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∈ ∈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 5log2 n)-time algorithm for k∈=∈2, and O(n 3k∈+∈2)-time algorithm for k∈¥∈3. These results are the first on the generalization of the monotonicity problem. © 2008 Springer Berlin Heidelberg.