When is a planar rod configuration infinitesimally rigid?

TL;DR

This paper introduces a method to determine the infinitesimal rigidity of planar rod configurations by relating geometric realizations to cone graph rigidity, generalizing the Molecular conjecture.

Contribution

It establishes a criterion linking the infinitesimal rigidity of planar rod configurations to cone graph rigidity, extending previous conjectures.

Findings

Infinitesimal rigidity in regular position is equivalent to cone graph rigidity in generic position.

Provides a new approach to analyze rigidity using cone over point sets.

Generalizes the Molecular conjecture to broader configurations.

Abstract

We provide a way of determining the infinitesimal rigidity of rod configurations realizing a rank two incidence geometry in the Euclidean plane. We model each rod with a cone over its point set and prove that the resulting geometric realization of the incidence geometry is infinitesimally rigid in regular position if and only if the resulting cone graph is infinitesimally rigid in generic position. This is a generalization of the Molecular conjecture.