Generating the Complement of a Staircase Starshaped Orthogonal Polygon from Staircase Convex Cones

Let \( S \) be an orthogonal polygon in the plane, bounded by a simple closed curve, and let \( R \) be the smallest rectangular region containing \( S \). Assume that \( S \) is star-shaped via staircase paths. For every point \( p \) in \( \mathbb{R}^2 \setminus (\text{int} \, S) \), there is a corresponding point \( q \) in \( \text{bdry} \, S \) such that \( p \) lies in a maximal staircase convex cone \( C_q \) at \( q \) in \( \mathbb{R}^2 \setminus (\text{int} \, S) \). Furthermore, point \( q \) may be selected to satisfy these requirements:

1. If \( p \in \mathbb{R}^2 \setminus (\text{int} \, R) \), then \( q \) is an endpoint of an extreme edge of \( S \).
2. If \( p \in (\text{int} \, R) \setminus (\text{int} \, S) \), then \( q \) is a point of local nonconvexity of \( S \) and \( C_q \) is unique. Moreover, there is a neighborhood \( N \) of \( q \) such that, for \( s \) in \( (\text{bdry} \, S) \cap N \) and for \( C_s \) any staircase cone at \( s \) in \( \mathbb{R}^2 \setminus (\text{int} \, S) \), \( C_s \subseteq C_q \).

Thus we obtain a finite family of staircase convex cones whose union is \( \mathbb{R}^2 \setminus (\text{int} \, S) \).