A volume formula for hyperbolic tetrahedra in terms of edge lengths by Murakami and Ushijima

In this paper the authors  give a closed formula for volumes of generic hyperbolic tetrahedra in terms of edge lengths.

Volume formulae for hyperbolic tetrahedra

For complex numbers z, a1, a2, . . . , a6, define a complex valued function U = U(a1, a2, a3, a4, a5, a6, z) as follows:

hypvolequli2 where Li2 is the dilogarithm function

hypvolequli2a

Then define z+ and z- by,

hypvolequz

where

hypvolequq

Define a complex-valued function V(a1, a2, a3, a4, a5, a6,) as follows:

hypvolequV

For the  tetrahedron

hypvolfig1

hypvoltheorem2.1

in this case:

hypvolequzz

where

hypvolequqa

and

hypvolequ2.1

is the Gram matrix defined in terms of dihedral angles.

Then

hypvoltheorem2.2

and

hypqq

where,

ql

and

Gl is the Gram matrix in terms of edge lengths defined as

gl

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