## Abstract

Let γ_{1},..., γ_{m} be m simple Jordan curves in the plane, and let K_{1},..., K_{m} be their respective interior regions. It is shown that if each pair of curves γ_{i}, γ_{j}, i ≠j, intersect one another in at most two points, then the boundary of K=∩_{i}^{=1 m}K_{i} contains at most max(2,6 m - 12) intersection points of the curves γ_{1}, and this bound cannot be improved. As a corollary, we obtain a similar upper bound for the number of points of local nonconvexity in the union of m Minkowski sums of planar convex sets. Following a basic approach suggested by Lozano Perez and Wesley, this can be applied to planning a collision-free translational motion of a convex polygon B amidst several (convex) polygonal obstacles A_{1},..., A_{m}. Assuming that the number of corners of B is fixed, the algorithm presented here runs in time O (n log^{2}n), where n is the total number of corners of the A_{i}'s.

Original language | English |
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Pages (from-to) | 59-71 |

Number of pages | 13 |

Journal | Discrete and Computational Geometry |

Volume | 1 |

Issue number | 1 |

DOIs | |

State | Published - 1 Dec 1986 |

Externally published | Yes |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics