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

Abstract

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 ${\mathbb{R}}^2\setminus (\text{int}S)$.