Immobilizing Hinged Polygons

The immobilization of non-rigid objects is a largely unaddressed subject. We explore the problem by studying the immobilization of a serial chain of polygons connected by rotational joints, or hinges, in a given placement with frictionless point contacts. We show that n + 2 such contacts along edge interiors or at concave vertices suffice to immobilize any serial chain of n ≠ 3 polygons without parallel edges; it remains open whether five contacts can immobilize three hinged polygons. At most n + 3 contacts suffice to immobilize a serial chain of n polygons when the polygons are allowed to have parallel edges. We also study a robust version of immobility, comparable to the classical notion of form closure, which is insensitive to perturbations. The robustness is achieved at the cost of a small increase in the number of contacts: [Formula: see text] and [Formula: see text] frictionless point contacts suffice for a chain of n hinged polygons without and with parallel edges respectively.