Epsilon local rigidity and numerical algebraic geometry

Andrew Frohmader and Alexander Heaton

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Submission date: 06. Nov. 2020
Pages: 18
Bibtex
MSC-Numbers: 70B15, 65D17, 14Q99
Keywords and phrases: rigidity, kinematics, homotopy continuation
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Link to arXiv: See the arXiv entry of this preprint.
DOI number (of the published article): 10.1142/S0219498822500098

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 5 and its corresponding Algorithm 1 which decide if a configuration is epsilon-locally rigid, a notion we define. A configuration which is epsilon-locally rigid may be locally rigid or flexible, but any continuous deformations remain within a sphere of radius epsilon in configuration space. Deciding epsilon-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.