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Volume Frameworks and Deformation Varieties

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Automated Deduction in Geometry (ADG 2014)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9201))

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Abstract

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 hyperdges 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, we study several properties of deformation spaces, including singularities, for families of volume frameworks associated to polygons.

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Acknowledgement

This research was supported by the NSF grants CCF-1016988 and CCF-1319366.

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Authors and Affiliations

  1. Department of Mathematics, Rider University, Lawrenceville, NJ, 08648, USA

    Ciprian S. Borcea

  2. Computer Science Department, Smith College, Northampton, MA, 01063, USA

    Ileana Streinu

Authors
  1. Ciprian S. Borcea
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  2. Ileana Streinu
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Correspondence to Ileana Streinu .

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Editors and Affiliations

  1. University of Vigo, Pontevedra, Spain

    Francisco Botana

  2. University of Coimbra, Coimbra, Portugal

    Pedro Quaresma

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Borcea, C.S., Streinu, I. (2015). Volume Frameworks and Deformation Varieties. In: Botana, F., Quaresma, P. (eds) Automated Deduction in Geometry. ADG 2014. Lecture Notes in Computer Science(), vol 9201. Springer, Cham. https://doi.org/10.1007/978-3-319-21362-0_2

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  • DOI: https://doi.org/10.1007/978-3-319-21362-0_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21361-3

  • Online ISBN: 978-3-319-21362-0

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