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The hyperbolic triangle defect. (English) Zbl 1062.51013
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 5th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 5–12, 2003. Sofia: Bulgarian Academy of Sciences (ISBN 954-84952-8-7/pbk). 225-236 (2004).
The author introduces hyperbolic trigonometry by means of gyrovector spaces (for a comprehensive introduction into the theory of gyrogroups and gyrovector spaces see the author’s book Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession. The Theory of Gyrogroups and Gyrovector Spaces (2001; Zbl 0972.83002)). A special kind of gyrovector spaces is used to describe the Poincaré ball model of hyperbolic geometry [Comput. Math. Appl. 41, No.1–2, 135–147 (2001; Zbl 0988.51017)]. Within this setting the author derives formulae for the hyperbolic law of sines and cosines, the Pythagorean Theorem, and the angular defect of a hyperbolic triangle.
MSC:
51M09Elementary problems in hyperbolic and elliptic geometries