can some one give the formula of VOLUME OF HYPERBOLIC solid. with derivation.
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11-Feb-1998 — Euclidean n-cube can be triangulated. 1. Introduction. Three types of regular solid, namely the simplex, ...
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Introduction. Calculating the volume of a polyhedron in three-dimensional space is a very old and difficult problem. The first known result in this direction is due to Tartaglia (1494), who found the volume of a Euclidean tetrahedron, a result now known as the Cayley–Menger formula. It has recently been shown [1], [2] that the volume of any Euclidean polyhedron is a root of an algebraic equation whose coefficients depend on the lengths of the edges and are completely determined by the combinatorial type of the polyhedron. In hyperbolic and spherical spaces the situation is more complicated. Formulae for the volume of a biorthogonal tetrahedron in these spaces have been known since the time of Lobachevskii and Schläfli [3]. The volume of a regular tetrahedron in hyperbolic space was found in [4], and the case of a hyperbolic tetrahedron with some ideal vertices was studied in [3]. A formula for the volume of an arbitrary tetrahedron remained unknown for a long time. A general algorithm for finding such a formula was outlined in [5]. A complete solution of the problem was obtained fairly recently by several authors [6]–[8]. All the results are expressed in terms of a combination of 16 dilogarithmic or Lobachevskii functions depending on the dihedral angles of the tetrahedron and certain auxiliary parameters that are roots of a rather
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