A Volume Formula for Generalized Hyperbolic Tetrahedra∗
نویسنده
چکیده
A generalized hyperbolic tetrahedra is a polyhedron (possibly noncompact) with finite volume in hyperbolic space, obtained from a tetrahedron by the polar truncation at the vertices lying outside the space. In this paper it is proved that a volume formula for ordinary hyperbolic tetrahedra devised by J. Murakami and M. Yano can be applied to such ones. There are two key tools for the proof; one is so-called Schläfli’s differential formula for hyperbolic polyhedra, and the other is a necessary and sufficient condition for given numbers to be the dihedral angles of a generalized hyperbolic simplex with respect to their dihedral angles.
منابع مشابه
symmetries of hyperbolic tetrahedra
We give a rigorous geometric proof of the Murakami-Yano formula for the volume of a hyperbolic tetrahedron. In doing so, we are led to consider generalized hyperbolic tetrahedra, which are allowed to be non-convex, and have vertices ‘beyond infinity’; and we uncover a group, which we call 22.5K, of 23040 = 30 · 12 · 2 scissors-class-preserving symmetries of the space of (suitably decorated) gen...
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