Epsilon local rigidity and numerical algebraic geometry

Abstract

A well-known combinatorial algorithm can decide generic rigidity in the plane by determining if the graph is of Pollaczek–Geiringer–Laman type. Methods from matroid theory have been used to prove other interesting results, again under the assumption of generic configurations. However, configurations arising in applications may not be generic. We present Theorem 4.2 and its corresponding Algorithm 1 which decide if a configuration is [Formula: see text]-locally rigid, a notion we define. A configuration which is [Formula: see text]-locally rigid may be locally rigid or flexible, but any continuous deformations remain within a sphere of radius [Formula: see text] in configuration space. Deciding [Formula: see text]-local rigidity is possible for configurations which are smooth or singular, generic or non-generic. We also present Algorithms 2 and 3 which use numerical algebraic geometry to compute a discrete-time sample of a continuous flex, providing useful visual information for the scientist.