On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles Klara Kedem Ron Livne János Pach Micha Sharir Article First Online: 01 March 1986 Received: 21 February 1985 Revised: 25 November 1985 DOI :
10.1007/BF02187683

Cite this article as: Kedem, K., Livne, R., Pach, J. et al. Discrete Comput Geom (1986) 1: 59. doi:10.1007/BF02187683 Abstract Let γ_{1} ,..., γ_{m} bem simple Jordan curves in the plane, and letK _{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 ofK =∩_{i} ^{=1m} K _{i} contains at most max(2,6m − 12) intersection points of the curvesγ _{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 ofm 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 polygonB amidst several (convex) polygonal obstaclesA _{1} ,...,A _{m} . Assuming that the number of corners ofB is fixed, the algorithm presented here runs in timeO (n log^{2} n ), wheren is the total number of corners of theA _{i} 's.

Work on this paper by the second author has been supported in part by a grant from the Bat-Sheva Fund at Israel. Work by the fourth author has been supported in part by a grant from the U.S.-Israeli Binational Science Foundation.

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.

2.

R. V. Benson, Euclidean Geometry and Convexity, McGraw-Hill, 1966, pp. 97–113.

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.

CrossRef Google Scholar 4.

P. Erdős and B. Grünbaum, Osculation vertices in arrangements of curves, Geometriae Dedicata 1 (1973) 322–333.

MathSciNet CrossRef Google 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.

MATH Google 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.

MathSciNet CrossRef MATH Google 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.

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.

9.

D. G. Kirkpatrick, Optimal search in planar subdivisions, SIAM J. Computing 12 (1983), 28–35.

MathSciNet CrossRef Google 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.

11.

T. Lozano-Perez and M. Wesley, An algorithm for planning collision-free paths among polyhedral obstacles, Comm. ACM 22 (1979), 560–570.

CrossRef Google Scholar 12.

E. E. Moise, Geometric Topology in Dimension 2 and 3, Springer-Verlag, New York, 1977.

CrossRef Google Scholar 13.

J. Nievergelt and F. P. Preparata, Plane sweeping algorithms for intersecting geometric figures, Comm. ACM 25 (1982), 739–747.

CrossRef MATH Google 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.

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.

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.

CrossRef Google 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.

MATH Google 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.

MathSciNet CrossRef MATH Google Scholar 19.

M. Sharir, Intersection and closest-pair problems for a set of planar discs, SIAM J. Computing 14 (1985), 448–468.

MathSciNet CrossRef MATH Google 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.

21.

G. T. Toussaint, On translating a set of spheres, Tech. Rept. SOCS-84.4, School of Computer Science, McGill University, March 1984.

22.

E. Welzl, Constructing the visibility graph for

n line segments in

O (

n
^{2} ) time, Inf. Proc. Letters 20 (1985), 167–172.

MathSciNet CrossRef MATH Google Scholar 23.

F. Aurenhammer, Power diagrams: properties, algorithms, and applications, Tech. Rept. F120, IIG, Tech. Univ. of Graz, Austria, 1983.

Google Scholar © Springer-Verlag New York Inc. 1986

Authors and Affiliations Klara Kedem Ron Livne János Pach Micha Sharir 1. School of Mathematical Sciences Tel Aviv University Israel 2. Courant Institute of Mathematical Sciences New York University New York USA 3. Mathematical Institute of the Hungarian Academy of Sciences Hungary