Exploring the infinitesimal rigidity of planar configurations of points and rods

TL;DR

This paper investigates the rigidity of planar point-line configurations, extending known results, providing necessary conditions for minimal rigidity, and exploring the rigidity and flexibility of specific geometric configurations.

Contribution

It extends Whiteley's results to 3-uniform hypergraphs, offers necessary conditions for minimal rigidity, and analyzes rigidity properties of v_k-configurations in the plane.

Findings

Necessary conditions for minimal rigidity of 3-uniform hypergraphs.

Examples of rigid and flexible v_3-configurations.

Discussion of the relationship between incidence geometry and graph rigidity.

Abstract

This article is concerned with the rigidity properties of geometric realizations of incidence geometries of rank two as points and lines in the Euclidean plane; we care about the distance being preserved among collinear points. We discuss the rigidity properties of geometric realizations of incidence geometries in relation to the rigidity of geometric realizations of other well-known structures, such as graphs and hypergraphs.The $2$-plane matroid is also discussed. Further, we extend a result of Whiteley to determine necessary conditions for an incidence geometry of points and lines with exactly three points on each line, or 3-uniform hypergraphs, to have a minimally rigid realization as points and lines in the plane. We also give examples to show that these conditions are not sufficient. Finally, we examine the rigidity properties of $v_k$-configurations. We provide several examples of rigid $v_3$-configurations, and families of flexible geometric $v_3$-configurations. The exposition of the material is supported by many figures.