Reconfiguring convex polygons

We prove that there is a motion from any convex polygon to any convex polyg on with the same counterclockwise sequence of edge lengths, that preserves the lengths of the edges, and keeps the polygon convex at all times. Furthe rmore, the motion is "direct" (avoiding any intermediate canonical configur ation like a subdivided triangle) in the sense that each angle changes mono tonically throughout the motion. In contrast, we show that it is impossible to achieve such a result with each vertex-to-vertex distance changing mono tonically. We also demonstrate that there is a motion between any two such polygons using three-dimensional moves known as pivots, although the comple xity of the motion cannot be bounded as a function of the number of vertice s in the polygon. (C) 2001 Elsevier Science B.V. All rights reserved.