Laguerre geometries and some connections to generalized quadrangles

Matthew R. Brown
2007 Journal of the Australian Mathematical Society  
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
more » ... 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.
doi:10.1017/s1446788700037964 fatcat:xlti5hpcjrfsnovjq7owa5lzma