## On the number of critical free contacts of a convex polygonal object moving in two-dimensional polygonal space.(English)Zbl 0616.52009

A convex k-sided polygonal area B moving (translations and rotations) amidst polygonal areas $$A_ 1,...,A_ m$$ composed of a total number of n straight line segments is allowed to have border points incident to such a straight line segment (a touching contact), but not allowed to have common points with the interior of one of these polygonal areas $$A_ 1,...,A_ m$$. In this paper estimates of the number of positions of B are given where B makes simultaneously three touching contacts with the areas $$A_ 1,...,A_ m$$. Such a position is called a critical free contact. It is shown that the number of critical free contacts is O(kn $$\lambda$$ $${}_ s(kn))$$ where $$\lambda_ s$$ is an almost linear functions, and that there exists an example of areas $$B,A_ 1,...,A_ m$$ where the number of critical free contacts is $$\Omega (k^ 2n^ 2)$$. The applications of these results to the design of motion-planning algorithms is described in the paper of K. Kedem and the second author [”An efficient motion-planning algorithm for a convex polygonal object in two-dimensional polygonal space” (Techn. Report 253, Comput. Sci. Dep., Courant Institute) (1986)].
Reviewer: R.Klette

### MSC:

 52A37 Other problems of combinatorial convexity 52Bxx Polytopes and polyhedra 68Q25 Analysis of algorithms and problem complexity 68R99 Discrete mathematics in relation to computer science 68T20 Problem solving in the context of artificial intelligence (heuristics, search strategies, etc.)
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### References:

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