Paper by Erik D. Demaine

Reference:

Robert Connelly, Erik D. Demaine, and Günter Rote, “Straightening Polygonal Arcs and Convexifying Polygonal Cycles”, Technical Report B 02-02, Fachbereich Mathematik und Informatik, Serie B - Informatik, Freie Universität Berlin, February 2002.

BibTeX

@TechReport{LinkageTR,
  AUTHOR        = {Robert Connelly and Erik D. Demaine and G\"unter Rote},
  TITLE         = {Straightening Polygonal Arcs and Convexifying Polygonal
                   Cycles},
  SERIES        = {Fachbereich Mathematik und Informatik, Serie B - Informatik},
  INSTITUTION   = {Freie Universit\"at Berlin},
  NUMBER        = {B 02-02},
  MONTH         = {February},
  YEAR          = 2002,

  LENGTH        = {49 pages},
  PAPERS        = {Linkage; FOCS2000a; EuroCG2000},
  WEBPAGES      = {linkage},
  UPDATES       = {Ivars Peterson wrote an article describing these results,
                   &ldquo;<A HREF="http://www.sciencenews.org/20000923/bob1.asp">Unlocking
                   Puzzling Polygons</A>&rdquo;,
                   <I><A HREF="http://www.sciencenews.org/">Science
                   News</A></I> 158(13):200-201, September 23, 2000.
                   <P>
                   Joseph O'Rourke also wrote an article describing these
                   results,
                   &ldquo;<A HREF="http://arXiv.org/abs/cs.CG/0007042">Computational
                   Geometry Column 39</A>&rdquo;,
                   <I><A HREF="http://www.worldscinet.com/ijcga/ijcga.shtml">International
                   Journal of Computational Geometry and Applications</A></I>,
                   10(4):441-444, 2000.},
}

Abstract:

Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths. Furthermore, our motion is piecewise-differentiable, does not decrease the distance between any pair of vertices, and preserves any symmetry present in the initial configuration. In particular, this result settles the well-studied carpenter's rule conjecture.

Updates:

Ivars Peterson wrote an article describing these results, “Unlocking Puzzling Polygons”, Science News 158(13):200-201, September 23, 2000.

Joseph O'Rourke also wrote an article describing these results, “Computational Geometry Column 39”, International Journal of Computational Geometry and Applications, 10(4):441-444, 2000.

Length:

The paper is 49 pages.

Availability:

The paper is available in PDF (389k).

See information on file formats.

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Related papers:

Linkage (Straightening Polygonal Arcs and Convexifying Polygonal Cycles)

FOCS2000a (Straightening Polygonal Arcs and Convexifying Polygonal Cycles)

EuroCG2000 (Every Polygon Can Be Untangled)

Related webpages:

Carpenter's Rule Theorem