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Local, Dimensional and Universal Rigidities: A Unified Gram Matrix Approach

  • A. Y. AlfakihEmail author
Chapter
First Online:
Part of the Fields Institute Communications book series (FIC, volume 70)

Abstract

This chapter is a unified treatment, based on projected Gram matrices (PGMs), of the problems of local, dimensional, and universal rigidities of bar frameworks. This PGM-based approach makes these problems amenable to semidefinite programming methodology; and naturally gives rise to results expressed in terms of Gale matrices. We survey known results emphasizing numerical examples and proofs which highlight the salient aspects of this approach.

Keywords

Bar-and-joint frameworks Infinitesimal rigidity Rigidity matrix Dual rigidity matrix Dimensional rigidity Universal rigidity Gram matrix and Gale transform 

Subject Classifications

52C25 05C50 15A57 
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Notes

Acknowledgements

Research supported by the Natural Sciences and Engineering Research Council of Canada.The author would like to thank an anonymous referee for his/her remarks which improved the presentation of this chapter.

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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of WindsorWindsorCanada

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