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”, Discrete & Computational Geometry, volume 44, number 2, 2010, pages 439–462.

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 distance from each edge along the path along the boundary of the slender adornment to each hinge should be monotone. In contrast, we show that isosceles triangles with any desired apex angle < 90° admit locked chains, which is precisely the threshold beyond which the slender property no longer holds.

Comments:

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

Length:

The paper is 23 pages.

Availability:

The paper is available in PDF (509k).

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