Locked and Unlocked Chains of Planar Shapes

TL;DR

This paper generalizes linkage unfolding results from polygons to chains of arbitrary planar shapes, identifying conditions for locked configurations and universal foldability, with broad implications for shape reconfiguration.

Contribution

It introduces the concept of slender adornments that ensure universal foldability and characterizes shapes like isosceles triangles that admit locked chains.

Findings

Slender adornments guarantee universal foldability.

Isosceles triangles with apex angles less than 90° admit locked chains.

Polygonal linkage results extend to general planar shapes.

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 less than 90 degrees admit locked chains, which is precisely the threshold beyond which the inward-normal property no longer holds.