# Separability of sets of polygons

## Abstract

Recently, a growing interest in problems dealing with the movability of objects has been observed. Motion problems are manifold due to the variety of areas in which they may occur; among these areas are e.g. robotics, computer graphics, etc. One motion problem class recently being investigated is the separability problem.

The **separability problem** is as follows: Given a set ℙ={P_{1}, ...,P_{M}} of M n-vertex polygons in the Euclidean plane, with pairwise non-intersecting interiors. The polygons are to be separated by an arbitrarily large distance through a sequence of M-1 translations while collisions with the polygons yet to be separated are to be avoided. The **uni-directional separability problem** arises, when all polygons are translated in a common direction; the more general problem of separability through translations in arbitrary directions is referred to as the **multi-directional separability problem**.

Here a simple, novel approach is presented for solving an array of uni-directional and multi-directional separability problems for sets of arbitrary simple polygons. The algorithms presented here provide efficient solutions to these problems and when applied to restricted polygon classes further improvements in the time complexities are achieved.

## Keywords

Euclidean Plane Separability Problem Visibility Hull Segment Tree Separability Decidability## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Chazelle, B., T. Ottmann, E. Soisalon-Soinen, and D. Wood, "The complexity and decidability of Separation™", Tech. Rept. CS-83-34, Data Structuring Group, University of Waterloo, November 1983.Google Scholar
- [2]Dehne, F. and J.-R. Sack, "Separability of Sets of Polygons", Tech. Rept. SCS-82, School of Computer Science, Carleton University, Ottawa, Oct. 1985.Google Scholar
- [3]Guibas, L.J. and F.F. Yao, "On translating a set of rectangles",
*Proceedings 12*^{th}*Annual ACM Symposium on Theory of Computing*, 1980, pp. 154–160.Google Scholar - [4]Mansouri, M. and G.T. Toussaint, "Translation queries for convex polygons",
*Proceedings IASTED International Symposium on Robotics and Automation'85*, Lugano, Switzerland, June 1985.Google Scholar - [5]Mehlhorn, K.,
*Data Structures and Algorithms 3: Multidimensional Searching and Computational Geometry*, Springer Verlag, Heidelberg, 1984.Google Scholar - [6]Nurmi, Otto, "On translating a set of objects in 2 and 3 dimensional spaces", Bericht 141, Institut für angewandte Informatik und formale Beschreibungsverfahren, Universität Karlsruhe, Karlsruhe, Federal Republic of Germany.Google Scholar
- [7]Ottmann, T. and P. Widmayer, "On translating a set of line segments",
*Computer Vision, Graphics and Image Processing 24*, 1983, pp. 382–389.Google Scholar - [8]Sack, J.-R. and G.T. Toussaint, "Movability of objects", IEEE International Symposium on Information Theory, St. Jovite, Canada, September 1983.Google Scholar
- [9]Sack, J.-R. and G.T. Toussaint, "Translating polygons in the plane",
*Proceedings STACS'85*, Saarbrücken, Federal Republic of Germany, January 1985, pp. 310–321.Google Scholar - [10]Sack, J.-R., "A linear-time algorithm for computing separability of pairs of polygons", unpublished notes, Carleton University, Ottawa, 1986.Google Scholar
- [11]Sedgewick, R.,
*Algorithms*, Addison-Wesley, Reading MA, 1983.Google Scholar - [12]Toussaint, G.T. and J.-R. Sack, "Some new results on moving polygons in the plane",
*Proceedings Robotic Intelligence and Productivity Conference*, Detroit, MI., November 1983, pp. 158–163.Google Scholar - [13]Toussaint, G.T. and H. ElGindy, "Separation of two monotone polygons in linear time",
*Robotica*, Vol. 2, 1984, pp. 215–220.Google Scholar - [14]Toussaint, G.T., "On translating a set of spheres", Tech. Rept. SOCS-8.4, School of Computer Science, McGill University, Montréal, March 1984.Google Scholar
- [15]Toussaint, G.T. "Movable separability of sets", in
*Computational Geometry*, Ed. G.T. Toussaint, North Holland, 1985, pp.335–376.Google Scholar - [16]Toussaint, G.T., Privat communication, 1986.Google Scholar