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. This reduces to quadratic time if in addition we are given the location of one of the vertices of P .
After revision:  Furthermore, even if we only know that \omega < \pi/ 2, we can still reconstruct P , albeit in cubic time in the input \omega-cloud .