Distance-Monotone Motions
Note that although we can achieve an
angle-monotone
motion between any two convex polygons, we cannot achieve
a distance-monotone
one. A counterexample is provided below, where a polygon
with quadrangle V
1 ,
V 3
, V 4
, V 6
goes through motions that first decreases | V
1-V
4 |
and then increases | V
3 -
V 6
|, setting the labels of the vertices to zero. The distance
| V
2 -
V 5
|, however, increases and later decreases.
Thus we observe an increasing and then decreasing
distance between vertices 2 and 5, and we have a motion that follows the
algorithm and is angle-monotone but not distance-monotone.