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Generic Global Rigidity in Complex and Pseudo-Euclidean Spaces

  • Steven J. GortlerEmail author
  • Dylan P. Thurston
Chapter
Part of the Fields Institute Communications book series (FIC, volume 70)

Abstract

In this paper we study the property of generic global rigidity for frameworks of graphs embedded in d-dimensional complex space and in a d-dimensional pseudo-Euclidean space (\(\mathbb{R}^{d}\) with a metric of indefinite signature). We show that a graph is generically globally rigid in Euclidean space iff it is generically globally rigid in a complex or pseudo-Euclidean space. We also establish that global rigidity is always a generic property of a graph in complex space, and give a sufficient condition for it to be a generic property in a pseudo-Euclidean space. Extensions to hyperbolic space are also discussed.

Keywords

Metric geometry 

Subject Classifications

52C25 51M10 05C62 

Notes

Acknowledgements

We would like to thank Robert Connelly, Bill Jackson, John Owen, Louis Theran, and for helpful conversations and suggestions. We would especially like to thank Walter Whiteley for sharing with us his explanation of the Pogorelov map.

References

  1. 1.
    Basu, S., Pollack, R., Roy, M.: Algorithms in Real Algebraic Geometry, vol. 10. Springer, New York (2006)zbMATHGoogle Scholar
  2. 2.
    Bochnak, J., Coste, M., Roy, M.F.: Real Algebraic Geometry. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  3. 3.
    Connelly, R.: On Generic Global Rigidity. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, pp. 147–155. American Mathematical Society, Providence (1991)Google Scholar
  4. 4.
    Connelly, R.: Generic global rigidity. Discret. Comput. Geomtr. 33(4), 549–563 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Connelly, R., Whiteley, W.: Global rigidity: the effect of coning. Discret. Comput. Geometr. 43(4), 717–735 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Frenkel, P.: New constructions in classical invariant theory. PhD thesis, Budapest University of Technology and Economics (2007)Google Scholar
  7. 7.
    Gortler, S., Healy, A., Thurston, D.: Characterizing generic global rigidity. Am. J. Math. 132(4), 897–939 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Jackson, B., Owen, J.: The number of equivalent realisations of a rigid graph. Arxiv preprint arXiv:1204.1228 (2012)Google Scholar
  9. 9.
    Parshin, A., Shafarevich, I.: Algebraic geometry IV: linear algebraic groups, invariant theory, vol. 55. Springer, Berlin/New York (1994)CrossRefzbMATHGoogle Scholar
  10. 10.
    Saliola, F., Whiteley, W.: Some notes on the equivalence of first-order rigidity in various geometries. Arxiv preprint arXiv:0709.3354 (2007)Google Scholar
  11. 11.
    Whitney, H.: Elementary structure of real algebraic varieties. Ann. Math. 66(2), 545–556 (1957)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Computer SciencesHavard School of Engineering and Applied SciencesCambridgeUSA
  2. 2.Department of MathematicsIndiana UniversityBloomingtonUSA

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