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Defect and Area in Beltrami–Klein Model of Hyperbolic Geometry

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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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156, Iran

    Mahfouz Rostamzadeh & Sayed-Ghahreman Taherian

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  1. Mahfouz Rostamzadeh
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  2. Sayed-Ghahreman Taherian
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Correspondence to Sayed-Ghahreman Taherian.

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Dedicated to Hans Joachim Kroll on the occasion of his 69th birthday

The authors thank Prof. Helmut Karzel for his useful helps and suggestions.

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Rostamzadeh, M., Taherian, SG. Defect and Area in Beltrami–Klein Model of Hyperbolic Geometry. Results. Math. 63, 229–239 (2013). https://doi.org/10.1007/s00025-011-0191-0

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  • DOI: https://doi.org/10.1007/s00025-011-0191-0

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