Cross-Ratios and 6-Figures in some Moufang-Klingenberg Planes

Abstract

This paper deals with Moufang-Klingenberg planes $\boldsymbol{M}(\mathcal{A}) $ defined over a local\ alternative ring $\mathcal{A}$\ of dual numbers. The definition of cross-ratio is extended to $\boldsymbol{M}(\mathcal{A})$. Also, some properties of cross-ratios and 6-figures that arewell-known for Desarguesian planes are investigated in $\boldsymbol{M}(\mathcal{A})$; so we obtain relations between algebraic properties of $\mathcal{A}$ and geometric properties of $\boldsymbol{M}(\mathcal{A})$. In particular, we show that pairwise non-neighbour four points of the line $g$ are in harmonic position if and only if they are harmonic, and that $\mu $ is Menelaus or Ceva 6-figure if and only if $r\left( \mu \right) =-1$ or $r\left( \mu \right) =1, $ respectively.