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arXiv · 1001.0172

Characterizing the universal rigidity of generic frameworks

Steven J. Gortler·Dylan P. Thurston
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Abstract

A framework is a graph and a map from its vertices to E^d (for some d). A framework is universally rigid if any framework in any dimension with the same graph and edge lengths is a Euclidean image of it. We show that a generic universally rigid framework has a positive semi-definite stress matrix of maximal rank. Connelly showed that the existence of such a positive semi-definite stress matrix is sufficient for universal rigidity, so this provides a characterization of universal rigidity for generic frameworks. We also extend our argument to give a new result on the genericity of strict complementarity in semidefinite programming.

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Steven J. Gortler, Dylan P. Thurston. 2014-06-16. Characterizing the universal rigidity of generic frameworks. https://doi.org/10.1007/s00454-014-9590-9

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