Locked and Unlocked Polygonal Chains in Three Dimensions

• DOI: 10.1007/s00454-001-0038-7
• Publication Date: 2012-12-05T19:10:45Z
• Authors: T. Biedl; E. Demaine; M. Demaine; S. Lazard; A. Lubiw; J. O'Rourke; M. Overmars; S. Robbins; I. Streinu; G. Toussaint; S. Whitesides
• Keywords: mécanismes articulés; [INFO.INFO-OH] Computer Science [cs]/Other [cs.OH]; ACM: F.: Theory of Computation/F.2: ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY/F.2.2: Nonnumerical Algorithms and Problems; linkages; [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH]; 0102 computer and information sciences; 01 natural sciences; ACM: G.: Mathematics of Computing/G.2: DISCRETE MATHEMATICS/G.2.0: General

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.''