Covering a simple polygon by monotone directions

Abstract

In this paper we study the problem of finding a set of k directions for a given simple polygon P, such that for each point P is an element of P there is at least one direction in which the line through p intersects the polygon only once. For k = 1, this is the classical problem of finding directions in which the polygon is monotone, and all such directions can be found in linear time for a simple n-gon. For k > 1, this problem becomes much harder; we give an O(n(5) log(2)n)-time algorithm for k = 2, and O(n(3k+1) log n)-time algorithm for fixed k >= 3. (C) 2009 Elsevier B.V. All rights reserved.