Application of Contemporary Proof of the Sforza Formula to Computation of Volumes of Hyperbolic Tetrahedra of Special Kind

Cover Page

Abstract


In this paper, we use the contemporary proof (by Abrosimov and Mednykh) of the Sforza formula for volume of an arbitrary non-Euclidean tetrahedron to derive new formulas that express volumes of hyperbolic tetrahedra of special kind (orthoschemes and tetrahedra with the symmetry group S 4) via dihedral angles.


About the authors

V. A. Krasnov

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: krasnov_va@rudn.university
Moscow, Russia

References

  1. Абросимов Н.В., Выонг Хыу Б. Объем гиперболического тетраэдра с группой симметрий S4// Тр. Ин-та мат. и мех. УрО РАН. - 2017. -23, № 4. - С. 7-17.
  2. Винберг Э.Б. Объемы неевклидовых многогранников// Усп. мат. наук. - 1993. -48, № 2. - С. 17-46.
  3. Лобачевский Н.И. Воображаемая геометрия// В сб.: «Полное собр. соч. Т. 3». - M.-Л., 1949.
  4. Abrosimov N.V., Mednykh A.D. Volumes of polytopes in spaces of constant curvature// Rigidity and Symmetry. - 2014. -70. - С. 1-26.
  5. Bolyai J. Appendix. The theory of space// В сб.: «Janos Bolyai». - Budapest, 1987.
  6. Cho Yu., Kim H. On the volume formula for hyperbolic tetrahedra// Discrete Comput. Geom. - 1999. - 22. - С. 347-366.
  7. Derevnin D.A., Mednykh A.D. A formula for the volume of hyperbolic tetrahedron// Russ. Math. Surv. - 2005. -60, № 2. - С. 346-348.
  8. Kellerhals R. On the volume of hyperbolic polyhedra// Math. Ann. - 1989. -285. - С. 541-569.
  9. Kneser H. Der Simplexinhalt in der nichteuklidischen Geometrie// Deutsche Math. - 1936. -1. - С. 337- 340.
  10. Milnor J. Hyperbolic geometry: the first 150 years// Bull. Am. Math. Soc. - 1982. -6, № 1. - С. 307- 332.
  11. Murakami J. The volume formulas for a spherical tetrahedron// Arxiv. - 2011. - 1011.2584v4.
  12. Murakami J., Ushijima A. A volume formula for hyperbolic tetrahedra in terms of edge lengths// J. Geom. - 2005. -83, № 1-2. - С. 153-163.
  13. Murakami J., Yano M. On the volume of a hyperbolic and spherical tetrahedron// Comm. Anal. Geom. - 2005. -13. - С. 379-400.
  14. Schlafli L.¨ Theorie der vielfachen Kontinuitat// В сб.: «Gesammelte mathematische Abhandlungen». -¨ Basel: Birkhauser, 1950.¨
  15. Sforza G. Spazi metrico-proiettivi// Ric. Esten. Different. Ser. - 1906. -8, № 3. - С. 3-66.
  16. Ushijima A. A volume formula for generalized hyperbolic tetrahedra// Non-Euclid. Geom. - 2006. - 581. - С. 249-265.

Statistics

Views

Abstract - 88

PDF (Russian) - 43

Cited-By


PlumX

Dimensions

Refbacks

  • There are currently no refbacks.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies