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.

An angle-monotone but not distance-monotone motion exists

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.

Back--Java Applet    Main    Next--Pivots