Note that although we can achieve an
motion between any two convex polygons, we cannot achieve
one. A counterexample is provided below, where a polygon
with quadrangle V
, V 4
, V 6
goes through motions that first decreases | V
and then increases | V
|, setting the labels of the vertices to zero. The distance
|, 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.