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Universal and Dimensional Rigidities

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Euclidean Distance Matrices and Their Applications in Rigidity Theory

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Abstract

In this chapter, we study the universal rigidity problem of bar frameworks and the related problem of dimensional rigidity. The main tools in tackling these two problems are the Cayley configuration spectrahedron \(\mathcal{F}\), defined in (8.10), and Ω, the stress matrix, defined in (8.13). The more general problem of universally linked pair of nonadjacent nodes is also studied and the results are interpreted in terms of the Strong Arnold Property and the notion of nondegeneracy in semidefinite programming.

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Notes

  1. 1.

    Affine motions are often defined without the nonsingularity assumption on A. However, this assumption is more convenient for our purposes.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Windsor, Windsor, ON, Canada

    Abdo Y. Alfakih

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  1. Abdo Y. Alfakih
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Alfakih, A.Y. (2018). Universal and Dimensional Rigidities. In: Euclidean Distance Matrices and Their Applications in Rigidity Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-97846-8_10

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