Elementary formulas for a hyperbolic tetrahedron
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We derive some elementary formulas expressing the relation between the dihedral angles and edge lengths of a tetrahedron in hyperbolic space.
Original Russian Text Copyright © 2006 Mednykh A. D. and Pashkevich M. G.
The authors were supported by the Russian Foundation for Basic Research (Grant 06-01-00153), the INTAS (Grant 03-51-3663), the Project Fondecyt (Grants 7050189 and 1060378), and the Grant-in-Aid 17-05045 of the Japan Society for the Promotion of Sciences.
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 4, pp. 831–841, July–August, 2006.
- Cho Yu. and Kim H., “On the volume formula for hyperbolic tetrahedra,” Discrete Comput. Geom., 22, No. 3, 347–366 (1999). CrossRef
- Murakami J. and Yano M., On the Volume of Hyperbolic Tetrahedron [Preprint] (2001). Available at http: //faculty.web.waseda.ac.jp/murakami/papers/tetrahedronrev3.pdf.
- Ushijima A., A Volume Formula for Generalized Hyperbolic Tetrahedra [Preprint] (2002). Available at http: //www.math.titech.ac.jp/Users/ushijima/welcome-e.html.
- Derevnin D. A. and Mednykh A. D., “A formula for the volume of a hyperbolic tetrahedron,” Russian Math. Surveys, 60, No. 2, 346–348 (2005). CrossRef
- Coolidge J. L., The Elements of Non-Euclidean Geometry, Clarendon Press, Oxford (1909).
- Derevnin D. A., Mednykh A. D., and Pashkevich M. G., “On the volume of a symmetric tetrahedron in hyperbolic and spherical spaces,” Siberian Math. J., 45, No. 5, 840–848 (2004). CrossRef
- Rivin I., “A multidimensional theorem of sines.” Available at http://www.arxiv.org/abs/math.GM/0211261.
- Ratcliffe J. G., Foundations of Hyperbolic Manifolds, Springer-Verlag, New York (1994).
- Fenchel W., Elementary Geometry in Hyperbolic Space, de Gruyter, Berlin; New York (1989).
- Eriksson F., “The law of sines for tetrahedra and n-simplices,” Geom. Dedicata, 7, No. 1, 71–80 (1978). CrossRef
- Prasolov V. V., Problems and Theorems in Linear Algebra, Amer. Math. Soc., Providence, RI (1994).
- Prasolov V. V. and Sossinsky A. B., Knots, Links, Braids, and 3-Manifolds [in Russian], MTsNMO, Moscow (1997).
- Buser P., Geometry and Spectra of Compact Riemann Surfaces, Birkhauser, Boston; Basel; Berlin (1992) (Progress in Mathematics; 106).
- Elementary formulas for a hyperbolic tetrahedron
Siberian Mathematical Journal
Volume 47, Issue 4 , pp 687-695
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- hyperbolic tetrahedron
- n-dimensional hyperbolic simplex
- law of sines
- law of cosines
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