Robert Connelly, Erik D. Demaine, Martin L. Demaine, Sándor
P.
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 Symposium on
Computational Geometry, Sedona, June 5–7, 2006. Association
for Computing Machinery, 2006, pp. 61–70.
doi:10.1145/1137856.1137868.

Discrete and Computational
Geometry 44 (2010), 439–462. doi:10.1007/s0045401092623,
arXiv:cs/0604022 [cs.CG].
Abstract
We extend linkage unfolding results from the wellstudied 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
selfintersection, 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 less than 90 degrees admit locked chains, which is
precisely the threshold beyond which the inwardnormal property no longer
holds.
Last update: August 15, 2017.