Locked and Unlocked Polygonal Chains in Three Dimensions

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.''

[1]  James U. Korein,et al.  A geometric investigation of reach , 1985 .

[2]  Hazel Everett,et al.  Convexifying star-shaped polygons , 1998, CCCG.

[3]  G. T. Sallee.  Stretching chords of space curves , 1973 .

[4]  Jason H. Cantarella,et al.  NONTRIVIAL EMBEDDINGS OF POLYGONAL INTERVALS AND UNKNOTS IN 3-SPACE , 1998 .

[5]  Gordon M. Crippen,et al.  Distance Geometry and Molecular Conformation , 1988 .

[6]  Sue Whitesides,et al.  Reconfiguring closed polygonal chains in Euclideand-space , 1995, Discret. Comput. Geom..

[7]  Günter Rote,et al.  Straightening polygonal arcs and convexifying polygonal cycles , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[8]  J. Schwartz,et al.  On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds , 1983 .

[9]  Sue Whitesides,et al.  Algorithmic Issues in the Geometry of Planar Linkage Movement , 1992, Aust. Comput. J..

[10]  John E. Hopcroft,et al.  Movement Problems for 2-Dimensional Linkages , 1984, SIAM J. Comput..

[11]  C. Gibson,et al.  On the geometry of the planar 4-bar mechanism , 1986 .

[12]  John E. Hopcroft,et al.  On the movement of robot arms in 2-dimensional bounded regions , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[13]  Pat Morin,et al.  Convexifying polygons with simple projections , 2001, Inf. Process. Lett..

[14]  G. Budworth.  The Knot Book , 1983 .

[15]  Prosenjit Bose,et al.  Drawing Nice Projections of Objects in Space , 1995, Graph Drawing.

[16]  McCarthy,et al.  Geometric Design of Linkages , 2000 .

[17]  K. H. Hunt,et al.  Kinematic geometry of mechanisms , 1978 .

[18]  John Canny,et al.  The complexity of robot motion planning , 1988 .

[19]  Joseph O'Rourke,et al.  Polygonal chains cannot lock in 4d , 1999, CCCG.

[20]  Ileana Streinu,et al.  A combinatorial approach to planar non-colliding robot arm motion planning , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[21]  K. Millett.  KNOTTING OF REGULAR POLYGONS IN 3-SPACE , 1994 .

[22]  N. Madras,et al.  THE SELF-AVOIDING WALK , 2006 .

[23]  Marc J. van Kreveld,et al.  Folding rulers inside triangles , 1993, CCCG.

[24]  Erik D. Demaine,et al.  Locked and unlocked polygonal chains in 3D , 1998, SODA '99.

[25]  Vitit Kantabutra.  Reaching a Point with an Unanchored Robot Arm in a Square , 1997, Int. J. Comput. Geom. Appl..