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

TL;DR

This paper derives an explicit integral volume formula for hyperbolic tetrahedra based on edge lengths, extending classical formulas and providing new geometric insights.

Contribution

It introduces a new volume formula for hyperbolic tetrahedra expressed via edge lengths and establishes conditions for their existence in hyperbolic space.

Findings

Derived necessary and sufficient conditions for hyperbolic tetrahedron existence.

Established relations between dihedral angles and edge lengths.

Presented a new integral volume formula in terms of edge matrix.

Abstract

We consider a compact hyperbolic tetrahedron of a general type. It is a convex hull of four points called vertices in the hyperbolic space $\mathbb{H}^3$. 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 $\mathbb{H}^3$. 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.