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, two points inside a simple polygon in the plane. Given two points p,q inside P, we say that p \mathcal{O if there is an O -staircase contained in P that connects them .