5. Conclusion

Let P be a simple polygon of n vertices and let S be a set of n points called sites lying in the interior of P. The geodesic-distance between two sites x and y in P is defined as the length of the shortest polygonal path connecting x and y constrained to lie in P. It is a useful notion in graphics problems concerning visibility between objects, computer vision problems concerned with the description of shape, wire-layout problems in VLSI, and robotics problems concerning path planning and facility location. In this paper we propose efficient algorithms for solving a variety of geodesic-distance problems. The key result and tool used in the design of these algorithms is an O(n log n)-time algorithm for computing the geodesic convex-hull of S with respect to P, i.e., the shortest polygonal circuit containing S constrained to lie in P. We illustrate the use of this structure in computing the geodesic-diameter, the geodesic-center, and the geodesic-median of S in P, as well as the maximum geodesic distance between two sets S1 and S2 in P.