Reconstructing a convex polygon from its ω -cloud

Abstract

An ω -wedge is the closed set of points contained between two rays that are emanating from a single point (the apex), and are separated by an angle ω< π. Given a convex polygon P, we place the ω -wedge such that P is inside the wedge and both rays are tangent to P. The set of apex positions of all such placements of the ω -wedge is called the ω -cloud of P. We investigate reconstructing a polygon P from its ω -cloud. Previous work on reconstructing P from probes with the ω -wedge required knowledge of the points of tangency between P and the two rays of the ω -wedge in addition to the location of the apex. Here we consider the setting where the maximal ω -cloud alone is given. We give two conditions under which it uniquely defines P: (i) when