Restricted Orientation Visibility

Let O be some set of orientations, i.e., O 0 ; 360). In this paper we look at the consequences of deening visibility based on curves that are monotone w.r.t. to the orientations in O. We call such curves O-staircases. Two points p and q in a polygon P are said to O s-see each other if there exists an O-staircase from p to q that is completely contained in P. We investigate some of the structural properties of O s-visibility and then turn to the computation of the O s-kernel of a polygon. The O s-kernel of a polygon P is then the set of all points which O s-see all other points. We show that the O s-kernel of a simple polygon can be obtained as the intersection of all fg s-kernels, with 2 O. With the help of this observation we are able to develop an O(n log jOj) algorithm to compute the O s-kernel in a simple polygon, for nite O. We also show how to compute the external O s-kernel of a polygon in time O(n + jOj). Both algorithms can be combined to compute the O s-kernel of a polygon with holes in time O(n 2 + njOj).