Reconstructing a convex polygon from its ω-cloud

An ω-wedge consists of two rays emanating from a single 1 point (the apex), separated by an angle ω < π. Given a convex polygon 2 P , we place the ω-wedge such that P is inside the wedge and both rays 3 are tangent to P . The ω-cloud of P is the curve traced by the apex of 4 the ω-wedge as it rotates around P while maintaining tangency in both 5 rays. 6 Previous work crucially required knowledge of the points of tangency of 7 the ω-wedge to reconstruct P . We show that if ω is known, the ω-cloud 8 alone uniquely determines P , and we give a linear-time reconstruction 9 algorithm. Furthermore, even if we only know that ω < π/2, we can 10 still reconstruct P , albeit in cubic time in the number of vertices. This 11 reduces to quadratic time if in addition we are given the location of one 12 of the vertices of P . 13