Computing the Geodesic Center of a Simple Polygon

Abstract

This paper presents a polynomial-time algorithm for finding the geodesic center of a simple polygon, i.e., the point in the polygon whose greatest internal distance to any other point in the polygon is a minimum. The distance between two points in a simple polygon is measured as the length of the shortest internal path between them. The key idea is the construction of a geodesic farthest-point Voronoi diagram for a simple polygon which consists of straight lines and hyperbolic curve segments. The geodesic center of a simple polygon is either the center of its geodesic diameter or a vertex of the geodesic farthest-point Voronoi diagram. The proposed algorithm runs in O(n3log log n) time, where n is the number of vertices of the given polygon.