Discrete & Computational Geometry

, Volume 12, Issue 3, pp 367–384

Objects that cannot be taken apart with two hands

  • J. Snoeyink
  • J. Stolfi
Article

Abstract

It has been conjectured that every configurationC of convex objects in 3-space with disjoint interiors can be taken apart by translation with two hands: that is, some proper subset ofC can be translated to infinity without disturbing its complement. We show that the conjecture holds for five or fewer objects and give a counterexample with six objects. We extend the counterexample to a configuration that cannot be taken apart with two hands using arbitrary isometries (rigid motions).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. Chvátal,Linear Programming. Freeman, San Francisco, CA, 1983.MATHGoogle Scholar
  2. 2.
    S. T. Coffin.The Puzzling World of Polyhedral Dissections, Oxford University Press, Oxford, 1991.Google Scholar
  3. 3.
    R. H. Crowell and R. H. Fox.Introduction to Knot Theory, Blaisdell, New York, 1965.Google Scholar
  4. 4.
    R. Dawson. On removing a ball without disturbing the others.Mathematics Magazine, 57 (1): 27–30, 1984.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    N. G. de Bruijn. Problems 17 and 18.Nieuw Archief voor Wikskunde, 2: 67, 1954, Answers inWiskundige Opgaven met de oplossingen, 20: 19–20, 1955.Google Scholar
  6. 6.
    L. Fejes-Toth and A. Heppes. Uber stabile Körpersysteme,Compositio Mathematica, 15 (2): 119–126, 1963.MathSciNetGoogle Scholar
  7. 7.
    J. B. Fraleigh.A First Course in Abstract Algebra. Addison-Wesley, Reading, MA, 1982.Google Scholar
  8. 8.
    L. J. Guibas and F. F. Yao. On translating a set of rectangles.Proceedings of the 12th Annual ACM Symposium on Theory of Computing, pages 154–160, 1980.Google Scholar
  9. 9.
    P. McMullen and G. C. Shephard,Convex Polytopes and the Upper Bound Conjecture. Cambridge University Press, Cambridge, 1971.MATHGoogle Scholar
  10. 10.
    L. S. H. d. Mello.Computer-Aided Mechanical Assembly Planning. Kluwer, Boston, 1991.CrossRefGoogle Scholar
  11. 11.
    B. Mishra, J. T. Schwart, and M. Sharir. On the existence and synthesis of multifinger positive grips.Algorithmica, 2: 541–558, 1987.MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    B. K. Natarajan. On planning assembles.Proceedings of the Fourth Annual ACM Symposium on Computational Geometry, pages 299–308, 1988.Google Scholar
  13. 13.
    C. H. Papadimitriou and K. Steiglitz,Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Englewood Cliffs, NJ, 1982.MATHGoogle Scholar
  14. 14.
    J. Pertin-Troccaz. Grasping: A state of the art. In O. Khatib, J. J. Craig, and T. Lozano-Perez, editors,The Robotics Review 1, pages 71–98. MIT Press, Cambridge, MA, 1989.Google Scholar
  15. 15.
    F. P. Preparata. Planar point location revisited.International Journal of Foundations of Computer Science, 1 (1): 71–86, 1990.MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    J. Snoeyink. Video: Objects that cannot be taken apart with two hands.Proceedings of the Ninth Annual ACM Symposium on Computational Geometry, page 405, 1993. Video Review of Computational Geometry also available as DEC SRC Report 101. 4:39 animation.Google Scholar
  17. 17.
    R. H. Wilson and T. Matsui. Partitioning an assembly for infinitestimal motions in translation and rotation.IEEE International Conference on Intellegent Robots and Systems, pages 1311–1318, 1992.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1994

Authors and Affiliations

  • J. Snoeyink
    • 1
  • J. Stolfi
    • 2
  1. 1.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada
  2. 2.Computer Science DepartmentUniversidade Estadual de CampinasCampinasBrazil

Personalised recommendations