CHARACTERIZING GENERIC GLOBAL RIGIDITY
A d-dimensional framework is a graph and a map from its vertices to E d . Such a frame-work 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 pr...
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| Veröffentlicht in: | American journal of mathematics 2010-08, Vol.132 (4), p.897-939 |
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| container_title | American journal of mathematics |
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| creator | Gortler, Steven J. Healy, Alexander D. Thurston, Dylan P. |
| description | A d-dimensional framework is a graph and a map from its vertices to E d . Such a frame-work 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. |
| doi_str_mv | 10.1353/ajm.0.0132 |
| format | Article |
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Such a frame-work 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. 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Such a frame-work 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.</abstract><cop>Baltimore, MD</cop><pub>Johns Hopkins University Press</pub><doi>10.1353/ajm.0.0132</doi><tpages>43</tpages><oa>free_for_read</oa></addata></record> |
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| subjects | Algebra Coordinate systems Euclidean space Exact sciences and technology General mathematics General, history and biography Geometry Mathematical functions Mathematical manifolds Mathematical problems Mathematical vectors Mathematics Matrices Polynomials Randomized algorithms Sciences and techniques of general use Secant function Tangents Vertices |
| title | CHARACTERIZING GENERIC GLOBAL RIGIDITY |
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