Reconstructing a convex polygon from its $\omega$-cloud

An ω-wedge is the (closed) set of all 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 ω-cloud of P is the curve traced by the apex of the ω-wedge as it rotates around P while maintaining tangency in both rays. 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 the two rays of the ω-wedge. Here we show that if ω is known, the ω-cloud alone uniquely determines P , and we give a linear-time reconstruction algorithm. Furthermore, even if we only know that ω < π/2, we can still reconstruct P , albeit in cubic time in the number of vertices. This reduces to quadratic time if in addition we are given the location of one of the vertices of P .