Geodesic Disks and Clustering in a Simple Polygon

Let P be a simple polygon of n vertices and let S be a set of N points lying in the interior of P . A geodesic disk GD(p, r) with center p and radius r is the set of points in P that have a geodesic distance ≤ r from p (where the geodesic distance is the length of the shortest polygonal path connection that lies in P ). In this paper we present an output sensitive algorithm for finding all N geodesic disks centered at the points of S, for a given value of r. Our algorithm runs in O((n + (kn) 2 3 + k) logc n) time, for some constant c and output size k. It is the basis of a cluster reporting algorithm where geodesic distances are used.