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Journals / Hacettepe Journal of Mathematics and Statistics / 2011 / Cilt: 40 - Sayı: 1

Cross-ratios of points and lines in some Moufang-Klingenberg planes

Atilla AKPINAR
Basri ÇELİK
Süleyman ÇİFTÇİ
Pages
1–13
Publication Date
2022-10-18
DOI
—
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Namely, we show that four pairwise nonneighbour lines passing through the point U are in harmonic position if and only if they are harmonic.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:This paper deals with Moufang-Klingenberg planes M(A) defined over a local alternative ring A of dual numbers.
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Key Findings & Data:The cross-ratio of concurrent lines is defined in a special case, and extended to the whole plane M(A).
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Conclusion & Impact:Namely, we show that four pairwise nonneighbour lines passing through the point U are in harmonic position if and only if they are harmonic.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

This paper deals with Moufang-Klingenberg planes M(A) defined over a local alternative ring A of dual numbers. The cross-ratio of concurrent lines is defined in a special case, and extended to the whole plane M(A). So, some results related to cross-ratios of points of M(A) are carried over to lines of M(A). Namely, we show that four pairwise nonneighbour lines passing through the point U are in harmonic position if and only if they are harmonic.

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Cite This Article

AKPINAR, A., ÇELİK, B., ÇİFTÇİ, S. (2022). Cross-ratios of points and lines in some Moufang-Klingenberg planes. Hacettepe Journal of Mathematics and Statistics, Cilt: 40 - Sayı: 1, 1–13. https://ajindex.com/articles/530576

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References (18)

  1. Akpinar, A., Celik, B. and Ciftci, S. Cross–ratios and 6-figures in some Moufang- Klingenberg planes, Bulletin of the Belgian Math. Soc. - Simon Stevin 15, 49–64, 2008
  2. Baker, C.A., Lane, N.D. and Lorimer, J.W. A coordinatization for Moufang-Klingenberg Planes, Simon Stevin 65, 3–22, 1991
  3. Bell, J. L. The art of the intelligible (Kluwer Acad. Publishers, The Netherland, 2001
  4. Blunck, A. Cross-ratios in Moufang planes, J. Geometry 40, 20–25, 1991
  5. Blunck, A. Projectivities in Moufang-Klingenberg planes, Geom. Dedicata 40, 341–359, 1991
  6. Blunck, A. Cross-ratios over local alternative rings, Res. Math. 19, 246–256, 1991
  7. Blunck, A. Cross-ratios in Moufang-Klingenberg planes, Geom Dedicata 43, 93–107, 1992
  8. Celik, B. and Ciftci, S. Cross–ratios of points and lines in Moufang planes, Balkan Journal of Geometry and its Applications 5, 37–46, 2000
  9. C¸ ift¸ci, S. On 6-figures in Moufang projective planes, Commun. Fac. Sci. Univ. Ank. Series A1 38, 21–28, 1989
  10. Ciftci, S. and C¸ elik, B. On the cross–ratios of points and lines in Moufang planes, Journal of Geometry 71, 34–41, 2001
  11. Ferrar, J.C. Cross-ratios in projective and affine planes, in Plaumann, P. and Strambach, K., Geometry - von Staudt’s Point of View (Proceedings Bad Windsheim, 1980), Reidel, Dordrecht, 101–125, 1981
  12. Hopkins, E. J. and Hails, J. S. An introduction to plane projective geometry (Clarendon Press, Oxford, 1953
  13. Horadam, A.F. A guide to undergraduate projective geometry (Pergamon Press, 1970
  14. Hughes, D.R. and Piper, F.C. Projective Planes (Springer, New York, 1973
  15. Meserve, B.E. Fundamental Concepts of Geometry (Dover Publications, New York, 1983
  16. Pickert, G. Projective Ebenen (Springer, Berlin, 1955
  17. Schafer, R.D. An Introduction to Nonassociative Algebras (Dover Publications, New York, 1995
  18. Stevenson, F.W. Projective Planes (WH Freeman Co., San Francisco, 1972)