Characterizing the universal rigidity of generic frameworks
Metric Geometry
2014-06-17 v3 Optimization and Control
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.
Cite
@article{arxiv.1001.0172,
title = {Characterizing the universal rigidity of generic frameworks},
author = {Steven J. Gortler and Dylan P. Thurston},
journal= {arXiv preprint arXiv:1001.0172},
year = {2014}
}
Comments
18 pages, v2: updates throughout; v3: published version