# On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles

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## 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 } ^{=1m} *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.

## Keywords

Voronoi Diagram Jordan Curve Simple Polygon Convex Corner Motion Planning Problem## References

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