Locked and Unlocked Chains of Planar Shapes |
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Authors: | Robert Connelly Erik D Demaine Martin L Demaine Sándor P Fekete Stefan Langerman Joseph S B Mitchell Ares Ribó Günter Rote |
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Institution: | 1. Department of Mathematics, Cornell University, Ithaca, NY, 14853, USA 2. MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA, 02139, USA 3. Department of Computer Science, Braunschweig University of Technology, Mühlenpfordtstr. 23, 38106, Braunschweig, Germany 4. Chercheur qualifié du FNRS, Département d’informatique, Université Libre de Bruxelles, ULB CP212, Bruxelles, Belgium 5. Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY, 11794-3600, USA 6. Institut für Informatik, Freie Universit?t Berlin, Takustra?e 9, 14195, Berlin, Germany
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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 families 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. |
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