Locked and unlocked chains of planar shapes

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 inci...

• Type: Conference
• Year: 2006
• Where: COMPGEOM
• Authors: Robert Connelly, Erik D. Demaine, Martin L. Demaine, Sándor P. Fekete, Stefan Langerman, Joseph S. B. Mitchell, Ares Ribó, Günter Rote
• Keywords: Admit Locked Chains, COMPGEOM 2006, Planar Shapes, Universal Foldability