__
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.**