Proceedings 41st Annual Symposium on Foundations of Computer Science (2000)
Redondo Beach, California
Nov. 12, 2000 to Nov. 14, 2000
R. Connelly , Dept. of Math., Cornell Univ., Ithaca, NY, USA
E.D. Demaine , Dept. of Math., Cornell Univ., Ithaca, NY, USA
G. Rote , Dept. of Math., Cornell Univ., Ithaca, NY, USA
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.
computational geometry; graph theory; polygonal arc straightening; polygonal cycle convexifying; planar linkage; polygonal chains; convex cycles; piecewise-differentiable motion; symmetry; rule conjecture; computational geometry; graph theory
R. Connelly, G. Rote and E. Demaine, "Straightening polygonal arcs and convexifying polygonal cycles," Proceedings 41st Annual Symposium on Foundations of Computer Science(FOCS), Redondo Beach, California, 2000, pp. 432.