Abstract
A generalized problem is defined in terms of functions on sets and illustrated in terms of the computational geometry of simple planar polygons. Although its apparent time complexity is O(n 2), the problem is shown to be solvable for several cases of interest (maximum and minimum distance, intersection detection and rerporting) in O(n logn), O(n) or O(logn) time, depending on the nature of a specialized “selection” function. (Some of the cases can also be solved by the Voronoi diagram method; but time complexity increases with that approach.) A new use of monotonicity and a new concept of “locality” in set mappings contribute significantly to the derivation of the results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Bentley J, Ottmann T (1979) Algorithms for reporting and counting geometric intersections. IEEE Transactions on Computers c-28:643–647
Chazelle B, Dobkin D (1980) Detection is easier than computation. ACM Symposium on the Theory of Computing, pp 146–153
Chin F, Wang C (1983) Optimal algorithms for the intersection and the minimum distance problems between planar polygons. IEEE Transactions on Computers c-32:1203–1207
Chin F, Wang C (1984) Minimum vertex distance between two separable convex polygons. Information Processing Letters 17:41–45
Edelsbrunner H (1982) On computing the extreme distances between two convex polygons. IIG, Technische Universität Graz, Austria, Report 96
Garey M, Johnson D, Preparata F, Tarjan R (1978) Triangulating a simple polygon. Information Processing Letters 7:175–179
Kirkpatrick D (1979) Efficient computation of continuous skeletons. Proceedings of the 20th Annual Symposium on the Foundations of Computer Science, pp 18–27
Lee D (1981) Generalization of Voronoi diagrams in the plane. SIAM J Comput 10:73–87
Nievergelt J, Preparata F (1982) Plane-sweep algorithms for intersecting geometric figures. Communications of the ACM 25:739–747
Schwartz J (1981) Finding the minimum distance between two convex polygons. Information Processing Letters 13:168–170
Shamos M (1975) Geometric complexity. Proceedings of the 17th Annual ACM Symposium on the Theory of Computing, pp 224–233
Shamos M, Hoey D (1976) Geometric intersection problems. Proceedings of the 17th Annual Conference on the Foundations of Computer Science, pp 208–215
Toussaint G (1982) Complexity, convexity, and unimodality. Proceedings of the Second World Conference on Mathematics
Toussaint G (1983) An optimal algorithm for computing the minimum vertex distance between two crossing convex polygons. 21st Allerton Conference on Communication, Control, and Computing, pp 457–458
Wang C (1983) Intersection and minimum distance problems for planar polygons. M.Sc. Thesis, Department of Computing Science, University of Alberta
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chin, F., Sampson, J. & Wang, C.A. A unifying approach for a class of problems in the computational geometry of polygons. The Visual Computer 1, 124–132 (1985). https://doi.org/10.1007/BF01898356
Issue Date:
DOI: https://doi.org/10.1007/BF01898356