Abstract
AbstractA 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.
Publisher
Cambridge University Press (CUP)
Reference29 articles.
1. Eine Bemerkung �ber endliche Laguerre- und Minkowski-Ebenen
2. Un estensione del teorema di Segre-Kustaanheimo;Barlotti;Boll. Unione Mat. Ital.,1955
3. [9] Cherowitzo W. . ‘Bill Cherowitzo's hyperoval page’, http://www-math.cudenver.edu/-wcherowi/research/hyperoval/hypero.html.
4. Generalized Quadrangles and Flocks of Cones
5. Some generalized quadrangles with parametersq 2,q