Results in Mathematics

, Volume 57, Issue 3–4, pp 205–219 | Cite as

On Algebraic Structures Related to Beltrami–Klein Model of Hyperbolic Geometry

Article

Abstract

A new approach to the algebraic structures related to hyperbolic geometry comes from Einstein’s special theory of relativity in 1988 (cf. Ungar, in Found Phys Lett 1:57–89, 1988). Ungar employed the binary operation of Einsteins velocity addition to introduce into hyperbolic geometry the concepts of vectors, angles and trigonometry in full analogy with Euclidean geometry (cf. Ungar, in Math Appl 49:187–221, 2005). Another approach is from Karzel for algebraization of absolute planes in the sense of Karzel et al. (Einführung in die Geometrie, 1973). In this paper we are going to develop a formulary for the Beltrami–Klein model of hyperbolic plane inside the unit circle \({\mathbb D}\) of the complex numbers \({\mathbb C}\) with geometric approach of Karzel.

Mathematics Subject Classification (2000)

Primary 51A25 Secondary 20N05 

Keywords

Hyperbolic geometry K-loop gyrogroup Beltrami–Kleinmodel special relativity 

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References

  1. 1.
    Capodaglio R.: Two loops in the absolute plane. Mitt. Math. Ges. Hamburg 23, 95–104 (2004)MATHMathSciNetGoogle Scholar
  2. 2.
    Karzel H., Wefelscheid H.: A geometric construction of the K-loop of a hyperbolic space. Geom. Dedicata 58, 227–236 (1995)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Karzel H.: Recent developments on absolute geometries and algebraization by K-loops. Discrete Math. 208/209, 387–409 (1999)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Karzel H., Sörensen K., Windelberg D.: Einführung in die Geometrie. Vandenhoeck, Göttingen (1973)MATHGoogle Scholar
  5. 5.
    Karzel H., Marchi M.: Relation between the K-loop and the defect of an absolute plane. Resultate Math. 47, 305–326 (2005)MATHMathSciNetGoogle Scholar
  6. 6.
    Ungar A.A.: Thomas rotation and the parametrization of the Lorentz transformation group. Found. Phys. Lett. 1, 57–89 (1988)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Ungar A.A.: Thomas precession and its associated grouplike structure. Am. J. Phys. 59, 824–834 (1991)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Ungar A.A.: Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces. Kluwer, Dordrecht (2001)MATHGoogle Scholar
  9. 9.
    Ungar A.A.: Einstein’s special relativity: unleashing the power of its hyperbolic geometry. Comput. Math. Appl. 49, 187–221 (2005)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ungar A.A.: Analytic Hyperbolic Geometry: Mathematical Foundations and Applications. World Scientific, Hackensack (2005)MATHCrossRefGoogle Scholar
  11. 11.
    Varičak V.: Über die Nichteuklidische Interpretation der Relativitätstheorie. Jahresber. Dtsch. Math.-Ver. 21, 103–127 (1912)Google Scholar

Copyright information

© Birkhäuser/Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran

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