Abstract
We say that a polygon P is immobilized by a set of points I on its boundary if any rigid motion of P in the plane causes at least one point of I to penetrate the interior of P. Three immobilization points are always sufficient for a polygon with vertices in general positions, but four points are necessary for some polygons with parallel edges. An O(n log n) algorithm that finds a set of 3 points that immobilize a given polygon with vertices in general positions is suggested. The algorithm becomes linear for convex polygons. Some results are generalized for d-dimensional polytopes, where 2d points are always sufficient and sometimes necessary to immobilize. When the polytope has vertices in general position d+1 points are sufficient to immobilize.
This research is partially supported by NSERC
Preview
Unable to display preview. Download preview PDF.
References
A. Aggarwal, L.J. Guibas, J. Saxe, and P.W. Shor, A linear-time algorithm for computing the Voronoi diagram of a convex polygon, Discr. Comput. Geom., 4, 591–604, 1989.
B. S. Baker, S. Fortune, E. Grosse, Stable Prehension with Three Fingers, Proc. 17th Symp. on Theory of Computing, 1985, pp. 114–120.
J. Czyzowicz, I. Stojmenovic, J. Urrutia, Immobilizing a shape, Rapport de recherche RR 90/11-18, Dept. Informatique, Université du Québec à Hull, November 1990 (earlier version available as Technical Report TR-90-37, Dept. of Computer Science, Univ. of Ottawa, July 1990).
H. Edelsbrunner, Algorithms in Combinatorial Geometry, Springer-Verlag, 1987.
S. Fortune, A sweep-line algorithm for Voronoi diagrams, Proc. 2nd ACM Symp. on Computational Geometry, 1986, 313–322.
D.G. Kirkpatrick, Efficient computation of continuous skeletons, Proc. 20th IEEE Symp. on Found. of Comp. Sci., 1979, 18–27.
W. Kuperberg, DIMACS Workshop on Polytopes, Rutgers University, Jan. 1990.
X. Markenscoff and Ch. H. Papadimitriou, Optimal grip of a polygon, Int. J. Robotics Research, 8, 2, 1989, 17–29.
X. Markenscoff, L. Ni and Ch. H. Papadimitriou, The Geometry of Grasping, Int. J. Robotics Research, 9, 1, 1990, 61–74.
B. Mishra, J.T. Schwartz and M. Sharir, On the Existence and Synthesis of Multifinger Positive Grips, Algorithmica (1987) 2: 541–548.
B. Mishra and N. Silver, Some Discussions of Static Gripping and Its Stability, IEEE Transactions on Systems, Man and Cybernetics, 19, 4, July/August, 1989, pp. 783–796.
J. O'Rourke, Computational geometry column 9, SIGACT News, 21, 1, 1990, 18–20, Winter 1990, #74.
C.K. Yap, An O(n log n) algorithm for the Voronoi diagram of a set of simple curve segments, Discr. and Comput. Geom., 2, 1987, 365–393.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Czyzowicz, J., Stojmenovic, I., Urrutia, J. (1991). Immobilizing a polytope. In: Dehne, F., Sack, JR., Santoro, N. (eds) Algorithms and Data Structures. WADS 1991. Lecture Notes in Computer Science, vol 519. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0028264
Download citation
DOI: https://doi.org/10.1007/BFb0028264
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-54343-5
Online ISBN: 978-3-540-47566-8
eBook Packages: Springer Book Archive