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

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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.

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

    Sayed-Ghahreman Taherian

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  1. Sayed-Ghahreman Taherian
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Correspondence to Sayed-Ghahreman Taherian.

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To Professor Helmut Karzel on the occasion of his 86th birthday.

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Taherian, SG. On Algebraic Structures Related to Beltrami–Klein Model of Hyperbolic Geometry. Results. Math. 57, 205–219 (2010). https://doi.org/10.1007/s00025-010-0021-9

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  • DOI: https://doi.org/10.1007/s00025-010-0021-9

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