On laguerre geometry and generalized quadrangles

2011 ◽  
Author(s):  
Ying, Anna Pun
Keyword(s):  
Generalized Quadrangles ◽  
Laguerre Geometry

scholarly journals Laguerre geometries and some connections to generalized quadrangles

2007 ◽  
Vol 83 (3) ◽  
pp. 335-356
Author(s):  
Matthew R. Brown
Keyword(s):  
Three Dimensional ◽  
Line Pair ◽  
Generalized Quadrangles ◽  
Constant Number ◽  
Unique Point ◽  
Line Geometry ◽  
Laguerre Geometry ◽  
Point Line ◽  
Dimensional Projective Space ◽  
Unique Circle

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.

scholarly journals MOUFANG QUADRANGLES OF MIXED TYPE

2008 ◽  
Vol 50 (1) ◽  
pp. 143-161
Author(s):  
KOEN STRUYVE ◽  
HENDRIK VAN MALDEGHEM
Keyword(s):  
Mixed Type ◽  
Geometric Characterization ◽  
Weak Version ◽  
Generalized Quadrangles

AbstractIn this paper, we present some geometric characterizations of the Moufang quadrangles of mixed type, i.e., the Moufang quadrangles all the points and lines of which are regular. Roughly, we classify generalized quadrangles with enough (to be made precise) regular points and lines with the property that the dual nets associated to the regular points satisfy the Axiom of Veblen-Young, or a very weak version of the Axiom of Desargues. As an application we obtain a geometric characterization and axiomatization of the generalized inversive planes arising from the Suzuki-Tits ovoids related to a polarity in a mixed quadrangle. In the perfect case this gives rise to a characterization with one axiom less than in a previous result by the second author.

scholarly journals Surfaces in Laguerre geometry

2019 ◽  
Vol 117 ◽  
pp. 223-255 ◽  
Author(s):  
Emilio Musso ◽  
Lorenzo Nicolodi
Keyword(s):  
Laguerre Geometry
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