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Möbius transformation; hyperbolic geometry; gyrogroups; gyrovector spaces and hyperbolic trigonometry
In [Comput. Math. Appl. 41 (2001), 135--147], A. A. Ungar employs the Möbius gyrovector spaces for the introduction of the hyperbolic trigonometry. This Ungar's work plays a major role in translating some theorems from Euclidean geometry to corresponding theorems in hyperbolic geometry. In this paper we explore the theorems of Stewart and Steiner in the Poincaré disc model of hyperbolic geometry.
[1] Ungar A.A.: Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession. Fundamental Theories of Physics, 117, Kluwer Academic Publishers Group, Dordrecht, 2001. MR 1978122 | Zbl 0972.83002
[2] Ungar A.A.: Analytic Hyperbolic Geometry,: Mathematical Foundations and Applications. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. MR 2169236 | Zbl 1089.51003
[3] Ungar A.A.: From Pythagoras to Einstein: The hyperbolic Pythagorean theorem. Found. Phys. 28 (1998), no. 8, 1283--1321. DOI 10.1023/A:1018874826277 | MR 1653451
[4] Ungar A.A.: The hyperbolic square and Möbius transformations. Banach J. Math. Anal. 1 (2007), no. 1, 101--116. MR 2350199 | Zbl 1129.30027
[5] Ungar A.A.: Hyperbolic trigonometry and its application in the Poincaré ball model of hyperbolic geometry. Comput. Math. Appl. 41 (2001), 135--147. DOI 10.1016/S0898-1221(01)85012-4 | MR 1808511 | Zbl 0988.51017
[6] Ungar A.A.: Einstein's velocity addition law and its hyperbolic geometry. Comput. Math. App. 53 (2007), 1228--1250. DOI 10.1016/j.camwa.2006.05.028 | MR 2327676 | Zbl 1132.83301
[7] G.S. Birman and Ungar A.A.: The Hyperbolic Derivative in the Poincaré ball model of Hyperbolic Geometry. Journal of. Math. Anal. and Appl. 254, 2001, 321--333. DOI 10.1006/jmaa.2000.7280 | MR 1807904
[8] Ungar A.A.: Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. MR 2169236 | Zbl 1147.83004
[9] Ungar A.A.: From Möbius to gyrogroups. Amer. Math. Monthly 115 (2008), no. 2, 138--144. MR 2384266 | Zbl 1152.30045
[10] Bhumkar K.: Interactive visualization of Hyperbolic geometry using the Weierstrass model. A Thesis submitted to the Faculty of the Graduate School of University of Minnesota, 2006.
[11] Demirel O., Soytürk E.: The hyperbolic Carnot theorem in the Poincaré disc model of hyperbolic geometry. Novi Sad J. Math. 38 (2008), no. 2, 33--39. MR 2526025
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