1 research outputs found
A Generalized Carpenter's Rule Theorem for Self-Touching Linkages
The Carpenter's Rule Theorem states that any chain linkage in the plane can
be folded continuously between any two configurations while preserving the bar
lengths and without the bars crossing. However, this theorem applies only to
strictly simple configurations, where bars intersect only at their common
endpoints. We generalize the theorem to self-touching configurations, where
bars can touch but not properly cross. At the heart of our proof is a new
definition of self-touching configurations of planar linkages, based on an
annotated configuration space and limits of nontouching configurations. We show
that this definition is equivalent to the previously proposed definition of
self-touching configurations, which is based on a combinatorial description of
overlapping features. Using our new definition, we prove the generalized
Carpenter's Rule Theorem using a topological argument. We believe that our
topological methodology provides a powerful tool for manipulating many kinds of
self-touching objects, such as 3D hinged assemblies of polygons and rigid
origami. In particular, we show how to apply our methodology to extend to
self-touching configurations universal reconfigurability results for open
chains with slender polygonal adornments, and single-vertex rigid origami with
convex cones.Comment: 20 pages, 7 figure