Approximating Polygons and Subdivisions with Minimum Link Paths

A linear time algorithm for minimum link paths inside a simple polygon. Algorithms for the reduction of the number of points required to represent a line or its caricature. Computing the visibility polygon from a convex set and related problems. implementation of the Douglas-Peucker line simpli-cation algorithm using at most cn log n operations. In preparation, 1991. 16] H. Imai and M. Iri. Computational-geometric methods for polygonal approximations of a curve. 25 t j O i+1 O i O j t' t i a. O i+1 O i t' t b. Proof: We begin by nding the longest preex that can be stabbed by a line using the algorithm of section 4.1. If this preex has i objects, then the algorithm ends in O(i) time with a stabbing wedge, which bounded by two limiting lines and a portion of O i , that does not intersect O i+1. We now have two cases, illustrated in gure 23 ((To be completed.)) 5 Conclusions and open problems We have examined minimum link approximations that lie in convolutions or are ordered stabbers as part of a basic approach to approximating paths, polygons, and subdivisions. We have developed some eecient algorithms and indicated that others are unlikely to ever be developed. There are many avenues that we hope to explore further|the most important being practical studies of implementations of theoretically eecient approximation methods. A few of the many open questions that remain are: Is computing the minimum link simple polygon enclosing all holes NP-complete? What other restrictions on approximation can be handled in subquadratic time? For example, the vertices may be required to lie within some < " of the original path. Can subquadratic time algorithms be developed for ordered stabbing with the other deenitions of visiting order? and their outer common tangents. For turns in objects, the visiting order also plays a r^ ole|R j is the object O j under deenitions 1 or 4 and object O j+1 under deenition 2. For each j < i such that the chain-stabbing wedge W j is formed by stabbing paths with k ? 1 links, compute the stabbing wedge for lines that stab, in order, W j \ R j , O j+1 , O j+2 , : : :, O i. The union of these stabbing wedges is the chain-stabbing wedge W i .