Characterizing generic global rigidity

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Characterizing generic global rigidity

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Title: Characterizing generic global rigidity
Author: Gortler, Steven J.; Healy, A.; Thurston, D

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Citation: Steven J. Gortler, Alexander D. Healy, and Dylan P. Thurston. 2010. “Characterizing Generic Global Rigidity.” American Journal of Mathematics 132 (4): 897–939. doi:10.1353/ajm.0.0132.
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Abstract: A d-dimensional framework is a graph and a map from its vertices to E^d. Such a framework is globally rigid if it is the only framework in E^d with the same graph and edge lengths, up to rigid motions. For which underlying graphs is a generic framework globally rigid? We answer this question by proving a conjecture by Connelly, that his sufficient condition is also necessary: a generic framework is globally rigid if and only if it has a stress matrix with kernel of dimension d+1, the minimum possible. An alternate version of the condition comes from considering the geometry of the length-squared mapping l: the graph is generically locally rigid iff the rank of l is maximal, and it is generically globally rigid iff the rank of the Gauss map on the image of l is maximal. We also show that this condition is efficiently checkable with a randomized algorithm, and prove that if a graph is not generically globally rigid then it is flexible one dimension higher.
Published Version: doi:10.1353/ajm.0.0132
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