On integral expressions for volumes of hyperbolic tetrahedra
Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 49 (2013), pp. 89-98.

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In this paper, we derive the Derevnin–Mednykh integral expression for the volume of a hyperbolic tetrahedron in terms of dihedral angles from the Murakami–Yano expression. As a consequence, we derive an expression for the volume of a hyperbolic tetrahedron in terms of edge lengths.
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V. A. Krasnov. On integral expressions for volumes of hyperbolic tetrahedra. Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 49 (2013), pp. 89-98. http://geodesic.mathdoc.fr/item/CMFD_2013_49_a1/

[1] Abrosimov N. V., “Ob ob'emakh mnogogrannikov v prostranstve postoyannoi krivizny”, Vestn. Kemerovskogo gos. un-ta, 2011, no. 3/1(47), 7–13 | MR

[2] Alekseevskii D. V., Vinberg E. B., Solodovnikov A. S., “Geometriya prostranstv postoyannoi krivizny”, Itogi nauki i tekhn. Ser. Sovr. probl. mat. Fundam. naprav., 29, VINITI, M., 1988, 5–146 | MR | Zbl

[3] Derevnin D. A., Mednykh A. D., Pashkevich M. G., “Ob'em simmetrichnogo tetraedra v giperbolicheskom i sfericheskom prostranstvakh”, Sib. mat. zh., 45:5 (2004), 1022–1031 | MR | Zbl

[4] Derevnin D. A., Mednykh A. D., “Ob'em kuba Lamberta v sfericheskom prostranstve”, Mat. zametki, 86:2 (2009), 190–201 | DOI | MR | Zbl

[5] Lobachevskii N. I., “Voobrazhaemaya geometriya”, Polnoe sobr. soch., v. 3, M.–L., 1949

[6] Cho Yu., Kim H., “On the volume formula for hyperbolic tetrahedra”, Discrete Comput. Geom., 22 (1999), 347–366 | DOI | MR | Zbl

[7] Coxeter H. S. M., “The functions of Schläfli and Lobatschefsky”, Quartely J. Math. (Oxford), 6 (1935), 13–29 | DOI | Zbl

[8] Derevnin D. A., Mednykh A. D., “A formula for the volume of hyperbolic tetrahedron”, Rusian Math. Surveys, 60:2 (2005), 346–348 | DOI | MR | Zbl

[9] Kellerhals R., “On the volume of hyperbolic polyhedra”, Math. Ann., 285 (1989), 541–569 | DOI | MR | Zbl

[10] Mednykh A. D., Parker J., Vesnin A. Y., “On hyperbolic polyhedra arising as convex cores of quasi-Fuchsian punctured torus groups”, Bol. Soc. Mat. Mex., III Ser., 10:3 (2004), 357–381 | MR | Zbl

[11] Milnor J., “Hyperbolic geometry: the first 150 years”, Bull. Amer. Math. Soc., 6:1 (1982), 307–332 | DOI | MR

[12] Murakami J., Ushijima A., “A volume formula for hyperbolic tetrahedra in terms of edge lengths”, J. Geom., 83:1–2 (2005), 153–163 | DOI | MR | Zbl

[13] Murakami J., Yano M., “On the volume of a hyperbolic and spherical tetrahedron”, Comm. Anal. Geom., 13 (2005), 379–400 | DOI | MR | Zbl

[14] Schläfli L., “Theorie der vielfachen Kontinuität”, Gesammelte mathematische Abhandlungen, Birkhäuser, Basel, 1950

[15] Sforza G., “Spazi metrico-proiettivi”, Ric. Esten. Different. Ser., 8:3 (1906), 3–66

[16] Ushijima A., “A volume formula for generalized hyperbolic tetrahedra”, Non-Euclidean Geometries, 581 (2006), 249–265 | DOI | MR | Zbl