Abstract
A volume framework is a (d+1)-uniform hypergraph together with real numbers associated to its edges. A realization is a labeled point set in R d for which the volumes of the d-dimensional simplices corresponding to the hypergraph edges have the pre-assigned values. A framework realization (shortly, a framework) is rigid if its underlying point set is determined locally up to affine volume-preserving transformations. If it ceases to be rigid when any volume constraint is removed, it is called minimally rigid.
We present a number of results on volume frameworks: a counterexample to a conjectured combinatorial characterization of minimal rigidity and a first enumerative lower bound. We also give upper bounds for the number of realizations of generic minimally rigid volume frameworks, based on degrees of naturally associated Grassmann varieties.
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Borcea, C.S., Streinu, I. (2013). Realizations of Volume Frameworks. In: Ida, T., Fleuriot, J. (eds) Automated Deduction in Geometry. ADG 2012. Lecture Notes in Computer Science(), vol 7993. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40672-0_8
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DOI: https://doi.org/10.1007/978-3-642-40672-0_8
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