Cross-ratios and 6-figures in some Moufang-Klingenberg planes. (English) Zbl 1138.51002
This paper deals with Moufang-Klingenberg planes over a local alternative ring of dual numbers. The cross-ratio of a quadruple of pairwise non-neighboring points is defined and some properties of the cross-ratio are investigated. Finally some special configurations, called 6-figures, are generalized from ordinary desarguesian and Moufang projective planes to Moufang-Klingenberg planes and the ratio of such a figure is defined (using cross-ratios). It is then proved that a 6-figure is a Menelaus or Ceva configuration iff the corresponding ratio equals \(-1\) or 1 respectively.
Reviewer: D. F. J. Keppens (Gent)
MSC:
51C05 | Ring geometry (Hjelmslev, Barbilian, etc.) |
51A35 | Non-Desarguesian affine and projective planes |
17D05 | Alternative rings |
51A20 | Configuration theorems in linear incidence geometry |