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Laguerre geometries and some connections to generalized quadrangles

Part of: Finite geometry and special incidence structures

Published online by Cambridge University Press:  09 April 2009

Matthew R. Brown
Matthew R. Brown
Affiliation:
School of Mathematical SciencesUniversity of AdelaideSA [email protected]
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Abstract

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A Laguerre plane is a geometry of points, lines and circles where three pairwise non-collinear points lie on a unique circle, any line and circle meet uniquely and finally, given a circle C and a point Q not on it for each point P on C there is a unique circle on Q and touching C at P. We generalise to a Laguerre geometry where three pairwise non-collinear points lie on a constant number of circles. Examples and conditions on the parameters of a Laguerre geometry are given.

A generalized quadrangle (GQ) is a point, line geometry in which for a non-incident point, line pair (P. m) there exists a unique point on m collinear with P. In certain cases we construct a Laguerre geometry from a GQ and conversely. Using Laguerre geometries we show that a GQ of order (s. s2) satisfying Property (G) at a pair of points is equivalent to a configuration of ovoids in three-dimensional projective space.

MSC classification

Secondary: 51E20: Combinatorial structures in finite projective spaces 51E12: Generalized quadrangles, generalized polygons
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 83 , Issue 3 , December 2007 , pp. 335 - 356
Copyright
Copyright © Australian Mathematical Society 2007

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