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

TL;DR

This paper derives an explicit formula for the dual Jacobian matrix of hyperbolic tetrahedra, including truncated types, and extends volume calculations to hyperbolic n-gonal prisms, linking Jacobian analysis with volume formulas.

Contribution

It provides the first analytic expressions for the dual Jacobian of generalized hyperbolic tetrahedra and extends volume formulas to hyperbolic n-gonal prisms using Jacobian analysis.

Findings

Derived analytic dual Jacobian formulas for truncated hyperbolic tetrahedra.

Established volume formulas for hyperbolic n-gonal prisms.

Linked Jacobian behavior with volume calculations via the Schl"afli formula.

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 Schl\"afli formula.