Optimal computation of finitely oriented convex hulls

Abstract

We define four versions of the “convex hull” of a simple finitely oriented polygon (i.e., a polygon whose edge orientations all belong to some fixed finite set of angles) and give optimal algorithms to find them. Two of these generalize the notions of the orthogonal convex hull of an orthogonal polygon and the traditional “bounding box” of a polygon. Three of the hulls have worst-case time complexity Θ(n + f) and worst case space complexity θ(n) space, where n is the number of edges of a given polygon and f (≥2) is the number of allowed orientations. We also show that testing whether an arbitrary simple polygon is (finitely oriented) convex has worst-case time and space complexity θ(n + f) and θ(n), respectively.