Before revision: Given a set \mathcal{O of k orientations in the plane, two points inside a simple polygon P \mathcal{O   if there is an O -staircase contained in P that connects them .
After revision: Given a set \mathcal{O of k orientations in the plane, and let P be a simple polygon P \mathcal{O if there is an O -staircase contained in P that connects them .