Laguerre geometries and some connections to generalized quadrangles

Abstract

<jats:title>Abstract</jats:title><jats:p>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 <jats:italic>C</jats:italic> and a point <jats:italic>Q</jats:italic> not on it for each point <jats:italic>P</jats:italic> on <jats:italic>C</jats:italic> there is a unique circle on <jats:italic>Q</jats:italic> and touching <jats:italic>C</jats:italic> at <jats:italic>P</jats:italic>. 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.</jats:p><jats:p>A generalized quadrangle (GQ) is a point, line geometry in which for a non-incident point, line pair (<jats:italic>P</jats:italic>. <jats:italic>m</jats:italic>) there exists a unique point on <jats:italic>m</jats:italic> collinear with <jats:italic>P</jats:italic>. In certain cases we construct a Laguerre geometry from a GQ and conversely. Using Laguerre geometries we show that a GQ of order (<jats:italic>s</jats:italic>. <jats:italic>s</jats:italic><jats:sup>2</jats:sup>) satisfying Property (G) at a pair of points is equivalent to a configuration of ovoids in three-dimensional projective space.</jats:p>