A Proper Definition of Higher Order Rigidity

TL;DR

This paper proposes a rigorous definition of higher order rigidity in bar-and-joint frameworks based on elongation along configuration space paths, clarifying the relationship with classical derivative-based definitions.

Contribution

It introduces a proper, geometrically motivated definition of n-th order rigidity and establishes its connection to classical derivative-based criteria.

Findings

Classical derivative-based n-th order rigidity is sufficient for n-th flexibility.

For n=1, 2, classical criteria are also necessary.

The new definition resolves previous ambiguities in higher order rigidity concepts.

Abstract

[Connelly and Servatius, 1994] shows the difficulty of properly defining n-th order rigidity and flexiblity of a bar-and-joint framework for higher order (n >= 3) through the introduction of a cusp mechanism. The author proposes a "proper" definition of the order of rigidity by the order of elongation of the bars with respect to the arclength along the path in the configuration space. We show that the classic definition using formal n-th derivative of the length constraint is a sufficient condition for the n-th flexiblity in the proposed definition and also a necessary condition only for n = 1, 2.