Borcea, Ciprian S.; Streinu, Ileana Volume frameworks and deformation varieties. (English) Zbl 1437.52015 Botana, Francisco (ed.) et al., Automated deduction in geometry. 10th international workshop, ADG 2014, Coimbra, Portugal, July 9–11, 2014. Revised selected papers. Cham: Springer. Lect. Notes Comput. Sci. 9201, 21-36 (2015). Summary: A volume framework is a \((d+1)\)-uniform hypergraph with real numbers associated to its hyperedges. A realization is given by placing the vertices as points in \({\mathbb {R}}^d\) in such a way that the volumes of the simplices induced by the hyperedges have the assigned values. A framework realization is rigid if its underlying point set is determined locally up to a volume-preserving transformation, otherwise it is flexible and has a non-trivial deformation space. The study of deformation spaces is a challenging problem requiring techniques from real algebraic geometry. Complementing a previous paper on ‘Realizations of volume frameworks’ [Lect. Notes Comput. Sci. 7993, 110–119 (2013; Zbl 1306.05170)], we study several properties of deformation spaces, including singularities, for families of volume frameworks associated to polygons.For the entire collection see [Zbl 1316.68005]. MSC: 52C25 Rigidity and flexibility of structures (aspects of discrete geometry) 51M25 Length, area and volume in real or complex geometry 05C75 Structural characterization of families of graphs Keywords:volume framework; singularity; deformation space Citations:Zbl 1306.05170 PDFBibTeX XMLCite \textit{C. S. Borcea} and \textit{I. Streinu}, Lect. Notes Comput. Sci. 9201, 21--36 (2015; Zbl 1437.52015) Full Text: DOI References: [1] Asimov, L.; Roth, B., The rigidity of graphs, Trans. Amer. Math. Soc., 245, 279-289 (1978) · Zbl 0392.05026 [2] Bobenko, A., Kenyon, R., Sullivan, J.M., Ziegler, G.: Discrete differential geometry, Tech. 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