MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Epsilon local rigidity and numerical algebraic geometry

Andrew Frohmader and Alexander Heaton


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.

MSC Codes:
70B15, 65D17, 14Q99
rigidity, kinematics, homotopy continuation

Related publications

2022 Repository Open Access
Andrew Frohmader and Alexander Heaton

Epsilon local rigidity and numerical algebraic geometry

In: Journal of algebra and its applications, 21 (2022) 1, p. 2250009