Robert Connelly, Erik D. Demaine, and Gunter Rote, Straightening polygonal arcs and convexifying polygonal cycles. Discrete & Computational Geometry, to appear. Preliminary version in Proc. 41st Ann. Symp. on Found. of Computer Science (FOCS 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Primary 68U05, 65D18; Secondary 52C25, 52C40, 52C30, 52C45, 05C10, 13P10. Key words and phrases. Computational Geometry, Rigidity Theory, Computational Algebraic Geometry. Research supported by NSF RUI grant CCR 0105507 and NSF DARPA CARGO CCR0138374. c 0000 (copyright holder) and Rote [8], based on Rote s ground breaking idea of using expansive in nitesimal motions to avoid collisions. For nding algorithmically a path in con guration space between any two compatible positions, the author has proposed in [20] a combinatorial approach: the path consists of a nite number of ....

....guration space. Expansive motions guarantee that no collisions will occur. It is shown in [20] that pseudo triangulations with one convex hull edge removed (de ned in section 2) are (in nitesimally) expansive one degreeof freedom mechanisms. The in nitesimally expansive motions form a cone ([8] and [19] and linear programming can be used to nd a set of in nitesimal velocities of the moving points. Pseudo triangulations with a convex hull edge removed correspond to canonical basic feasible solutions found by such a linear program. They can be computed very eciently geometrically, ....

R. Connelly, E. Demaine and G. Rote. Straightening Polygonal Arcs and Convexifying Polygonal Cycles, Proc. 41st Symp. on Foundations of Computer Science (FOCS), 2000, pp. 432-442.

.... [9, 19, 38] visibility graph or visibility complex counting [37, 38, 36] covering and separation [41] collision detection [2, 25, 24, 23] stretchability and realizability of pseudoline arrangements [36, 38, 18] and, last but not least, planning expansive motions of polygonal chains [33, 12, 47]. A key feature of pseudo triangulations that is largely exploited in the applications mentioned above is the existence of a ip operation e.g. in Figure 2 the rightmost pseudo triangulation is obtained from the leftmost pseudo triangulation after a sequence of six ip operations (see also left ....

R. Connelly, E. D. Demaine, and G. Rote. Straightening polygonal arcs and convexifying polygonal cycles. In Proc. 41th Annu. IEEE Sympos. Found. Comput. Sci., pages 432-442, 2000.

....it is called locked. This terminology is borrowed from [BDD 99] and [BDD 98] 1 Most of the work in this area was fueled by the longstanding open problem of determining whether every open (or closed) chain in 2D can be straightened (or convexified) This was recently settled [CDR00] in the affirmative: 2D chains cannot lock. In contrast it was earlier established that trees in 2D [BDD 98] and both open and closed chains in 3D [CJ98, BDD 99] can lock. In this paper we prove that, for all dimensions d 4, neither chains (open or closed) nor trees can lock. We ....

R. Connelly, E. D. Demaine, and G. Rote. Straightening polygonal arcs and convexifying polygonal cycles. In Proc. 41st Annu. IEEE Sympos. Found. Comput. Sci. IEEE, November 2000. To appear.

....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 lock in 2D [CDR00] Their result is even more general: no collection of disjoint simple chains are locked (although of course a chain nested inside a polygon is forever confined) They prove that every such collection has an expansive motion: one during which the distance between every pair of vertices increases ....

R. Connelly, E. D. Demaine, and G. Rote. Straightening polygonal arcs and convexifying polygonal cycles. In Proc. 41st Annu. IEEE Sympos. Found. Comput. Sci. IEEE, November 2000. To appear.

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Robert Connelly, Erik D. Demaine, and Gunter Rote, Straightening polygonal arcs and convexifying polygonal cycles. Discrete & Computational Geometry, to appear. Preliminary version in Proc. 41st Ann. Symp. on Found. of Computer Science (FOCS 2000.

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Robert Connelly, Erik D. Demaine, and Gunter Rote. Straightening polygonal arcs and convexifying polygonal cycles. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pages 432-442, Redondo Beach, California, November 2000.

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R. Connelly, E. Demaine and G. Rote. Straightening polygonal arcs and convexifying polygonal cycles, Discrete and Comp. Geometry (2003), 205-- 239.

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R. Connelly, E. D. Demaine, and G. Rote. Straightening polygonal arcs and convexifying polygonal cycles. In Proc. 41st Annu. IEEE Sympos. Found. Comput. Sci., pages 432-442. IEEE, November 2000.

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Robert Connelly, Erik D. Demaine, and Gunter Rote, Straightening polygonal arcs and convexifying polygonal cycles. Discrete & Computational Geometry, to appear. Preliminary version in Proc. 41st Ann. Symp. on Found. of Computer Science (FOCS 2000.

....of this work is on the problem of which classes of chains can lock in the sense that they cannot be recon gured to straight or convex con gurations. In 3D, it is known that some chains can lock [4] but the exact class of chains that lock has not been delimited [3] In 2D, no chains can lock [5, 13]. All of these results concern chains with universal joints. Research initiated at the 17th Winter Workshop on Computational Geometry, Bellairs Research Institute of McGill University, Feb. 1 8, 2002. Motivation: Protein Folding. The backbone of a protein can be modeled as a polygonal chain, ....

R. Connelly, E. D. Demaine, and G. Rote. Straightening polygonal arcs and convexifying polygonal cycles. In Proc. 41st Annu. IEEE Sympos. Found. Comput. Sci., pages 432-442. IEEE, November 2000.

.... To be more precise, we consider the complexity of variants of the following Linkage recon guration problem: Given two linkages S and T in d space, can we recon gure S into T In 2D, every simple polygonal chain of xed length segments can be continuously moved to a straight con guration, [10, 5]. Nevertheless, the general question is nontrivial since it is known that simple trees of xed length segments cannot always be moved to con gurations that are essentially at, 5] and that there are polygonal chains in 3D that cannot be moved to a straightened con guration [3, 1] In Section ....

.... 2D, every simple polygonal chain of xed length segments can be continuously moved to a straight con guration, 10, 5] Nevertheless, the general question is nontrivial since it is known that simple trees of xed length segments cannot always be moved to con gurations that are essentially at, [5], and that there are polygonal chains in 3D that cannot be moved to a straightened con guration [3, 1] In Section 2 we observe that the linkage recon guration problem is decidable in polynomial space for any xed dimension d. In Section 3 we complement these upper bounds by showing that the ....

R. Connelly, E. D. Demaine, and G. Rote. Straightening polygonal arcs and convexifying polygonal cycles. Discrete and Computational Geometry, 2003. To appear.

....exploiting the latter. Expansive motions. An expansive motion on a set of points P is an infinitesimal motion of the points such that no distance between them decreases. Expansive motions were instrumental in the first proof of the Carpenter s Rule Theorem by Connelly, Demaine and Rote [6]: Every simple polygon or polygonal arc in the plane can be unfolded into convex position without collisions. Streinu [25] built on this work, realizing the importance of pseudo triangulations in connection with expansive motions and studying their rigidity properties. This paper provides a ....

....of (a) that the removal of an edge creates a (not necessarily expansive) 1DOF mechanism. The expansiveness of pte mechanisms (part (b) was proved in [25] using the Maxwell Cremona correspondence between self stresses and 3 d liftings of planar frameworks, a technique that was introduced in [6]. Self stresses. A self stress (or an equilibrium stress) on a framework G(P ) see [27] or [6, Section 3.1] is an assignment of scalars # ij to edges such #i # P , ij#E # ij (p i p j ) 0. That is, the self stresses are the row dependences of the rigidity matrix M . The proof of the ....

[Article contains additional citation context not shown here]

Robert Connelly, Erik D. Demaine, and Gunter Rote, Straightening polygonal arcs and convexifying polygonal cycles. Discrete & Computational Geometry, to appear. Preliminary version in Proc. 41st Ann. Symp. on Found. of Computer Science (FOCS 2000.

....pseudo triangulation. In particular, the underlying graphs of pointed pseudo triangulations have exactly 2n 3 edges and are in fact Laman graphs. Historical Perspective. Techniques from Rigidity Theory have been recently applied to problems such as collision free robot arm motion planning [9, 37], molecular conformations [21, 46, 43] and sensor and network topologies with distance and angle constraints [11] Laman graphs are the fundamental objects in 2 dimensional Rigidity Theory. Also known as isostatic or generically minimally rigid graphs, they characterize combinatorially the ....

R. Connelly, E. Demaine and G. Rote. Straightening polygonal arcs and convexifying polygonal cycles, to appear, Discrete and Comp. Geometry, 2003.

.... If edges are allowed to cross each other, then this is true in every dimension [12, 16] If edges are 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 ....

....a natural generalization of regular polygons. It is interesting to note that we cannot hope for a distance monotone motion between any two convex polygons, in which every distance between a pair of vertices varies monotonically with time. This is in direct contrast to convexi cation of a polygon [9], where all distances can be made to increase. An example is shown in Figure 3. Because the dotted lines are the same length in both con gurations, these distances must be preserved throughout the motion; in other words, the chains v 1 ; v 2 ; v 3 and v 4 ; v 5 ; v 6 must move rigidly. The ....

Robert Connelly, Erik D. Demaine, and Gunter Rote. Straightening polygonal arcs and convexifying polygonal cycles. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pages 432-442, Redondo Beach, California, November 2000.

....to lie on a straight line, and whether every closed chain can be convexified, i.e. reconfigured to form a planar convex polygon. In both cases, the chains are to remain simple throughout the motion. If a chain cannot be so reconfigured, it is called locked. Connelly, Demaine, and Rote [2] have recently shown that no chain or polygon is locked; Streinu [3] provides an alternative motion. There is a natural equivalence relation on the set of configurations of a linkage: two configurations are equivalent if there is a motion that takes the linkage from one configuration to the ....

R. Connelly, E. D. Demaine, and G. Rote. Straightening polygonal arcs and convexifying polygonal cycles. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pages 432--442, Redondo Beach, California, November 2000.

....and 1995: 20] and [16, p. 270] Initial computational geometry results focused on certain classes of configurations such as visible chains [4] star shaped polygons [9] and monotone polygons [3] Connelly, Demaine, and Rote have recently proved that in the plane, no chain or polygon is locked [7]; Streinu [28] provides an alternative proof. In three dimensions, while a complete characterization isn t known, there are configurations of open polygonal chains and of polygons that can be straightened, or convexified, respectively, and other configurations that can not be [1] In four or more ....

R. Connelly, E. D. Demaine, and G. Rote. Straightening polygonal arcs and convexifying polygonal cycles. In Symposium on Foundations of Computer Science, November 2000. In press.

....to lie on a straight line, and whether every closed chain can be convexified, i.e. reconfigured to form a planar convex polygon. In both cases, the chains are to remain simple throughout the motion. If a chain cannot be so reconfigured, it is called locked. Connelly, Demaine, and Rote [2] have recently shown that no chain or polygon is locked; Streinu [3] provides an alternative motion. There is a natural equivalence relation on the set of configurations of a linkage: two configurations are equivalent if there is a motion that takes the linkage from one configuration to the ....

R. Connelly, E. D. Demaine, and G. Rote. Straightening polygonal arcs and convexifying polygonal cycles. In Symposium on Foundations of Computer Science, November 2000. In press.

....Padualaan 14, De Uithof, 3584 CH Utrecht, The Netherlands, email: markov cs.uu.nl. k School of Computer 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 ....

....[14] in which the vertices are cocircular, a natural generalization of regular polygons. It is interesting to note that we cannot hope for a distance monotone motion between any two convex polygons, in which every distance between a pair of vertices varies monotonically with time (like [8]) An example is shown in Figure 3. Because the dotted lines are the same length in both configurations, these distance must be preserved throughout the motion; in other words, the chains v 1 ; v 2 ; v 3 and v 4 ; v 5 ; v 6 must move rigidly. The problem is thus reduced to moving a quadrangle v 1 ....

R. Connelly, E. D. Demaine, and G. Rote. Straightening polygonal arcs and convexifying polygonal cycles. In Proc. 41st Ann. Symp. Foundations of Computer Science, Redondo Beach, California, Nov. 2000, to appear.

No context found.

R. Connelly, E. D. Demaine, and G. Rote. Straightening polygonal arcs and convexifying polygonal cycles. In Proc. 41th Annu. Sympos. on Found. of Computer Science, pp. 432-442, 2000.

No context found.

R. Connelly, E. D. Demaine, and G. Rote. Straightening polygonal arcs and convexifying polygonal cycles. In Proc. 41th Annu. IEEE Sympos. Found. Comput. Sci., pages 432442, 2000.

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