# The theorems of Stewart and Steiner in the Poincaré disc model of hyperbolic geometry

Commentationes Mathematicae Universitatis Carolinae (2009)

- Volume: 50, Issue: 3, page 359-371
- ISSN: 0010-2628

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topDemirel, Oğuzhan. "The theorems of Stewart and Steiner in the Poincaré disc model of hyperbolic geometry." Commentationes Mathematicae Universitatis Carolinae 50.3 (2009): 359-371. <http://eudml.org/doc/33320>.

@article{Demirel2009,

abstract = {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.},

author = {Demirel, Oğuzhan},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {Möbius transformation; hyperbolic geometry; gyrogroups; gyrovector spaces and hyperbolic trigonometry; Möbius transformation; hyperbolic geometry; gyrogroups; gyrovector space; hyperbolic trigonometry},

language = {eng},

number = {3},

pages = {359-371},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {The theorems of Stewart and Steiner in the Poincaré disc model of hyperbolic geometry},

url = {http://eudml.org/doc/33320},

volume = {50},

year = {2009},

}

TY - JOUR

AU - Demirel, Oğuzhan

TI - The theorems of Stewart and Steiner in the Poincaré disc model of hyperbolic geometry

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2009

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 50

IS - 3

SP - 359

EP - 371

AB - 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.

LA - eng

KW - Möbius transformation; hyperbolic geometry; gyrogroups; gyrovector spaces and hyperbolic trigonometry; Möbius transformation; hyperbolic geometry; gyrogroups; gyrovector space; hyperbolic trigonometry

UR - http://eudml.org/doc/33320

ER -

## References

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- Ungar A.A., Analytic Hyperbolic Geometry,: Mathematical Foundations and Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. Zbl1089.51003MR2169236
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- Ungar A.A., The hyperbolic square and Möbius transformations, Banach J. Math. Anal. 1 (2007), no. 1, 101--116. Zbl1129.30027MR2350199
- Ungar A.A., 10.1016/S0898-1221(01)85012-4, Comput. Math. Appl. 41 (2001), 135--147. Zbl0988.51017MR1808511DOI10.1016/S0898-1221(01)85012-4
- Ungar A.A., 10.1016/j.camwa.2006.05.028, Comput. Math. App. 53 (2007), 1228--1250. Zbl1132.83301MR2327676DOI10.1016/j.camwa.2006.05.028
- G.S. Birman and Ungar A.A., 10.1006/jmaa.2000.7280, Journal of. Math. Anal. and Appl. 254, 2001, 321--333. MR1807904DOI10.1006/jmaa.2000.7280
- Ungar A.A., Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. Zbl1147.83004MR2169236
- Ungar A.A., From Möbius to gyrogroups, Amer. Math. Monthly 115 (2008), no. 2, 138--144. Zbl1152.30045MR2384266
- 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.
- 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. MR2526025
- http://zimmer.csufresno.edu/~larryc/proofs/proofs.contradict.html, .
- { http://www.cut-the-knot.org/pythagoras/index.shtml}, .

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