Offset-Polygon Annulus Placement Problems

r down tò 2 , it is a weakly monotone increasing function. But in a monotone function, a value can only appear once (or in one continuous range). Since the function is divided into two weakly monotone regions, a xed value can appear at most twice (or in two continuous ranges). This suuces to show that there can be at most two intersections between (I r) and O r , one in each weakly monotone region of the distance function. Note that each intersection can be either a single point or a connected portion of an edge. The edge intersection possibility follows from the fact that the functions are weakly monotone rather than strictly monotone. Next we note that the same holds for I l and O r except that with the left chain of I P;; , the distance function is rst weakly monotone increasing, and then weakly monotone decreasing, as y goes down from`1 tò 2. Thus there are at most two intersections (points or connected portions of segments) between I r and O r , and at most two intersections between I l and O r. It remains only to show that if (I r) intersects O r twice, then (I l) does not intersect O r at all, and vice versa. For this we need note only that in order for (I r) to intersect O r twice, must be less than the minimum of the distances from b to b 0 and from t to t 0 , which are the largest values of the two monotone portions of the width function. But this condition implies that (I l) does not intersect O r at all. Similarly, for I l to intersect O r twice we need to be greater than the maximum of the distances from b to b 0 and from t to t 0 , which would preclude an intersection between (I r) and O r. 2 20 t 0 b O r ` 1 t I l