

The Apollonius problem in four-dimensional Laguerre planes

The number of circles of a four-dimensional locally compact Laguerre plane touching three given circles or points depends only on the given geometric configuration but not on the Laguerre plane.

Half-regular and regular points in compact polygons

We investigate the derived structures of compact polygons at half-regular and at regular points. This enables us to give a geometric characterization of the real or complex split Cayley hexagon.

The apollonius problem in flat laguerre planes

The number of circles of a flat Laguerre plane touching three given circles or points depends only on the given geometric configuration but not on the Laguerre plane.

Semi-Biplanes and Antiregular Generalied Quadrangles

A natural method to construct semi-biplanes from antiregular generalized quadrangles is introduced. Properties of the semi-biplanes constructed are discussed. In the finite case and in the topological case the semi-biplanes that arise bear a strong resemblance to semi-biplanes that arise in the natural way from projective planes admitting an involutory homology.

Sisterhoods of flat Laguerre planes

Every flat Laguerre plane of shear type over a pair of skew parabolae is related to a flat Laguerre plane of translation type over a pair of skew parabolae and vice versa. The relationship is defined using the connection between flat Laguerre planes and three-dimensional generalized quadrangles.

Partial circle geometries and antiregular quadrangles

A new and rather general definition of circle geometries is given. This definition is such that circle planes and chain spaces are circle geometries. Also the geometry of points and traces of an antiregular quadrangle is a partial circle geometry. Orthogonal quadrangles can then be characterised as those antiregular generalised quadrangles where in the associated partial circle geometry the Miquel condition is satisfied.

Flat weakly miquelian laguerre planes are ovoidal

Every flat Laguerre plane that satisfies a certain variation of the Miquel Condition is ovoidal. Equivalently, in flat Laguerre planes a certain special version of the Bundle Theorem already implies the Bundle Theorem.