Straightening polygonal arcs and convexifying polygonal cycles
A planar polygonal arc or open polygonal chain is a sequence of finitely many line segments in the plane connected in a path without self-intersections. It has been an outstanding question as to whether it is possible to move a polygonal arc continuously in such a way that each edge remains a fixed length, there are no self-intersections during the motion, and at the end of the motion the arc lies on a straight line. This has come to be known as the carpenter's rule problem. A related question is whether it is possible to move a polygonal simple closed curve continuously in the plane, often called a closed polygonal chain or polygon, again without creating self-intersections or changing the lengths of the edges, so that it ends up a convex closed curve. The authors solve both problems by showing that in both cases there is such a motion.
scientific article; zbMATH DE number 2051390