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Local and Infinitesimal Rigidities

In: Euclidean Distance Matrices and Their Applications in Rigidity Theory

Author

Listed:
  • Abdo Y. Alfakih

    (University of Windsor, Department of Mathematics and Statistics)

Abstract

This chapter focuses on the problems of local rigidity and infinitesimal rigidity of bar frameworks. These problems have a long and rich history going back at least as far as Cauchy [51]. The main tools in tackling these problems are the rigidity matrix R and the dual rigidity matrix R ̄ $$\bar{R}$$ . While R is defined in terms of the underlying graph G and configuration p, R ̄ $$\bar{R}$$ is defined in terms of the complement graph Ḡ $$\bar{G}$$ and Gale matrix Z. Gale matrix Nonetheless, both matrices R and R ̄ $$\bar{R}$$ carry the same information. The chapter concludes with a discussion of generic local rigidity in dimension 2, where the local rigidity problem reduces to a purely combinatorial one depending only on graph G. The literature on the theory of local and infinitesimal rigidities is vast [59, 57, 66, 97, 194]. However, in this chapter, we confine ourselves to discussing only the basic results and the results pertaining to EDMs.

Suggested Citation

  • Abdo Y. Alfakih, 2018. "Local and Infinitesimal Rigidities," Springer Books, in: Euclidean Distance Matrices and Their Applications in Rigidity Theory, chapter 0, pages 185-210, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-97846-8_9
    DOI: 10.1007/978-3-319-97846-8_9
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