J. Aust. Math. Soc. 83 (2007), no. 3, pp. 335–355.

Laguerre geometries and some connections to generalized quadrangles

Matthew R. Brown
School of Mathematical Sciences University of Adelaide SA 5005 Australia
matthew.brown@adelaide.edu.au
Received 15 November 2005; revised 30 September 2006
Communicated by L. Batten

Abstract

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,s^2) satisfying Property (G) at a pair of points is equivalent to a configuration of ovoids in three-dimensional projective space.

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2000 Mathematics Subject Classification: primary 51E20, 51E12
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