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The dual Jacobian of a generalised hyperbolic tetrahedron, and volumes of prisms. (English) Zbl 1358.51014
A generalized hyperbolic tetrahedron is called mildly truncated (resp. prism truncated) if no two (resp. exactly two) of the four truncating planes associated with its ultra-ideal vertices intersect. The dual Jacobian of a generalized hyperbolic tetrahedron is the Jacobian matrix of the dihedral angles considered as functions of the edge lengths. The authors derive an analytic formula for the dual Jacobian matrix of a mildly truncated generalized hyperbolic tetrahedron, and of a prism truncated generalized hyperbolic tetrahedron. The dual Jacobian of a prism truncated generalized hyperbolic tetrahedron is seen to be an analytic continuation of that of a mildly truncated generalized hyperbolic tetrahedron. The authors also obtain a volume formula for a hyperbolic \(n\)-gonal prism.

MSC:
51M20 Polyhedra and polytopes; regular figures, division of spaces
51M25 Length, area and volume in real or complex geometry
52A38 Length, area, volume and convex sets (aspects of convex geometry)
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