Discrete and Computational Geometry
This paper studies movements of polygonal chains in three dimensions whose links are not allowed to cross or change length. Our main result is an algorithmic proof that any simple closed chain that initially takes the form of a planar polygon can be made convex in three dimensions. Other results include an algorithm for straightening open chains having a simple orthogonal projection onto some plane, and an algorithm for making convex any open chain initially configured on the surface of a polytope. All our algorithms require only O (n) basic "moves.".
Biedl, T.; Demaine, E.; Demaine, M.; Lazard, S.; Lubiw, A.; O'Rourke, J.; Overmars, M.; Robbins, S.; Streinu, I.; Toussaint, G.; and Whitesides, S., "Locked and Unlocked Polygonal Chains in Three Dimensions" (2001). Computer Science: Faculty Publications, Smith College, Northampton, MA.