Stably placing piecewise smooth objects

  • Chao-Kuei Hung
  • Doug Ierardi
Submitted Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1148)


Given a piecewise smooth object, its stable poses consist of all the orientations into which other initial orientations of the object will eventually converge under dissipative forces. The capture region for each stable pose is the set of initial orientations converging to the stable pose in question.

We employ duality to solve these two related problems. Our approach produces non-trivial combinatorial bounds on the complexity of these problems as well as asymptotically efficient algorithms. It also allows us to remove the non-singularity constraints in Kriegman's previous work on the latter problem, and to enumerate the degenerate cases in a systematic way. Our analysis leads to a significant reduction in the algebraic complexity for objects consisting of quadratic surface patches cut by planes. The practical value of this approach is demonstrated by the implementation of an efficient approximation algorithm for this subclass of objects.


Surface Patch Potential Energy Function Lower Envelope Quadratic Surface Primal Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [BG92]
    J. W. Bruce and P. J. Giblin. Curves and Singularities. Cambridge University Press, 2nd edition, 1992.Google Scholar
  2. [Cai68]
    Stewart Scott Cairns. Introductory Topology. Ronald Press Company, 1968.Google Scholar
  3. [Col75]
    G. E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In Second GI Conference on Automata Theory and Formal Languages, volume 33, pages 134–183. Srpinger-Verlag, 1975.Google Scholar
  4. [Hun95]
    Chao-Kuei Hung. Convex Hull of Surface Patches: Construction and Applications. PhD thesis, Computer Science Department, University of Southern California, 1995.Google Scholar
  5. [IK93]
    Doug Ierardi and Dexter Kozen. Parallel resultant computation. In John H. Reif, editor, Synthesis of Parallel Algorithms. Morgan Kaufmann, 1993.Google Scholar
  6. [Kri91]
    David J. Kriegman. Computing stable poses of piecewise smooth objects. In Computer Vision Graphics Image Processing: Image Understanding, 1991.Google Scholar
  7. [Kri95]
    David J. Kriegman. Let them fall where they may: Capture regions of curved objects and polyhedra. Technical Report 9508, Yale University, June 1995. Submitted to the International Journal of Robotics Research.Google Scholar
  8. [Lay92]
    Steven R. Lay. Convex Sets and Their Applications. John Wiley & Sons, Inc., 1992. original edition 1982.Google Scholar
  9. [Man93]
    Dinesh Manocha. Solving polynomial systems for curves, surface, and solid modeling. In Proc. of ACM/SIGGRAPH, 1993.Google Scholar
  10. [Sha93]
    Micha Sharir. Almost tight upper bounds for lower envelopes in higher dimensions. In Symposium on Foundations of Computer Science, pages 498–507, 1993.Google Scholar
  11. [SS85]
    Otto Schreier and Emanuel Sperner. Projective Geometry of n Dimensions. Chelsea Publishing Company, New York, 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Chao-Kuei Hung
    • 1
  • Doug Ierardi
    • 1
  1. 1.Department of Computer ScienceUniversity of Southern CaliforniaUSA

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