Abstract
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.
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References
J. W. Bruce and P. J. Giblin. Curves and Singularities. Cambridge University Press, 2nd edition, 1992.
Stewart Scott Cairns. Introductory Topology. Ronald Press Company, 1968.
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.
Chao-Kuei Hung. Convex Hull of Surface Patches: Construction and Applications. PhD thesis, Computer Science Department, University of Southern California, 1995.
Doug Ierardi and Dexter Kozen. Parallel resultant computation. In John H. Reif, editor, Synthesis of Parallel Algorithms. Morgan Kaufmann, 1993.
David J. Kriegman. Computing stable poses of piecewise smooth objects. In Computer Vision Graphics Image Processing: Image Understanding, 1991.
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.
Steven R. Lay. Convex Sets and Their Applications. John Wiley & Sons, Inc., 1992. original edition 1982.
Dinesh Manocha. Solving polynomial systems for curves, surface, and solid modeling. In Proc. of ACM/SIGGRAPH, 1993.
Micha Sharir. Almost tight upper bounds for lower envelopes in higher dimensions. In Symposium on Foundations of Computer Science, pages 498–507, 1993.
Otto Schreier and Emanuel Sperner. Projective Geometry of n Dimensions. Chelsea Publishing Company, New York, 1985.
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© 1996 Springer-Verlag Berlin Heidelberg
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Hung, CK., Ierardi, D. (1996). Stably placing piecewise smooth objects. In: Lin, M.C., Manocha, D. (eds) Applied Computational Geometry Towards Geometric Engineering. WACG 1996. Lecture Notes in Computer Science, vol 1148. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0014492
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DOI: https://doi.org/10.1007/BFb0014492
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