Hyperbolic Tetrahedron: Volume Calculation with Application to the Proof of the Schläfli Formula
https://doi.org/10.18255/1818-1015-2013-6-149-161
Abstract
We propose a new approach to the problem of calculations of volumes in the Lobachevsky space, and we apply this method to tetrahedra. Using some integral formulas, we present an explicit formula for the volume of a tetrahedron in the function of the coordinates of its vertices as well as in the function of its edge lengths. Finally, we give a direct analitic proof of the famous Schläfli formula for tetrahedra.
About the Author
I. Kh. Sabitov
Lomonosov Moscow State University;
P.G. Demidov Yaroslavl State University
Russian Federation
Russian Federation
профессор;
ведущий научный сотрудник,
Sovetskaya str., 14, Yaroslavl, 150000, Russia
References
1. Abrosimov N.V., Mednykh A.D. Volumes of polytops in spaces of constant curvature // Fields Institut Communications, 2013 (in press) // arXiv:1302.4919 [math.MG].
2. Сабитов И.Х. Алгебраические методы решения многогранников // Успехи мат. наук. 2011. 66:3. С. 3–66. (English transl.: Sabitov I.Kh. Algebraic methods for solution of polyhedra // Russian Math. Surveys. 2011. 66:3. P. 445–505.)
3. Sabitov I.Kh. On an approach to the calculation of volumes in spaces of constant curvature // Yaroslavl Intrernatinal Conference «Geometry, Topology and Applications», September 23–27: Abstracts. Yaroslavl, 2013. P. 98–100.
4. Сабитов И.Х. Об одном методе вычисления объемов в пространствах постоянной кривизны // Крымская Международная Математическая Конференция. Судак, 22 сентября – 4 октября 2013: Сборник тезисов. Судак, 2013. Т. 2. С. 77–78. (Sabitov I.Kh. Ob odnom metode vychisleniya obyomov v prostranstvah postoyannoi krivizny // Crimea International Mathematical Conference. Sudak, Ukraine, September, 22—October, 4, 2013: Book of Abstracts. Sudak, 2013. Vol. 2. P. 77–78 [in Russian].)
5. Сабитов И.Х. Об одном методе вычисления объемов тел // Сибирские электронные математические известия. 2013. Т. 10. С. 615–626. (Sabitov I.Kh. Ob odnom metode vychisleniya obyemov tel // Sibirskie elektronnye matematicheskie izvestiya. 2013. Vol. 10. P. 615–626 [in Russian].)
6. Murakami J. and Ushijima A. A volume formula for hyperbolic tetrahedral in terms of edge lengths // Journal of Geometry. 2005. Vol. 83, No 1–2. P. 153—163.
Review
For citations:
Sabitov I.Kh. Hyperbolic Tetrahedron: Volume Calculation with Application to the Proof of the Schläfli Formula. Modeling and Analysis of Information Systems. 2013;20(6):149-161. (In Russ.) https://doi.org/10.18255/1818-1015-2013-6-149-161
Views:
970
Email this article 