Results in Mathematics

, Volume 63, Issue 1–2, pp 229–239 | Cite as

Defect and Area in Beltrami–Klein Model of Hyperbolic Geometry

Article

Abstract

Ungar (Beyond the Einstein addition law and its gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrouector Spaces, 2001; Comput Math Appl 49:187–221, 2005; Comput Math Appl 53, 2007) introduced into hyperbolic geometry the concept of defect based on relativity addition of A. Einstein. Another approach is from Karzel (Resultate Math. 47:305–326, 2005) for the relation between the K-loop and the defect of an absolute plane in the sense (Karzel in Einführung in die Geometrie, 1973). Our main concern is to introduce a systematical exact definition for defect and area in the Beltrami–Klein model of hyperbolic geometry. Combining the ideas and methods of Karzel and Ungar give an elegant concept for defect and area in this model. In particular we give a rigorous and elementary proof for the defect formula stated (Ungar in Comput Math Appl 53, 2007). Furthermore, we give a formulary for area of circle in the Beltrami–Klein model of hyperbolic geometry.

Mathematics Subject Classification (2000)

Primary 51A25 Secondary 20N05 

Keywords

Area defect absolute geometry hyperbolic geometry Beltrami–Klein model special relativity 

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References

  1. 1.
    Karzel H., Sörensen K., Windelberg D.: Einführung in die Geometrie. Vandenhoeck, Göttingen (1973)MATHGoogle Scholar
  2. 2.
    Karzel H.: Recent developments on absolute geometries and algebraization by K-loops. Discret. Math. 208/209, 387–409 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Karzel H., Marchi M.: Relation between the K-loop and the defect of an absolute plane. Resultate Math. 47, 305–326 (2005)MathSciNetMATHGoogle Scholar
  4. 4.
    Karzel H., Wefelscheid H.: A geometric construction of the K-loop of a hyperbolic space. Geom. Dedicata. 58, 227–236 (1995)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Taherian S-Gh.: On algebraic structures related to Beltrami–Klein model of hyperbolic geometry. Results. Math. 57, 205–219 (2010)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Ungar A.A.: Thomas precession and its associated grouplike structure. Am. J. Phys. 59, 824–834 (1991)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ungar, A.A.: Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: the Theory of Gyrogroups and Gyrouector Spaces. Fundamental Theories of Physics, vol. 117. Kluwer Academic Publishers Group, Dordrecht (2001)Google Scholar
  8. 8.
    Ungar A.A.: Einstein’s special relativity: unleashing the power of its hyperbolic geometry. Comput. Math. Appl. 49, 187–221 (2005)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Ungar, A.A.: Einstein’s velocity addition law and its hyperbolic geometry. Comput. Math. Appl. 53 (2007)Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran

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