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The inner and outer space of 2-dimensional Laguerre planes

Part of: Topological geometry Nonlinear incidence geometry

Published online by Cambridge University Press:  09 April 2009

B. Polster  and
G. F. Steinke
B. Polster
Affiliation:
Department of Mathematics and Statistics University of CanterburyChristchurchNew Zealand e-mail: bmp and [email protected]
G. F. Steinke
Affiliation:
Department of Mathematics and Statistics University of CanterburyChristchurchNew Zealand e-mail: bmp and [email protected]
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Abstract

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The classical 2-dimensional Laguerre plane is obtained as the geometry of non-trivial plane sections of a cylinder in R3 with a circle in R2 as base. Points and lines in R3 define subsets of the circle set of this geometry via the affine non-vertical planes that contain them. Furthemore, vertical lines and planes define partitions of the circle set via the points and affine non-vertical lines, respectively, contained in them.

We investigate abstract counterparts of such sets of circles and partitions in arbitrary 2-dimensional Laguerre planes. We also prove a number of related results for generalized quadrangles associated with 2-dimensional Laguerre planes.

Keywords

Laguerre planetopological incidence geometrygeneralized quadrangle

MSC classification

Secondary: 51B15: Laguerre geometries 51H15: Topological nonlinear incidence structures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 62 , Issue 1 , February 1997 , pp. 104 - 127
Copyright
Copyright © Australian Mathematical Society 1997

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