The Dual Jacobian of a Generalised Hyperbolic Tetrahedron, and Volumes of Prisms

Alexander Kolpakov (1) , Jun Murakami (2)

1 Department of Mathematics [University of Toronto]
2 Departement of Mathematics [Waseda]

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.

Mots clés: hyperbolic polyhedron; Gram matrix; volume

Domaines: Géométrie métrique [math.MG]

Alexander Kolpakov, Jun Murakami. The Dual Jacobian of a Generalised Hyperbolic Tetrahedron, and Volumes of Prisms. Tokyo Journal of Mathematics, 2016, 39, pp.45 - 67. ⟨10.3836/tjm/1471873312⟩. ⟨hal-01396110⟩