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Chain Reconfiguration The Ins and Outs, Ups and Downs of Moving Polygons and Polygonal Linkages

  • Sue Whitesides
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2223)

Abstract

Apolygonal linkage or chain is a sequence of segments of fixed lengths, free to turn about their endpoints, which act as joints. This paper reviews some results in chain reconfiguration and highlights several open problems1

Keywords

Computational Geometry Simple Polygon Polygonal Chain Algorithmic Motion Planning Rigidity Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    T. Biedl, E. Demaine, M. Demaine, S. Lazard, A. Lubiw, J. O’Rourke, S. Robbins, I. Streinu, G. Toussaint and S. Whitesides. On reconfiguring tree linkages: treescan lock. Accepted in Discrete Applied Math., Feb. 2001, to appear; conference abstract in Proc. of the 10th Canadian Conf. on Computational Geometry CCCG’98, McGill University, Montreal, Canada, Aug. 10–12, 1998, pp. 4–5.Google Scholar
  2. 2.
    T. Biedl, E. Demaine, M. Demaine, S. Lazard, A. Lubiw, J. O’Rourke, M. Overmars, S. Robbins, I. Streinu, G. Toussaint, and S. Whitesides. Locked and unlocked polygonal chains in 3D. Accepted in Discrete and Computation Geom., May, 2001,to appear; conference abstract in Proc. of the 10th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), Baltimore MD, USA, Jan. 1999, pp. 866–867.Google Scholar
  3. 3.
    T. Biedl, E. Demaine, S. Lazard, S. Robbins, and M. Soss. Convexifying monotone polygons. Proc. of the 10th Annual International Symp. on Algorithms and Computation (ISAAC’99), Chennai, India, Dec. 16–18, 1999, Springer-Verlag Lecture Notes in Computer Science, pp. 415–424.zbMATHGoogle Scholar
  4. 4.
    J. Cantarella and H. Johnston. Nontrivial embeddings of polygonal intervals and unknots in 3-space. J. of Knot Theory and its Ramifications, vol. 7 (8), pp. 1027–1039, 1998.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. Cauchy. Sur les polygones et les polyèdres, seconde mémoire. Journal Ecole Polytechnique, vol. 16 (9); pp. 26–38, 1813.Google Scholar
  6. 6.
    R. Cocan and J. O’Rourke. Polygonal chains cannot lock in 4D. Proc. 11th Canadian Conf. on Computational Geometry (CCCG), 1999.Google Scholar
  7. 7.
    R. Connelly, E. Demaine, G. Rote. Straightening polygonal arcs and convexifying polygonal cycles. Proc. of the 41st IEEE Symp. on Foundations of Computer Sciences (FOCS), 2000, pp. 432–442.Google Scholar
  8. 8.
    T. Cormen, C. Leiserson, and R. Rivest. Introduction to Algorithms. Undergraduate textbook, MIT Press and McGraw Hill, 1990.Google Scholar
  9. 9.
    R. G. Downey and M. R. Fellows. Parameterized complexity. Springer-Verlag, New York, 1999.CrossRefzbMATHGoogle Scholar
  10. 11.
    H. Everett, S. Lazard, S. Robbins, H. Schröder and S. Whitesides. Convexifying star-shaped polygons. Proc. of the 10th Canadian Conf. on Computational Geometry CCCG’ 98, McGill University, Montreal, Canada, Aug. 10–12, 1998, pp. 2–3.Google Scholar
  11. 12.
    P. Finn, D. Halperin, L. Kavraki; J-C. Latombe; R. Motwani; C. Shelton, and S. Venkatasubramanian. Geometric manipulation of flexible ligands. Applied Computational Geometry, Springer-Verlag, pp. 67–78, 1996.Google Scholar
  12. 13.
    Aviezri Frankel. Complexity of protein folding. Bulletin of Mathematical Biology, vol. 55 (6), pp. 1199–1210, 1993.CrossRefzbMATHGoogle Scholar
  13. 15.
    J. Hopcroft, D. Joseph, and S. Whitesides. On the movement of robotic arms in 2-dimensional bounded regions. SIAM J. on Computingvol. 14, May1985, pp. 315–333.Google Scholar
  14. 16.
    J. Hopcroft, D. Joseph, and S. Whitesides. Movement problems for 2-dimensional linkages. SIAM J. on Computingvol. 13, Aug. 1984, pp. 610–629.Google Scholar
  15. 17.
    J. Hopcroft, D. Joseph and S. Whitesides. On the movement of robot arms in two-dimensional bounded regions. Proc. of the IEEE 23rd Annual Symp. on the Foundations of Computer Science (FOCS), Chicago IL, USA, Nov. 3–5, 1982, pp. 280–289.Google Scholar
  16. 18.
    V. Kantabutra. Motions of a short-linked robot arm in a square. Discrete Comput. Geom. vol. 7, 1992, pp. 69–76.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 19.
    Vitit Kantabutra. Reaching a point with an unanchored robot arm in a square. Int. J. of Computational Geometry and Applications vol. 7 (6), pp. 539–550, 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 20.
    V. Kantabutra and R. Kosaraju. New algorithms for multilink robot arms. J. Comput. System Sci., vol. 32, pp. 136–153, 1986.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 21.
    Dexter Kozen. The Design and Analysis of Algorithms. Graduate textbook, Springer-Verlag, 1992.Google Scholar
  20. 22.
    M. van Kreveld, J. Snoeyink and S. Whitesides. Folding rulers inside triangles. Discrete and Computational Geometry vol. 15, 1996, pp. 265–285; conference abstract in Proc. of the 5th Candadian Conf. on Computational Geometry, Queen’sU., Kingston, Canada, Aug. 5–10, 1993, pp. 1-6.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 23.
    J. Kutcher. Coordinated Motion Planning of Planar Linkages. Ph.D. thesis, John Hopkins U., 1992.Google Scholar
  22. 24.
    Jean-Claude Latombe. Robot Motion Planning. Kluwer Academic Publishers, Boston, 1991.Google Scholar
  23. 25.
    W. Lenhart and S. Whitesides. Reconfiguring closed polygonal chains in Euclidean d-space. Discrete and Computational Geometry, vol. 13, 1995, pp. 123–140; conference abstracts in Proc. of the 3rd Canadian Conf. on Computational Geometry, Vancouver, Canada, Aug. 6–10, 1991, pp. 66-69(“Turning a Polygon Insideout”), and in Proc. of the 4th Canadian Conf. on Computational Geometry, St. John’s, Newfoundland, Canada, Aug. 10–14, 1992, pp. 198-203 (“Reconfiguring with Linetracking Motions”); see also Reconfiguring Simple Polygons, technical report, McGill University, School of Computer Science SOCS-93.3, 1993.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 26.
    A. Lubiw and J. O’Rourke. When can a polygon fold to a polytope? Technical Report 048, Dept. of Computer Science, Smith College, June 1996.Google Scholar
  25. 27.
    Joseph O’Rourke. Chapter 8.6, Computational Geometry in C. Cambridge University Press, 1998.Google Scholar
  26. 28.
    J. O’Rourke. Folding and unfolding in computational geometry. Proc. Japan Conf. Discrete Comput. Geom., Dec. 1998, LNCS vol. 1763, pp. 258–266, 1999.zbMATHGoogle Scholar
  27. 29.
    Naixun Pei. On the Reconfiguration and Reachability of Chains. Ph.D. thesis, School of Computer Science, McGill U., 1996.Google Scholar
  28. 30.
    N. Pei and S. Whitesides. On folding rulers in regular polygons. Proc. of the 9th Canadian Conf. on Computational Geometry CCCG’ 97, Queen’s University, Kingston, Ontario, Canada, Aug. 11–14, 1997, pp. 11–16.Google Scholar
  29. 31.
    N. Pei and S. Whitesides. On the reachable regions of chains. Proc. of the 8th Canadian Conf. on Computational Geometry CCCG’ 96, Carleton University, Ottawa, Ontario, Canada, Aug. 12–15, 1996, pp. 161–166.Google Scholar
  30. 32.
    S. Whitesides and N. Pei. On the reconfiguration of chains. Computing and Combinatorics, Proc. of the 2nd Annual International Conf., COCOON’ 96, Hong Kong, June 17–19, 1996, J-Y Cai and C-K Wong, eds., Springer-Verlag Lecture Notes in Computer Science LNCSvol. 1090, pp. 381–390.Google Scholar
  31. 33.
    Micha Sharir. Algorithmic motion planning. J. E. Goodman and J. O’Rourke, eds., Handbook of Discrete and Computational Geometry, chapter 40, pp. 733–754, CRC Press, Boca Raton FL, 1997.Google Scholar
  32. 34.
    J. Schwartz and M. Sharir. On the “piano mover’s” problem, II. General techniques for computing topological properties of real algebraic manifolds. Advances in Applied Math. vol. 4, pp. 298–351, 1983.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 35.
    Ileana Streinu A combinatorial approach to planar non-colliding robot arm mtion planning. Proc. of the 41st IEEE Symp. on Foundations of Computer Sciences (FOCS), 2000, pp. 443–453. 2, 10Google Scholar
  34. 36.
    Godfried Toussaint. The Erdős-Nagy theorem and its ramifications. Proc. 11th Canadian Conf. on Computational Geometry, Vancouver, Aug. 1999.Google Scholar
  35. 37.
    Sue Whitesides. Algorithmic issues in the geometry of planar linkage movement. Australian Computer Journal, vol. 24 (2), pp. 42–50, 1992.Google Scholar
  36. 38.
    S. Whitesides and R. Zhao. Algorithmic and complexity results for drawing Euclidean trees. Advanced Visual Interfaces, Proc. of the InternationalWorkshop AVI’ 92, Rome, Italy, May 25–29, 1992, T. Catarci, M. F. Costabile, and S. Levialdi, eds.. World Scientific Series in Computer Science vol. 36, 1992, pp. 395–410.Google Scholar
  37. 39.
    Rongyao Zhao. Placements of EuclideanTrees. Ph. D. thesis, School of Computer Science, McGill U., 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Sue Whitesides
    • 1
  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada

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