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**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.) |

### Keywords:

convex polygon; geometric complexity; computational geometry; critical free contact; motion-planning
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\textit{D. Leven} and \textit{M. Sharir}, Discrete Comput. Geom. 2, 255--270 (1987; Zbl 0616.52009)

### References:

[1] | M. Atallah, Dynamic computational geometry,Proceedings of the 24th Symposium on Foundations of Computer Science, 92-99, 1983. |

[2] | Hart, S.; Sharir, M., Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes, Combinatorica, 6, 175-201, (1986) · Zbl 0636.05003 |

[3] | K. Kedem and M. Sharir, An Efficient Motion-Planning Algorithm for a Convex Polygonal Object in Two-Dimensional Polygonal Space, Technical Report 253, Computer Science Department, Courant Institute, 1986. · Zbl 0688.68039 |

[4] | Kedem, K.; Livne, R.; Pach, J.; Sharir, M., On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles, Discrete Comput. Geom., 1, 59-71, (1986) · Zbl 0594.52004 |

[5] | D. Leven and M. Sharir, An efficient and simple motion-planning algorithm for a ladder moving in two-dimensional space amidst polygonal barriers,Proceedings of the ACM Symposium on Computational Geometry, 221-227, 1985 (also to appear inJ. Algorithms). |

[6] | D. Leven and M. Sharir, Planning a Purely Translational Motion for a Convex Object in Two-Dimensional Space Using Generalized Voronoi Diagrams, Technical Report 34/85, The Eskenasy Institute of Computer Science, Tel Aviv University, 1985 (also to appear inDiscrete Comput. Geom.). · Zbl 0606.52002 |

[7] | Schwartz, J. T.; Sharir, M., On the piano movers’ problem: I. The case of a two-dimensional rigid polygonal body moving amidst polygonal barriers, Comm. Pure Appl. Math., 36, 345-398, (1983) · Zbl 0554.51007 |

[8] | M. Sharir, Almost Linear Upper Bounds on the Length of Generalized Davenport-Schinzel Sequences, Technical Report 29/85, The Eskenasy Institure of Computer Science, Tel-Aviv University, 1985 (also to appear inCombinatorica7 (1987).) |

[9] | M. Sharir, Improved Lower Bounds on the Length of Davenport-Schinzel Sequences, Technical Report 204, Computer Science Department, Courant Institute, 1986. · Zbl 0672.05015 |

[10] | Szemeredi, E., On a problem by Davenport and Schinzel, Acta Arith., 25, 213-224, (1974) · Zbl 0291.05003 |

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