R. Cocan and J. O'Rourke. Polygonal chains cannot lock in 4D. In Proceedings of the Eleventh Canadian Conference on Computational Geometry, Vancouver, August 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....They describe con gurations of pairs of short three dimensional chains such that the two chains cannot be separated. In the same year, Soss [67] independently discovered short chains which are interlocked under the restriction that the angles between edges are xed. In 1999, Cocan and O Rourke [18] proved that all chains, polygons, and trees can be unfolded in dimensions four and higher. 5. Kinematic engineers unfolding linkages The study of kinematic linkages appears to have been rst mathematically codi ed in 1874 by the engineer Franz Reuleaux in his work Theoretische Kinematik [59] ....

R. Cocan and J. O'Rourke. Polygonal chains cannot lock in 4D. In Proceedings of the Eleventh Canadian Conference on Computational Geometry, Vancouver, August 1999.

....here. In three dimensions, it has recently been shown that there exist open (and closed) chains that can lock [7, 10] which is a relevant result for the protein folding problem. Finally, in dimensions higher than three, it has recently been established that neither open nor closed chains can lock [11]. The randomized motion planning approach we advocate here is somewhat different in nature to the previous approaches used to study these problems in the computational geometry community. In particular, as the methods we employ are not complete (i.e. they are not guaranteed to find a solution if ....

R. Cocan and J. O'Rourke. Polygonal chains cannot lock in 4D. In Proc. 11th Canad. Conf. Comput. Geom., pages 5--8, 1999.

....here. In three dimensions, it has recently been shown that there exist open (and closed) chains that can lock [8, 12] which is a relevant result for the protein folding problem. Finally, in dimensions higher than three, it has recently been established that neither open nor closed chains can lock [13]. The randomized motion planning approach we advocate here is somewhat di erent in nature to the previous approaches used to study these problems in the computational geometry community. In particular, as the methods we employ are not complete (i.e. they are not guaranteed to nd a solution if ....

R. Cocan and J. O'Rourke. Polygonal chains cannot lock in 4D. In Proc. 11th Canad. Conf. Comput. Geom., pages 5-8, 1999.

.... shortly thereafter with a proof that O(n 2 ) motions of constant complexity suce and an algorithm to easily compute them [25] Other results include demonstrating that some three dimensional chains are unstraightenable [6, 7, 26] and that all chains can be straightened in four dimensions [9]. The applications of these questions are numerous, including robot arm path planning [16] and wire and sheet metal bending [2] but the application with which we are concerned here is of mapping conformation spaces of molecules. The chemistry, biology, and physics communities have long been ....

Roxana Cocan and Joseph O'Rourke. Polygonal chains cannot lock in 4D. In Proc. 11th Canadian Conference on Computational Geometry, 1999.

.... not allowed to cross, universality does not hold in general for polygons in 3D [2, 5] but has been shown for polygons in the plane and motions in 3D [1, 2] for polygons and motions in the plane [9] for polygons in 3D with simple projections [4] and for all polygons in 4D and higher dimensions [8]. All of these papers show universality by proving that every polygon can be convexi ed, that is, moved to a convex (planar) polygon while preserving edge lengths. Convex polygons are used as an intermediate state; because motions can be reversed and concatenated, all that remains is to show that ....

Roxana Cocan and Joseph O'Rourke. Polygonal chains cannot lock in 4D. In Proceedings of the 11th Canadian Conference on Computational Geometry, Vancouver, Canada, August 1999. http://www.cs.ubc.ca/conferences/CCCG/elec_proc/c17.ps.gz.

.... that is, there are some pairs of configurations of the linkage which cannot be connected if the links are not allowed to cross [4] In three dimensions, there exist open (and closed) chains that can lock [4, 5] while, in dimensions higher than three, neither open nor closed chains can lock [6]. Protein Folding. The protein folding problem is to predict a protein s three dimensional conformation based solely on its amino acid sequence. Many different approaches for predicting protein structure have been explored. In folding simulations, several computational approaches have been ....

R. Cocan and J. O'Rourke. Polygonal chains cannot lock in 4D. In Proc. 11th Canad. Conf. Comput. Geom., pages 5--8, 1999.

....straightened (or convexified) in the plane This question was recently answered in the affirmative [10] by Connelly, Demaine, and Rote. Other results include demonstrating that some three dimensional chains are unstraightenable [6, 7, 25] and that all chains can be straightened in four dimensions [9]. The applications of these questions are numerous, including robot arm path planning [16] and wire and sheet metal bending [2] but the application with which we are concerned here is of mapping conformation spaces of molecules. The chemistry, biology, and physics communities have long been ....

Roxana Cocan and Joseph O'Rourke. Polygonal chains cannot lock in 4D. In Proc. 11th Canadian Conference on Computational Geometry, 1999.

....want to convexify it by motion through three dimensions, then it is surely not convexifiable if it is knotted. But there are unknotted polygons that cannot be convexified [1] The complexity of determining whether a polygon in 3D can be convexified also remains open. Amazingly, Cocan and O Rourke [4] have shown that every polygon in d dimensions can be convexified through d dimensions for any d 4. 1.2 Outline The rest of this paper is organized as follows. Section 2 begins with a more formal description of the problem. Section 3 describes our algorithm for computing the motion. Sections 4 ....

Roxana Cocan and Joseph O'Rourke. Polygonal chains cannot lock in 4D. In Proc. 11th Canadian Conf. Comput. Geom., Vancouver, Aug. 1999.

....for more animations. In contrast to this result, in dimension three there are arcs that cannot be straightened and polygons that cannot be convexified [3, 7] In dimension four, all arcs and cycles unlock, i.e. can be straightened and convexified, respectively [8]. In the plane, there are examples of trees embedded in the plane that are locked in the sense that they cannot be properly moved so that the vertices lie nearly on a line [4] In other words, there are two embeddings of the tree such that there is no proper motion from one configuration to the ....

....there is no proper motion from one configuration to the other. The important difference between trees and arc and cycle sets is that arc and cycle sets have maximum degree two. Arcs and Cycles Trees 2 D Not lockable (this paper) Lockable [4] 3 D Lockable [3, 7] Lockable 4 D Not lockable [8] Not lockable Table 1. Summary of what types of linkages can be locked. The question mark denotes a conjecture. Whether every arc in the plane can be straightened, and whether every polygon in the plane can be convexified, have been outstanding open questions until now. The problems are ....

R. Cocan and J. O'Rourke. Polygonal chains cannot lock in 4D. In Proc. 11th Canadian Conf. Comput. Geom., Vancouver, Aug. 1999. http://www.cs.ubc.ca/conferences/ CCCG/elec proc/c17.ps.gz.

....[4] In three dimensions, it has recently been shown that there exist open (and closed) chains that can lock [4, 7] which is a relevant result for the protein folding problem. Finally, in dimensions higher than three, it has recently been established that neither open nor closed chains can lock [8]. The randomized motion planning approach we advocate here is somewhat di erent in nature to the previous approaches used to study these problems in the computational geometry community. In particular, as the methods we employ are not complete (i.e. they are not guaranteed to nd a solution if ....

R. Cocan and J. O'Rourke. Polygonal chains cannot lock in 4D. In Proc. 11th Canad. Conf. Comput. Geom., pages 5-8, 1999.

.... Following the McGill University Workshop on Folding in Barbados, organized by Anna Lubiw and Sue Whitesides in 1998, several results on recon guring with non crossing motions have appeared: planar linkages using 3d motions ( BD 99] AGP] trees, 3 and higher dimensional linkages ( BD 98] [COR]) The decisive result came in the beginning of this year, when Connelly et al. CDR] announced the positive answer to the planar convexi cation problem: all chains can be convexi ed, all linkages can be straightened. Their approach is to rst prove (using linear programming duality and Maxwell s ....

R. Cocan and J. O'Rourke. Polygonal Chains Cannot Lock in 4d, Proc. 11th Canad.Conf. Comput. Geometry, Vancouver, 1999, pp. 5-8.

....Science, McGill University, 3480 University Street, Montr eal, Qu ebec H3A 2A7, Canada, email: fsoss, godfriedg cs.mcgill.ca. and motions in 3 D [1, 2] for polygons and motions in the plane [8] for polygons in 3 D with simple projections [4] and for all polygons in 4 D and higher dimensions [7]. All of these papers show universality by proving that every polygon can be convexified, that is, moved to a convex (planar) polygon while preserving edge lengths. Convex polygons are used as an intermediate state; because motions can be reversed and concatenated, all that remains is to show ....

R. Cocan and J. O'Rourke. Polygonal chains cannot lock in 4D. In Proc. 11th Canadian Conf. Comput. Geom., Vancouver, Canada, Aug. 1999. http://www.cs.ubc.ca/conferences/ CCCG/elec proc/c17.ps.gz.

....want to convexify it by motion through three dimensions, then it is surely not convexifiable if it is knotted. But there are unknotted polygons that cannot be convexified [1] The complexity of determining whether a polygon in 3D can be convexified also remains open. Amazingly, Cocan and O Rourke [6] have shown that every polygon in d dimensions can be convexified through d dimensions for any d 4. 1.2 Outline The rest of this paper is organized as follows. Section 2 begins with a more formal description of the problem. Section 3 describes our algorithm for computing the motion, and Section ....

R. Cocan and J. O'Rourke. Polygonal chains cannot lock in 4D. Draft, February 1999. Based on an undergraduate thesis by R. Cocan.

....of any number of edges between A4 and A5 as long as their total length does not exceed the length l 4 . As a final remark we add that although hexagons in 3D can be knotted, any simple polygon with less than six edges is unknotted [22] In contrast to the result in 3D, recently Cocan and O Rourke [10] have shown that for all dimensions greater than 3 every simple open chain may be straightened, and every simple closed chain may be convexified. The algorithms run in polynomial time. We close this section by mentioning a couple of additional open problems. 1) A natural generalization of ....

Roxana Cocan and Joseph O'Rourke. Polygonal chains cannot lock in 4D. In Proc. 11th Canadian Conference on Computational Geometry, 1999.

....maximum degree three. It is easy to replace a high degree joint with a number of degreethree joints joined by a chain of tiny links. Do equilateral locked tree linkages with maximum degree three exist Finally, many interesting questions can be posed for linkages moving in higher dimensions. See [1, 6] for recent work on chain and cycle linkages moving under simple motion in three and more dimensions. 11 Acknowledgments The research reported here was initiated at the International Workshop on Wrapping and Folding, co organized by Anna Lubiw and Sue Whitesides, at the Bellairs Research ....

R. Cocan and J. O'Rourke. Polygonal chains cannot lock in 4d. In Proceedings of the Eleventh Canadian Conference on Computational Geometry, pages 5--8. University of British Columbia, Vancouver, B.C., Canada, Aug. 1999. (Extended Abstract). 14

....we faced in Step 1 of Algorithm 1a. This permits replacing the O(n 3 log n) moves per step from motion planning, with at most two moves. We now proceed to describe this. Because this represents a computational improvement only, the proofs are only sketched. More detailed proofs are contained in [Coc99] Algorithm 1b distinguishes three possibilities: 9 w v 1 v 0 v g w 0 w g v 2 Figure 4: The goal direction vector w defines the direction that w 0 should be rotated to reach w g . The shaded triangle cone 1 4(v 0 ; v g ) is not crossed by any links of the chain if w is unobstructed. 1. ....

R. Cocan. Polygonal chains cannot lock in 4D. Undergraduate thesis, Smith College, 1999. 28

....chains can lock. Smith Technical Report 063 (Major revision of the August 1999 version with the same report number. Dept. of Computer Science, Smith College, Northampton, MA 01063, USA. frcocan, orourkeg cs.smith.edu. Research supported by NSF Grant CCR 9731804. Results first reported in [CO99] i Contents 1 Introduction 1 1.1 Summary . 1 1.2 Background . 2 2 Straightening Open Chains in 4D 3 2.1 Algorithm 1a . ....

R. Cocan and J. O'Rourke. Polygonal chains cannot lock in 4D. In Proc. 11th Canad. Conf. Comput. Geom., pages 5--8, 1999. Full version: Smith College Tech. Rep. 063; LANL arXive cs.CG/9908005.

.... by a different algorithm which uses only O(n) basic moves [BDD 99] 5 In dimensions d 4, it has recently been established that neither open nor closed chains can lock: Every open chain can be straightened in O(n) moves, and every closed chain can be convexified in O(n 6 ) moves [CO99a,CO99b] Thus less is known about locking in two dimensions than in three and higher dimensions. But with recent increased scrutiny by the community, perhaps P3 will soon be settled. Acknowledgements I thank Erik Demaine for comments, and Marty Demaine for the Durer reference. 4 This application was ....

R. Cocan and J. O'Rourke. Polygonal chains cannot lock in 4D. In Proc. 11th Canad. Conf. Comput. Geom., pages 5--8, 1999.

.... in 3D by a different algorithm which uses only O(n) basic moves [BDD 99] 5 In dimensions d 4, it has recently been established that neither open nor closed chains can lock: Every open chain can be straightened in O(n) moves, and every closed chain can be convexified in O(n 6 ) moves [CO99a,CO99b] Thus less is known about locking in two dimensions than in three and higher dimensions. But with recent increased scrutiny by the community, perhaps P3 will soon be settled. Acknowledgements I thank Erik Demaine for comments, and Marty Demaine for the Durer reference. 4 This ....

R. Cocan and J. O'Rourke. Polygonal chains cannot lock in 4D. Technical Report 063, Smith College, Northampton, MA, July 1999. Full version of proceedings abstract. LANL archive paper number cs.CG/9908005.

....literature (e.g. 9 13, 15 20, 22 24] allow the links to cross or to pass through or over one another. In other words, the links represent distance constraints between joints, not physical obstacles that must avoid each other. Recently, other results for free motion have been obtained; see [1, 3, 4, 6, 8]. There is also the very general algebraic approach to motion planning of [5] and [21] since the constraint that the links not cross can be specified algebraically. For related work from a topological point of view, see [14] and references therein; again, this work allows links to cross. Even ....

.... [3] In three dimensions, while a complete characterization isn t known, there are linkage configurations that can be straightened or convexified, and other configurations that cannot be [1] In four or more dimensions, any path linkage can be straightened, and any cycle linkage can be convexified [6]. An equivalence relation on the set of simple configurations can be defined as follows. One simple configuration is equivalent to another means that the one can be moved to the other by a free motion. The basic problem we consider in this paper is whether the simple configurations of a tree ....

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R. Cocan and J. O'Rourke. Polygonal chains cannot lock in 4d. In Proceedings of the Eleventh Canadian Conference on Computational Geometry, pages 5-- 8. University of British Columbia, Vancouver, B.C., Canada, Aug. 1999. (Extended Abstract).

....the 1970 s, 1 and were the subject of intense investigation by the late 1990 s. It was first established that chains can lock in three dimensions (3D) both locked open chains, and locked closed but unknotted chains are possible [CJ98, BDD 99] In 4D, neither open nor closed chains can lock [CO99] But aside from some special cases (e.g. star shaped polygons cannot lock [ELR 98] polygonal trees can lock [BDD 98] the problems remained unresolved for chains in 2D. Connelly, Demaine, and Rote have now settled the questions, establishing that neither open nor closed chains can ....

R. Cocan and J. O'Rourke. Polygonal chains cannot lock in 4D. In Proc. 11th Canad. Conf. Comput. Geom., pages 5--8, 1999. Full version: Smith College Tech. Rep. 063; LANL arXive cs.CG/9908005.

....in dimensions greater than three. We partition the results this way because the proofs of these three theorems are rather different. In particular, the proof of Theorem 3 is not difficult. In this extended abstract, we will sketch the proofs of the theorems. More detailed proofs are contained in [Coc99] and we are currently preparing a full version for publication. We summarize our results in the context of earlier work in the table below. Dimension (Un)Locked 2 3 9 locked chains d 4 Cannot lock 2 Straightening Open Chains in 4D Let P be a simple, open polygonal chain in 4D with n 2 ....

Roxana Cocan. Polygonal chains cannot lock in 4D. Undergraduate thesis, Smith College, 1999.

....question of whether every planar, simple open chain can be straightened in the plane while maintaining simplicity has circulated in the computational geometry community for years, but remains open at this writing. Whether locked chains exist in dimensions d 4 was only settled (negatively, in [CO99] as a result of the open problem we posed in a preliminary version of this paper [BDD 99] In piecewise linear knot theory, complete classification of the 3D embeddings of closed chains with n edges has been found to be difficult, even for n = 6. These types of questions are basic to the ....

R. Cocan and J. O'Rourke. Polygonal chains cannot lock in 4d. In Proc. 11th Canad. Conf. Comput. Geom., 1999. Extended abstract. Full version:

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