Approximate motion planning and the complexity of the boundary of the union of simple geometric figures
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We study rigid motions of a rectangle amidst polygonal obstacles. The best known algorithms for this problem have a running time of Ω(n 2), wheren is the number of obstacle corners. We introduce thetightness of a motion-planning problem as a measure of the difficulty of a planning problem in an intuitive sense and describe an algorithm with a running time ofO((a/b · 1/ɛcrit + 1)n(logn)2), wherea ≥b are the lengths of the sides of a rectangle and ɛcrit is the tightness of the problem. We show further that the complexity (= number of vertices) of the boundary ofn bow ties (see Figure 1) isO(n). Similar results for the union of other simple geometric figures such as triangles and wedges are also presented.
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- Approximate motion planning and the complexity of the boundary of the union of simple geometric figures
Volume 8, Issue 1-6 , pp 391-406
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- Computational geometry
- Motion planning
- Boundary complexity
- Combinatorial geometry
- Analysis of algorithms
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- Author Affiliations
- 1. Fachbereich Mathematik, Freie Universität Berlin, Arnimallee 2-6, 1000, Berlin 33, Federal Republic of Germany
- 2. Max-Planck-Institut für Informatik, Im Stadtwald, W-6600, Saarbrücken, Federal Republic of Germany