A polygon is determined by its angles

We study the problem of reconstructing a simple polygon from angles measured at the vertices of the polygon. We assume that at each vertex v   a sensing device returns a list of angles α1,α2,…α1,α2,…, where αiαi is the angle between the i  -th and the (i+1)(i+1)-th vertices visible to v in counterclockwise (ccw) order starting with the ccw neighbor of v along the boundary. We prove that the angle measurements at all vertices of a simple polygon together with the order of the vertices along the boundary uniquely determine the polygon (up to similarity). In addition, we give an algorithm for reconstructing the polygon from this information in polynomial time.