Journal of Geometry

, 110:3 | Cite as

Immobilization of convex bodies in \({\mathbb {R}}^n\)

  • Anthony David GilbertEmail author
  • Saul Hannington Nsubuga
Open Access


We extend to arbitrary finite n the notion of immobilization of a convex body O in \({\mathbb {R}}^n\) by a finite set of points \({\mathcal {P}}\) in the boundary of O. Because of its importance for this problem, necessary and sufficient conditions are found for the immobilization of an n-simplex. A fairly complete geometric description of these conditions is given: as n increases from \(n = 2\), some qualitative difference in the nature of the sets \({\mathcal {P}}\) emerges.


Immobilization n-simplex contact points 

Mathematics Subject Classification

Primary 52A20 Secondary 52A15 



The authors wish to thank Elmer Rees for his generous advice towards this paper.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


  1. 1.
    Bracho, J., Fetter, H., Mayer, D., Montejano, L.: Immobilization of solids and mondriga quadratic forms. J. Lond. Math. Soc. 51(1), 189–200 (1995)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bracho, J., Montejano, L., Urrutia, J.: Immobilization of smooth convex figures. Geom. Dedic. 53(2), 119–131 (1994)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bracho, J., Montejano, L.: Rotors in triangles and tetrahedra. J. Geom. 108(3), 851–859 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Czyzowicz, J., Stojmenovic, I., Urrutia, J.: Immobilizing a shape. Int. J. Comput. Geom. Appl. 9(2), 181–206 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gilbert, A.D., Nsubuga, S.H.: Immobilization of Convex Bodies in \({\mathbb{R}}^{n}\). arXiv:1810.11381 (2018)
  6. 6.
    Kuperberg, W.: Dimacs Workshop on Polytopes. Rutgers University, Camden (1990)Google Scholar
  7. 7.
    Markenscoff, X., Ni, L., Papadimitriou, C.H.H.: The geometry of grasping. Int. J. Robot. Res. 9(1), 61–74 (1990)CrossRefGoogle Scholar
  8. 8.
    Markenscoff, X., Papadimitriou, C.H.: Optimum grip of a polygon. Int. J. Robot. Res. 8(2), 17–29 (1989)CrossRefGoogle Scholar
  9. 9.
    O’Neill, B.: Elementary Differential Geometry. Academic Press, London (1997)zbMATHGoogle Scholar
  10. 10.
    Rimon, E., Burdick, J.W.: New bounds on the number of frictionless fingers requied to immobilize. J. Robot. Syst. 12(6), 433–451 (1995)CrossRefGoogle Scholar
  11. 11.
    Rimon, E., Burdick, J.W.: Mobility of bodies in contact. I. A 2nd-order mobility index for multiple-finger grasps. IEEE Trans. Robot. Autom. 14(5), 696–708 (1998)CrossRefGoogle Scholar
  12. 12.
    Rimon, E., Burdick, J.W.: Mobility of bodies in contact. II. How forces are generated by curvature effects. IEEE Trans. Robot. Autom. 14(5), 709–717 (1998)CrossRefGoogle Scholar
  13. 13.
    van der Stappen, A.F.: Immobilization: analysis, existence, and output-sensitive synthesis a. frank van der stappen. In: Dimacs Workshop Computer Aided Design and Manufacturing, vol. 67, p. 165 (2005)Google Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Anthony David Gilbert
    • 1
    Email author
  • Saul Hannington Nsubuga
    • 2
  1. 1.School of MathematicsThe University of EdinburghEdinburghUK
  2. 2.Department of MathematicsMakerere UniversityKampalaUganda

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