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Computer Science: Faculty Publications Computer Science

1999 Locked and Unlocked Polygonal Chains in 3D Therese Biedl University of Waterloo

Erik D. Demaine University of Waterloo

Martin L. Demaine University of Waterloo

Sylvain Lazard Institut National de Recherche en Informatique et en Automatique, Lorraine

Anna Lubiw University of Waterloo

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Recommended Citation Biedl, Therese; Demaine, Erik D.; Demaine, Martin L.; Lazard, Sylvain; Lubiw, Anna; O'Rourke, Joseph; Overmars, Mark; Robbins, Steve; Streinu, Ileana; Toussaint, Godfried; and Whitesides, Sue, "Locked and Unlocked Polygonal Chains in 3D" (1999). Computer Science: Faculty Publications, Smith College, Northampton, MA. https://scholarworks.smith.edu/csc_facpubs/84

This Article has been accepted for inclusion in Computer Science: Faculty Publications by an authorized administrator of Smith ScholarWorks. For more information, please contact [email protected] Authors Therese Biedl, Erik D. Demaine, Martin L. Demaine, Sylvain Lazard, Anna Lubiw, Joseph O'Rourke, Mark Overmars, Steve Robbins, Ileana Streinu, Godfried Toussaint, and Sue Whitesides

This article is available at Smith ScholarWorks: https://scholarworks.smith.edu/csc_facpubs/84 arXiv:cs/9811019v1 [cs.CG] 11 Nov 1998 h egho ahln n h ipiiyo h hi are is chain chains the con- closed of analogous for simplicity The cept the movement. and the both throughout link that maintained each manner a of such length in the segments chain of sequence polygonal polygonal straight simple simple a open, of an that be movements say can We study 3D. we in chains paper, this In Abstract n ntae tteBlar eerhIsiueo cilU McGill of Institute Research Bellairs to the Correspondence at initiated and rrcngrscoe hiswt rsiglnsi di- in links crossing with chains mensions obstacles; closed of presence reconfigures the or in moving chains links, re- planar crossing concerns with the typically on chains of research writing. configuration this geometry at geometry computational open computational remains Previous but the years, in for community sim- circulated maintaining while has plane plicity the chain in open straightened simple be planar, can every example, whether For of question difficult. the surprisingly proved closed have and chains open of reconfiguration concerning questions hogottermtos h cwrzSai elde- cell Schwartz-Sharir simple The remain chains motions. that their require throughout and 3D in work we per osctvl ondsget o edges) (or segments joined consecutively closed aho,ie,if i.e., fashion, xdlengths fixed A Introduction 1 a with 3D moves. in of number simple polynomial algo- planar an are a provide convexifying there we for that and rithm chains, show closed can we locked it chains, but plane, unknotted chain closed some For open onto an straightened. if projection be that orthogonal show simple we a locked, has are straight- that 3D be in cannot called chains are that convexified Chains or ened polygon. convex planar a ¶ ∗ oyoa chain polygonal § ‡ † teh nvriy teh,TeNetherlands. 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[CJ99]. in closed independently for established result same the hogotaymto.Ti emt opeigatre- to a exterior completing knot permits foil This motion. any throughout tcnb hw that shown be can it fln egh n radius and lengths link of eesrihee.B otaito,then, contradiction, By straightened. were odo length of cord enx hwta o l pncan a estraight- chain be the may chains Consider open all ened. not that show next We Chains Locked 3 in polynomial time si i.1 n a hn of think can One needles, knitting 1. rigid Fig. in as ri iso h ufc facne oyoe hni may it then polytope, convex a in plane, of straightened surface a be the onto on projection lies orthogonal it or simple a has either bu nai hog joint through axis an about xdi eeec rm tahdt oeedges. some to 2.1. attached Theorem frame reference a in fixed u,weefrec a oc- each moves for joint where individual cur, of move, number each constant During (small) “moves.” reconfigura- a of open sequences compute projec- are two algorithms that straighten tions of Our to one conditions. either algorithms tion satisfy that are chains results Projections polygonal Simple first with Chains Our Open 2 algorithms. in polynomial-time exponential therefore singly is are paper goal that this [Can87] solutions in algorithm to roadmap consider leads Canny’s we and problems decidable, the are all robotics that algorithmic shows from [SS83] approach composition † y“doubling” By .Whitesides S. ‡ L ¶ hr xs okdoe n locked and open locked exist There fa pnplgnlcanof chain polygonal open an If = O .Lazard S. .Robbins S. n ℓ K B ( 1 . n + hc ol eukotdif unknotted be would which , e ) n onn npit,w prove we endpoints, joining and v 0 0 † ℓ oe.Teagrtm u in run algorithms The moves. and 2 and r K + v faball a of v i ℓ e i ihteai frotation of axis the with , † ( = 3 +1 4 v yaporaechoice appropriate By . ∗ once yaflexible a by connected , 5 K † oae monotonically rotates v eanetro to exterior remain 0 scmoe ftwo of composed as v , . . . , B etrdon centered 5 configured ) K slocked. is n n Our . links v K B 1 , 2

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Figure 2: The arch A after the ith step, i.e., after “picking up” P (0,i) into A. (The planes Πxy and Πε are not distinguished in this figure.) Figure 1: A locked open chain K (“knitting needles”). ones) may be convexified in its plane in O(i) moves. After convexification, the arch is rotated up into the pockets, i.e., rotating the pocket chain into 3D and ′ ′ back down to the plane. This simple procedure was vertical plane containing the new arch base v0vi+1, and suggested by Erd˝os [Erd35] and proved to work by de the procedure is repeated. Sz. Nagy [dSN39]. The number of flips, however, cannot Theorem 4.1. The “St. Louis Arch” Algorithm con- be bound as a function of the number of vertices n of the 2 vexifies a planar of n vertices in O(n ) polygon, as first proved by Joss and Shannon [Gr¨u95]. moves; it runs in time polynomial in n. We offer a new algorithm for convexifying planar closed chains, which we call the “St. Louis Arch” algo- 5 Open Problems rithm. It is more complicated than flipping but uses a bounded number of moves. It models the intuitive ap- Two of the most prominent among the many open proach of picking up the polygon into 3D. We discretize problems suggested by our work are: this to lifting vertices one by one, accumulating the at- 1. What is the complexity of deciding whether a chain tached links into a convex “arch” A in a vertical half- (open or closed) in 3D is locked? plane above the remaining polygonal chain. Although 2. Can a closed chain with a simple projection always the algorithm is conceptually simple, some care is re- be convexified? quired to make it precise, and to then establish that simplicity is maintained throughout the motions. References Let P be a simple polygon in the xy-plane, Πxy. Let Πε be the plane z = ε parallel to Πxy, for ε > 0. [Can87] J. Canny. The Complexity of Robot Motion Plan- The value of ε is determined by the initial geometry of ning. ACM – MIT Press Doctoral Dissertation Award P in a complex way. We use this plane to convexify Series. MIT Press, Cambridge, MA, 1987. the arch safely above the portion of the polygon not [CJ99] J. Cantarella and H. Johnston. Nontrivial embed- yet picked up. We use primes to indicate positions of dings of polygonal intervals and unknots in 3-space. J. Knot Theory Ramifications, 1999. To appear. moved (raised) vertices. Let P [i, j] represent the chain [dSN39] B. de Sz. Nagy. Solution to problem 3763. Amer. v , v ,...,v v v i < ( i i+1 j ), including i and j (where 0 ≤ Math. Monthly, 46:176–177, 1939. j