Paper by Erik D. Demaine

Reference:

Robert Connelly, Erik D. Demaine, Martin L. Demaine, Sándor Fekete, Stefan Langerman, Joseph S. B. Mitchell, Ares Ribó, and Günter Rote, “Locked and Unlocked Chains of Planar Shapes”, in Proceedings of the 22nd Annual ACM Symposium on Computational Geometry (SoCG 2006), Sedona, Arizona, June 5–7, 2006, pages 61–70.

Abstract:

We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged together sequentially at rotatable joints. Our goal is to characterize the familes of planar shapes that admit locked chains, where some configurations cannot be reached by continuous reconfiguration without self-intersection, and which families of planar shapes guarantee universal foldability, where every chain is guaranteed to have a connected configuration space. Previously, only obtuse triangles were known to admit locked shapes, and only line segments were known to guarantee universal foldability. We show that a surprisingly general family of planar shapes, called slender adornments, guarantees universal foldability: roughly, the inward normal from any point on the shape's boundary should intersect the line segment connecting the two incident hinges. In constrast, we show that isosceles triangles with any desired apex angle < 90° admit locked chains, which is precisely the threshold beyond which the inward-normal property no longer holds.

Comments:

This paper is also available as arXiv:cs.CG/0604022 of the Computing Research Repository (CoRR).

Length:

The paper is 10 pages.

Availability:

The paper is available in PostScript (656k), gzipped PostScript (199k), and PDF (301k).

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LockedShapes_DCG (Locked and Unlocked Chains of Planar Shapes)