Laguerre geometries and some connections to generalized quadrangles

• Autores: Matthew R. Brown
• Localización: Journal of the Australian Mathematical Society, ISSN 1446-7887, Vol. 83, Nº 3, 2007, págs. 335-356
• Idioma: inglés
• DOI: 10.1017/s1446788700037964
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• Resumen

  • 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 (Pm) 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 (ss2) satisfying Property (G) at a pair of points is equivalent to a configuration of ovoids in three-dimensional projective space.