Non-Euclidean Geometries pp 249-265 | Cite as
A Volume Formula for Generalised Hyperbolic Tetrahedra
Abstract
A generalised hyperbolic tetrahedron 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 generalised tetrahedra. There are two key tools for the proof; one is the 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 generalised hyperbolic simplex with respect to their dihedral angles.
Keywords
Hyperbolic tetrahedron Gram matrix volume formulaPreview
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Bibliography
- [BH]Johannes Böhm and Eike Hertel, Polyedergeometrie in n-dimensionalen Räumen konstanter Krümmung, Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften. Mathematische Reihe 70, Birkhäuser Verlag, 1981.Google Scholar
- [CK]Yunhi Cho and Hyuk Kim, On the Volume Formula for Hyperbolic Tetrahedra, Discrete & Computational Geometry 22 (1999), 347–366.MathSciNetCrossRefMATHGoogle Scholar
- [HM]Uffe Haagerup and Hans J. Munkholm, Simplices of maximal volume in hyperbolic n-space, Acta Mathematica 147 (1981), 1–11.MathSciNetCrossRefMATHGoogle Scholar
- [Hs]Wu-Yi Hsiang, On infinitesimal symmetrization and volume formula for spherical or hyperbolic tetrahedrons, The Quarterly Journal of Mathematics, Oxford, Second Series 39 (1988), 463–468.MATHMathSciNetGoogle Scholar
- [Ka]R. M. Kashaev, The Hyperbolic Volume of Knots from the Quantum Dilogarithm, Letters in Mathematical Physics 39 (1997), 269–275.MATHMathSciNetCrossRefGoogle Scholar
- [Ke]Ruth Kellerhals, On the volume of hyperbolic polyhedra, Mathematische Annalen 285 (1989), 541–569.MATHMathSciNetCrossRefGoogle Scholar
- [Ki]Anatol N. Kirillov, Dilogarithrn identities, Progress of Theoretical Physics. Supplement 118 (1995), 61–142.MATHMathSciNetGoogle Scholar
- [Lo]N. I. Lobatschefskij, Imaginäre Geometrie und ihre Anwendung auf einige Integrale, translated into German by H. Liebmann (Leipzig, 1904).Google Scholar
- [Lu]Feng Luo, On a Problem of Fenchel, Geometriae Dedicata 64 (1997), 277–282.MATHMathSciNetCrossRefGoogle Scholar
- [Mi]John Milnor, Hyperbolic geometry: the first 150 years, Bulletin (New Series) of the American Mathematical Society 6 (1982), 9–24.MATHMathSciNetCrossRefGoogle Scholar
- [MY]Jun Murakami and Masakazu Yano, On the volume of a hyperbolic tetrahedron, available at http://www.f.waseda.jp/murakami/papers/tetrahedronrev4.pdf.Google Scholar
- [Pe]Norbert Peyerimhoff, Simplices of maximal volume or minimal total edge length in hyperbolic space, Journal of the London Mathematical Society (2) 66 (2002), 753–768.MATHMathSciNetGoogle Scholar
- [Pr]V. V. Prasolov, Problems and Theorems in Linear Algebra, Translations of Mathematical Monographs 134, American Mathematical Society, 1994.Google Scholar
- [Us]Akira Ushijima, The Tilt Formula for generalised Simplices in Hyperbolic Space, Discrete & Computational Geometry 28 (2002), 19–27.MATHMathSciNetCrossRefGoogle Scholar
- [Ve]Andrei Vesnin, On Volumes of Some Hyperbolic 3-manifolds, Lecture Notes Series 30, Seoul National University, 1996.Google Scholar
- [Vi]E. B. Vinberg (Ed.), Geometry II, Encyclopaedia of Mathematical Sciences 29, Springer-Verlag, 1993.Google Scholar