Cross-ratios and 6-figures in some Moufang-Klingenberg planes

AKPINAR A., ÇELİK B., Ciftci S.

BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY-SIMON STEVIN, vol.15, no.1, pp.49-64, 2008 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 15 Issue: 1
  • Publication Date: 2008
  • Doi Number: 10.36045/bbms/1203692446
  • Journal Name: BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY-SIMON STEVIN
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.49-64
  • Keywords: Moufang-Klingenberg planes, local alternative ring, cross-ratio, 6-figure
  • Bursa Uludag University Affiliated: Yes

Abstract

This paper deals with Moufang-Klingenberg planes M(A) defined over a local alternative ring A of dual numbers. The definition of cross-ratio is extended to M(A). Also, some properties of cross-ratios and 6-figures that are well-known for Desarguesian planes are investigated in M(A); so we obtain relations between algebraic properties of A and geometric properties of M(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 p is Menelaus or Ceva 6-figure if and only if r (mu) = - 1 or r (mu) = 1, respectively.