Before revision:  Furthermore, even if we only know that \omega < \pi/ 2, we can still reconstruct P , albeit in cubic time in the number of vertices.
After revision:  Furthermore, even if we only know that \omega < \pi/ 2. We show that if neither of these two conditions hold, then P may not be unique. We show that, when the uniqueness conditions hold, the polygon P can be reconstructed in O(n) time with O(1) working space in addition to the input, where n is the number of arcs in the number of vertices.