A Generalization of staircase visibility

Abstract

Let $\mathcal{O}$be some set of orientations, i.e., $\mathcal{O}$ $\subseteq$[0°,360°). In this paper we look at the consequences of defining visibility based on curves that are monotone w.r.t. to the orientations in $\mathcal{O}$. We call such curves $\mathcal{O}$-staircases. Two points p and q in a polygon P are said to $\mathcal{O}$-see each other if there exists an $\mathcal{O}$-staircase from p to q that is completely contained in P. The $\mathcal{O}$-kernel of a polygon P is then the set of all points which $\mathcal{O}$-see all other points. We show that the $\mathcal{O}$-kernel of a simple polygon can be obtained as the intersection of all θ-kernels, with θ∈$\mathcal{O}$. With the help of this observation we are able to develop an O(n log |$\mathcal{O}$|) algorithm to compute the $\mathcal{O}$-kernel in a simple polygon, for finite $\mathcal{O}$.

This work was supported by the Deutsche Forschungsgemeinschaft under Grant No. Ot 64/5-4 and by Natural Sciences and Engineering Research Council Grant No. A-5692.