TY - JOUR

T1 - Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths

AU - Abrosimov, Nikolay

AU - Vuong, Bao

N1 - The authors are grateful to Alexander Mednykh for useful remarks and comments. This work was supported by the Ministry of Science and Higher Education of Russia (Agreement No. 075-02-2021-1392). Publisher Copyright: © 2021 World Scientific Publishing Company.

PY - 2021/9/1

Y1 - 2021/9/1

N2 - We consider a compact hyperbolic tetrahedron of a general type. It is a convex hull of four points called vertices in the hyperbolic space H3. It can be determined by the set of six edge lengths up to isometry. For further considerations, we use the notion of edge matrix of the tetrahedron formed by hyperbolic cosines of its edge lengths. We establish necessary and sufficient conditions for the existence of a tetrahedron in H3. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formula expressing the volume of a hyperbolic tetrahedron in terms of the edge lengths. The latter volume formula can be regarded as a new version of classical Sforza's formula for the volume of a tetrahedron but in terms of the edge matrix instead of the Gram matrix.

AB - We consider a compact hyperbolic tetrahedron of a general type. It is a convex hull of four points called vertices in the hyperbolic space H3. It can be determined by the set of six edge lengths up to isometry. For further considerations, we use the notion of edge matrix of the tetrahedron formed by hyperbolic cosines of its edge lengths. We establish necessary and sufficient conditions for the existence of a tetrahedron in H3. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formula expressing the volume of a hyperbolic tetrahedron in terms of the edge lengths. The latter volume formula can be regarded as a new version of classical Sforza's formula for the volume of a tetrahedron but in terms of the edge matrix instead of the Gram matrix.

KW - edge matrix

KW - Hyperbolic tetrahedron

KW - hyperbolic volume

KW - Sforza's formula

UR - http://www.scopus.com/inward/record.url?scp=85121263044&partnerID=8YFLogxK

U2 - 10.1142/S0218216521400071

DO - 10.1142/S0218216521400071

M3 - Article

AN - SCOPUS:85121263044

VL - 30

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 10

M1 - 2140007

ER -