The dual Jacobian of a generalised tetrahedron, and volumes of prisms

Abstract

We derive an analytic formula for the dual Jacobian matrix of a generalised hyperbolic tetrahedron. Two cases are considered: a mildly truncated and a prism truncated tetrahedron. The Jacobian for the latter arises as an analytic continuation of the former, that falls in line with a similar behaviour of the corresponding volume formulae. Also, we obtain a volume formula for a hyperbolic $n$-gonal prism: the proof requires the above mentioned Jacobian, employed in the analysis of the edge lengths behaviour of such a prism, needed later for the Schläfli formula.

arXiv comment26 pages, 11 figures, 1 table; to appear in Tokyo Journal of Mathematics

References in corpus (4)

• The volume formulas for a spherical tetrahedron
• Trigonometry in extended hyperbolic space and extended de Sitter space
• 3-Dimensional Schlaefli Formula and Its Generalization
• Calculus of generalized hyperbolic tetrahedron