Before revision:  Here we show that if \omega is known, the \omega-cloud alone uniquely determines P, and we give a linear-time reconstruction algorithm. 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:  Here we show that if \omega is known, the \omega-cloud alone is given. We give two conditions under which it uniquely defines P: (i) when \omega < \pi is fixed/given, or (ii) when what is known is that \omega < \pi/ 2, we can still reconstruct P , albeit in cubic time in the number of vertices.