Robert Connelly, Erik D. Demaine, and Günter Rote, “Straightening Polygonal Arcs and Convexifying Polygonal Cycles”, in Proceedings of the 41st Annual Symposium on Foundations of Computer Science (FOCS 2000), Redondo Beach, California, November 12–14, 2000, pages 432–442.
Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of
rigid bars joined at incident endpoints (polygonal chains), with the property
that no cycle surrounds another arc or cycle. We prove that the linkage can be
continuously moved so that the arcs become straight, the cycles become convex,
and no bars cross while preserving the bar lengths. Furthermore, our motion is
piecewise-differentiable, does not decrease the distance between any pair of
vertices, and preserves any symmetry present in the initial configuration. In
particular, this result settles the well-studied carpenter's rule conjecture.