Polygon casting - Removing from mold by rotation
Consider the marked edge of the following object (light gray) in its mold (dark gray) - 


Where must $p$ be such that an arbitrary points on the edge will not be pushed against the mold? This figure shows the direction a point on the edge will be moving in when rotated around a point $p$ - 


Notice that the point will be moving perpendicular to the line connecting it with $p$. Now recall that in order not to push on the mold, the points of the edge must move in the direction of the inner half-plane defined by the edge. This means that the possible locations for $p$ lie on a half-plane perpendicular to the edge. Here's the half-plane (in green) corresponding to an inner point of the edge - 


The half-plane corresponding to the upper vertex of the edge, in this case, is contained in all of the half-planes for inner points, and so a rotation around any point in it moves the entire edge such that it does not push on the mold.

Whether $P$ can be removed with a rotation around $p$, is thus equivalent to whether $p$ is in the common intersection of the half-planes defined by $P$'s edges.