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.
Publié le : 2008-02-15
Classification:
Moufang-Klingenberg planes,
local alternative ring,
cross-ratio,
6-figure,
51C05,
51A35,
17D05
@article{1203692446,
author = {Akpinar, Atilla and Celik, Basri and Ciftci, S\"uleyman},
title = {Cross-Ratios and 6-Figures in some Moufang-Klingenberg Planes},
journal = {Bull. Belg. Math. Soc. Simon Stevin},
volume = {15},
number = {1},
year = {2008},
pages = { 49-64},
language = {en},
url = {http://dml.mathdoc.fr/item/1203692446}
}
Akpinar, Atilla; Celik, Basri; Ciftci, Süleyman. Cross-Ratios and 6-Figures in some Moufang-Klingenberg Planes. Bull. Belg. Math. Soc. Simon Stevin, Tome 15 (2008) no. 1, pp. 49-64. http://gdmltest.u-ga.fr/item/1203692446/