Characterizing the Universal Rigidity of Generic Frameworks

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...

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Bibliographic Details
Published inDiscrete & computational geometry Vol. 51; no. 4; pp. 1017 - 1036
Main Authors Gortler, Steven J., Thurston, Dylan P.
Format Journal Article
LanguageEnglish
Published Boston Springer US 01.06.2014
Springer Nature B.V
Subjects
Applied mathematics
Combinatorics
Computational geometry
Computational Mathematics and Numerical Analysis
Geometry
Graphs
Mathematical programming
Mathematics
Mathematics and Statistics
Rigidity
Stresses
Texts
Semidefinite programming
Graph rigidity
Strict complementarity
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Summary: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.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
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ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-014-9590-9