Covering a simple polygon by monotone directions

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∈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(n5log2n)-time algorithm for k=2, and O(n3k+1logn)-time algorithm for fixed k⩾3.