Rigidity of Circle Packings with Flexible Radii

TL;DR

This paper investigates the stability of circle packings with flexible radii, providing dual conditions for first-order rigidity and exploring challenges beyond first-order analysis.

Contribution

It introduces a dual condition for first-order rigidity of circle packings with flexible radii and discusses complexities beyond first-order rigidity analysis.

Findings

Provides a dual condition for first-order rigidity.

Establishes a sufficient condition for rigidity.

Explores difficulties in higher-order rigidity analysis.

Abstract

Circle packings are arrangement of circles satisfying specified tangency requirements. Many problems about packing of circles and spheres occur in nature particularly in material design and protein structure. Surprisingly, little is known about the stability and rigidity of circle packings. In this paper, we study the rigidity of circle packings representing a given planar graph. The radii of circles are flexible with equality and inequality constraints. We provide a dual condition for the packing to be rigid in the first order. This gives us a sufficient condition to show a packing is rigid. Then we will explore the difficulties on rigidity problems beyond the first order.