Geodesic
The volume of a trirectangular hyperbolic tetrahedron
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 1, pp. 275-284.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider a three-parameter family of tetrahedra in the hyperbolic space, which three edges at one vertex are pairwise orthogonal. It is convenient to determine such tetrahedra by the lengths of these edges. We obtain relatively simple formulas for them expressing the volume and the surface area. This allows us to find normalized volume and investigate its asymptotics.
Keywords: hyperbolic volume, normalized volume, Poincaré upper half-space model, hyperbolic tetrahedron, infinite cone.
Mots-clés : trirectangular tetrahedron 
@article{SEMR_2023_20_1_a20,
     author = {N. Abrosimov and S. Stepanishchev},
     title = {The volume of a trirectangular hyperbolic tetrahedron},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {275--284},
     publisher = {mathdoc},
     volume = {20},
     number = {1},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2023_20_1_a20/}
}
TY  - JOUR
AU  - N. Abrosimov
AU  - S. Stepanishchev
TI  - The volume of a trirectangular hyperbolic tetrahedron
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2023
SP  - 275
EP  - 284
VL  - 20
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2023_20_1_a20/
LA  - en
ID  - SEMR_2023_20_1_a20
ER  - 
%0 Journal Article
%A N. Abrosimov
%A S. Stepanishchev
%T The volume of a trirectangular hyperbolic tetrahedron
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2023
%P 275-284
%V 20
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2023_20_1_a20/
%G en
%F SEMR_2023_20_1_a20
N. Abrosimov; S. Stepanishchev. The volume of a trirectangular hyperbolic tetrahedron. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 1, pp. 275-284. http://geodesic.mathdoc.fr/item/SEMR_2023_20_1_a20/

[1] N.V. Abrosimov, G.A. Baigonakova, “Hyperbolic octahedron with $mmm$-symmetry”, Sib. Èlectron. Mat. Izv., 10 (2013), 123–140 | MR | Zbl

[2] N.V. Abrosimov, A.D. Mednykh, “Area and volume in non-Euclidean geometry”, Eighteen essays in non-Euclidean geometry, IRMA Lect. Math. Theor. Phys., 29, eds. Alberge Vincent et al., Eur. Math. Soc., Zürich, 2019, 151–189 | DOI | MR | Zbl | Zbl

[3] N.V. Abrosimov, A.D. Mednykh, “Volumes of polytopes in spaces of constant curvature”, Rigidity and symmetry, Fields Inst. Commun., 70, eds. Connelly Robert et al., Springer, New York, 2014, 1–26 | DOI | MR | Zbl

[4] N.V. Abrosimov, B. Vuong, “Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths”, J. Knot Theory Ramifications, 30:10 (2021), 2140007 | DOI | MR | Zbl

[5] D.V. Alekseevskii, E.B. Vinberg, A.S. Solodovnikov, “Geometry of spaces of constant curvature”, Geometry, v. II, Encycl. Math. Sci., 29, Spaces of constant curvature, Springer, Berlin, 1993, 1–138 | MR | Zbl

[6] G.A. Baigonakova, Calculation of volumes of polyhedra in non-Euclidean geometry, PhD thesis, Gorno-Altaisk, 2012

[7] S. Bilinski, “Zur Begründung der elementaren Inhaltslehre in der hyperbolischen Ebene”, Math. Ann., 180 (1969), 256–268 | DOI | MR | Zbl

[8] Yu. Cho, H. Kim, “On the volume formula for hyperbolic tetrahedra”, Discrete Comput. Geom., 22:3 (1999), 347–366 | DOI | MR | Zbl

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

[10] W. Fenchel, Elementary geometry in hyperbolic space, Walter de Gruyter, Berlin etc, 1989 | MR | Zbl

[11] F. Lannér, “On complexes with transitive groups of automorphisms”, Meddel. Lunds Univ. Mat. Semin., 11 (1950) | MR | Zbl

[12] N.I. Lobatschefskij (N.I. Lobachevsky), Imaginäre Geometrie und ihre Anwendung auf einige Integrale, German translation by H. Liebmann, Teubner, Leipzig, 1904

[13] B. McConnell, The laws of cosines for non-Euclidean tetrahedra, 2022 http://daylateanddollarshort.com/mathdocs/The-Laws-of-Cosines-for-Non-Euclidean-Tetrahedra.pdf

[14] J. Murakami, M. Yano, “On the volume of a hyperbolic and spherical tetrahedron”, Comm. Anal. Geom., 13:2 (2005), 379–400 | DOI | MR | Zbl

[15] O. Nemoul, N. Mebarki, Volume and boundary face area of a regular tetrahedron in a constant curvature space, 2018, arXiv: 1803.10809 [physics.gen-ph]

[16] G. Sforza, “Ricerche di estensionimetria negli spazi metrico-proiettivi”, Modena Mem. (3), 8 (1907), 21–66 (Appendice) | Zbl

[17] P. Stäckel, Geometrische Untersuchungen von Wolfgang Bolyai und Johann Bolyai, v. I.-II, B.G. Teubner, Leipzig, 1913 | Zbl

[18] A. Ushijima, “A volume formula for generalized hyperbolic tetrahedra”, Non-Euclidean Geometries, Mathematics and Its Applications, 581, eds. Prékopa András et al., Springer, New York, 2006, 249–265 | DOI | MR | Zbl