GENERATING THE COMPLEMENT OP A STAIRCASE STARSHAPED ORTHOGONAL POLYGON FROM STAIRCASE CONVEX CONES

摘要

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 starshaped via staircase paths. For every point p in R~2(int S), there is a corresponding point g in bdry S such that p lies in a maximal staircase convex cone C_q at q in R~2(int S). Furthermore, point g may be selected to satisfy these requirements: 1)If p∈R~2(ini R), then q is an endpoint of an extreme edge of S. 2)If p∈(int R)(int S), then q is a point of local nonconvexity of S and C_q is unique. Moreover, there is a neighborhood N of g such that, for s in (bdry S)∩N and for C_s any staircase cone at s in R~2(int S),C_s is contained in C_q. Thus we obtain a finite family of staircase convex cones whose union is R~2(int S).