Advertisement

Lobachevskii Journal of Mathematics

, Volume 39, Issue 1, pp 1–12 | Cite as

Generalizations of Casey’s Theorem for Higher Dimensions

  • N. V. Abrosimov
  • V. V. Aseev
Article
  • 28 Downloads

Abstract

We give generalizations of Casey’s theorem and its converse for higher dimensions. We also present a multidimensional generalization for the problem of Apollonius. To do this we introduce a notion of ψ-tangent for a generalized k-sphere that touches a number of generalized n-balls in proper manner.

Keywords and phrases

Casey’s theorem Ptolemy’s theorem problem of Apollonius 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ya. P. Ponarin, Elementary Geometry, Vol. 1: Planimetry (MCCME, Moscow, 2004) [in Russian].Google Scholar
  2. 2.
    J. Casey, A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry, with Numerous Examples, 5th ed. (Hodges, Figgis, Dublin, 1888).zbMATHGoogle Scholar
  3. 3.
    R. A. Johnson, Advanced Euclidean Geometry. An Elementary Treatise on the Geometry of the Triangle and the Circle (Dover, New York, 1960).zbMATHGoogle Scholar
  4. 4.
    W. J. McClelland and T. Preston, A Treatise on Spherical Trigonometry with Application to Spherical Geometry and Numerous Examples. Part II (Macmillian, London, 1886).Google Scholar
  5. 5.
    T. Kubota, “On the extended Ptolemy’s theorem in hyperbolic geometry,” Sci. Rep. Tohoku Univ., Ser. 1: Phys., Chem., Astron. 2, 131–156 (1912).Google Scholar
  6. 6.
    P. A. Shirokov, “Etudes on the Lobachevskii geometry,” Izv. Fiz.-Mat. Ob-va KGU, Ser. 2 24 (1), 26–32 (1924).Google Scholar
  7. 7.
    J. E. Valentine, “An analogue of Ptolemy’s theorem and its converse in hyperbolic geometry,” Pacif. J. Math. 34, 817–825 (1970).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    N. V. Abrosimov and L. A. Mikaiylova, “Casey’s theorem in hyperbolic geometry,” Sib. Elektron. Mat. Izv. 12, 354–360 (2015).MathSciNetzbMATHGoogle Scholar
  9. 9.
    A. V. Kostin and N. N. Kostina, “An interpretation of Casey’s theorem and of its hyperbolic analogue,” Sib. Elektron.Mat. Izv. 13, 242–251 (2016).MathSciNetzbMATHGoogle Scholar
  10. 10.
    A. F. Berdon, The Geometry of Discrete Groups (Springer, New York, 1995).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

Personalised recommendations