Moving an Angle Around a Region

摘要

Let D be a connected region inside a simple polygon, P. We define the angle hull, AH(D), of D to be the set of all points in P that can see two points of D at a right angle. We show that the perimeter of AH(D) cannot exceed the perimeter of the relative convex hull of D by more than a factor of 2. A special case occurs when P equals the full plane. Here we prove a bound of #pi#/2. Both bounds are tight, and corresponding results are obtained for any other angle.