Characterizing the universal rigidity of generic frameworks

Details Characterizing the universal rigidity of generic frameworks Characterizing the universal rigidity of generic frameworks

2009-12-31

arxiv.org/abs/1001.0172v2

Mathematics

Metric Geometry

18 pages, v2: updates throughout

Scientific paper

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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