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Volumes of Polytopes in Spaces of Constant Curvature

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Rigidity and Symmetry

Part of the book series: Fields Institute Communications ((FIC,volume 70))

Abstract

We overview volume calculations for polyhedra in Euclidean, spherical and hyperbolic spaces. We prove the Sforza formula for the volume of an arbitrary tetrahedron in H 3 and S 3. We also present some results, which provide a solution for the Seidel problem on the volume of non-Euclidean tetrahedron. Finally, we consider a convex hyperbolic quadrilateral inscribed in a circle, horocycle or one branch of equidistant curve. This is a natural hyperbolic analog of the cyclic quadrilateral in the Euclidean plane. We find several versions of the Brahmagupta formula for the area of such quadrilateral. We also present a formula for the area of a hyperbolic trapezoid.

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Notes

  1. 1.

    By an obtuse tetrahedron we mean a tetrahedron with at least one dihedral angle \(> \frac{\pi } {2}\).

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Acknowledgements

The work is supported by Russian Foundation for Basic Research (grants 12-01-00210, 12-01-33058, 12-01-31006 and 13-01-00513) and Council for Grants of President of the Russian Federation (grants MK-4447.2012.1 and SS-921.2012.1).

The authors would like to express their gratitude to a referee for careful consideration of the paper and useful remarks and suggestions.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia

    Nikolay Abrosimov & Alexander Mednykh

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  1. Nikolay Abrosimov
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  2. Alexander Mednykh
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Correspondence to Nikolay Abrosimov .

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Editors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, New York, USA

    Robert Connelly

  2. Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada

    Asia Ivić Weiss

  3. Department of Mathematics & Statistics, York University, Toronto, Ontario, Canada

    Walter Whiteley

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Abrosimov, N., Mednykh, A. (2014). Volumes of Polytopes in Spaces of Constant Curvature. In: Connelly, R., Ivić Weiss, A., Whiteley, W. (eds) Rigidity and Symmetry. Fields Institute Communications, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0781-6_1

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