New Similarity Measures between Polylines with Applications to Morphing and Polygon Sweeping

We introduce two new related metrics, the geodesic width and the link width, for measuring the “distance” between two non-intersecting polylines in the plane. If the two polylines have n vertices in total, we present algorithms to compute the geodesic width of the two polylines in O(n2 log n) time using O(n2) space and the link width in O(n3 log n) time using O(n2) working space where n is the total number of edges of the polylines. Our computation of these metrics relies on two closelyrelated combinatorial strutures: the shortest-path diagram and the link diagram of a simple polygon. The shortest-path (resp., link) diagram encodes the Euclidean (resp., link) shortest path distance between all pairs of points on the boundary of the polygon. We use these algorithms to solve two problems: • Compute a continuous transformation that “morphs” one polyline into another polyline. Our morphing strategies ensure that each point on a polyline moves ∗Preliminary versions of this paper appeared in the Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms [EGH00] and the Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms [EGHM01] †Address: Department of Computer Science, University of Arizona, Tucson AZ 85721-0077. Email: [email protected]. WWW: http://www.cs.arizona.edu/people/alon/. The author did part of this research while affiliated with Stanford University. ‡Address: Computer Science Dept., Stanford University, Stanford CA 94305. Email: [email protected]. WWW: http://graphics.stanford.edu/~guibas. Partially supported by NSF (CCR-9910633), by U.S. Army Research Office MURI grant DAAH04-96-1-007, and a grant from the Intel Corporation. §Address: Dept. of Computer Science, University of Illinois, Urbana, IL 61801-2987. Email: [email protected]. WWW: http://valis.cs.uiuc.edu/~sariel/. The author did part of this research while affiliated with Duke University. ¶Address: Applied Mathematics and Statistics, University at Stony Brook, Stony Brook, NY 117943600. Email: [email protected]. WWW: http://www.ams.sunysb.edu/~jsbm. Partially supported by NSF (CCR-9732220), a DARPA subcontract from HRL Laboratories, NASA Ames Research (NAG21325), Northrop-Grumman Corporation, and Sun Microsystems. ‖Address: Bioinformatics Program, Boston University, Boston MA 02215. Email: [email protected], WWW: http://people.bu.edu/murali. The author did part of this research while affiliated with Stanford University.