Bulletin of the Belgian Mathematical Society - Simon Stevin

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

Atilla Akpinar, Basri Celik, and Süleyman Ciftci

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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.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 15, Number 1 (2008), 49-64.

Dates
First available in Project Euclid: 22 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1203692446

Digital Object Identifier
doi:10.36045/bbms/1203692446

Mathematical Reviews number (MathSciNet)
MR2406086

Zentralblatt MATH identifier
1138.51002

Subjects
Primary: 51C05: Ring geometry (Hjelmslev, Barbilian, etc.) 51A35: Non-Desarguesian affine and projective planes 17D05: Alternative rings

Keywords
Moufang-Klingenberg planes local alternative ring cross-ratio 6-figure

Citation

Akpinar, Atilla; Celik, Basri; Ciftci, Süleyman. Cross-Ratios and 6-Figures in some Moufang-Klingenberg Planes. Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 1, 49--64. doi:10.36045/bbms/1203692446. https://projecteuclid.org/euclid.bbms/1203692446


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