# 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

- 1.T. Asano, T. Asano, L. Guibas, J. Hershberger, and H. Imai, Visibility polygon search and Euclidean shortest paths, Proc. 26th IEEE Symp. on Foundations of Computer Science, 1985, pp. 155–164.Google Scholar
- 2.R. V. Benson, Euclidean Geometry and Convexity, McGraw-Hill, 1966, pp. 97–113.Google Scholar
- 3.J. L. Bentley and A. Ottmann, Algorithms for reporting and counting geometric intersections, IEEE Trans. on Computers, Vol. C-28 (1979), pp. 643–647.CrossRefGoogle Scholar
- 4.P. Erdős and B. Grünbaum, Osculation vertices in arrangements of curves, Geometriae Dedicata 1 (1973) 322–333.MathSciNetCrossRefGoogle Scholar
- 5.B. Grünbaum, Arrangements and Spreads, Regional Conference Series in Mathematics, Vol. 10, Conference Board of the Mathematical Sciences, American Mathematical Society, Providence R.I. 1972.zbMATHGoogle Scholar
- 6.M. D. Guay and D. C. Kay, On sets having finitely many points of local nonconvexity, Israel J. Math. 8 (1970), 39–52.MathSciNetCrossRefzbMATHGoogle Scholar
- 7.L. Guibas, L. Ramshaw, and J. Stolfi, A kinetic approach to computational geometry, Proc. 24th IEEE Symp. on Foundations of Computer Science, 1983, 100–111.Google Scholar
- 8.H. Imai, M. Iri, and K. Murota, Voronoi diagram in the Laguerre geometry and its applications, Tech. Rept. RMI 83-02, Dept. of Mathematical Engineering and Instrumentation Physics, University of Tokyo.Google Scholar
- 9.D. G. Kirkpatrick, Optimal search in planar subdivisions, SIAM J. Computing 12 (1983), 28–35.MathSciNetCrossRefGoogle Scholar
- 10.D. Leven and M. Sharir, An efficient and simple motion planning algorithm for a ladder moving in two-dimensional space amidst polygonal barriers, Tech. Rept. 84-014, The Eskenasy Institute of Computer Science, Tel Aviv University, November 1984.Google Scholar
- 11.T. Lozano-Perez and M. Wesley, An algorithm for planning collision-free paths among polyhedral obstacles, Comm. ACM 22 (1979), 560–570.CrossRefGoogle Scholar
- 12.E. E. Moise, Geometric Topology in Dimension 2 and 3, Springer-Verlag, New York, 1977.CrossRefGoogle Scholar
- 13.J. Nievergelt and F. P. Preparata, Plane sweeping algorithms for intersecting geometric figures, Comm. ACM 25 (1982), 739–747.CrossRefzbMATHGoogle Scholar
- 14.C. Ó'Dúnlaing, M. Sharir, and C. Yap, Generalized Voronoi diagrams for a ladder: I. Topological analysis, Tech. Rept. 139, Computer Science Dept., Courant Institute, November 1984.Google Scholar
- 15.C. Ó'Dúnlaing, M. Sharir, and C. Yap, Generalized Voronoi diagrams for a ladder: II. Efficient construction of the diagram, Tech. Rept. 140, Computer Science Dept., Courant Institute, November 1984.Google Scholar
- 16.C. Ó'Dúnlaing and C. K. Yap, A ‘retraction’ method for planning the motion of a disc, J. of Algorithms 6 (1985) 104–111.CrossRefGoogle Scholar
- 17.T. Ottmann, P. Widmeyer, and D. Wood, A fast algorithm for boolean mask operations, Inst. f. Angewandte Mathematik und Formale Beschreibungsverfahren, D-7500 Karlsruhe, Rept. No. 112, 1982.zbMATHGoogle Scholar
- 18.J. T. Schwartz and M. Sharir, On the “Piano Movers” problem: I. The case of a two dimensional rigid polygonal body moving amidst polygonal barriers, Comm. Pure and Appl. Math. 36 (1983), 345–398.MathSciNetCrossRefzbMATHGoogle Scholar
- 19.M. Sharir, Intersection and closest-pair problems for a set of planar discs, SIAM J. Computing 14 (1985), 448–468.MathSciNetCrossRefzbMATHGoogle Scholar
- 20.S. Sifrony and M. Sharir, A new efficient motion planning algorithm for a rod in two-dimensional polygonal space, Tech. Rept. 40/85, The Eskenasy Institute of Computer Sciences, Tel Aviv University, August 1985.Google Scholar
- 21.G. T. Toussaint, On translating a set of spheres, Tech. Rept. SOCS-84.4, School of Computer Science, McGill University, March 1984.Google Scholar
- 22.E. Welzl, Constructing the visibility graph for
*n*line segments in*O*(*n*^{2}) time, Inf. Proc. Letters 20 (1985), 167–172.MathSciNetCrossRefzbMATHGoogle Scholar - 23.F. Aurenhammer, Power diagrams: properties, algorithms, and applications, Tech. Rept. F120, IIG, Tech. Univ. of Graz, Austria, 1983.Google Scholar