Affine Rigidity and Conics at Infinity

TL;DR

This paper establishes a connection between affine rigidity, conics at infinity, and the existence of affine flexes in frameworks, providing new insights into rigidity properties and invariance under projective transformations.

Contribution

It proves that affine flexes occur if and only if the framework is ruled on a single quadric, simplifying previous results and unifying concepts in rigidity theory.

Findings

Affine flexes correspond to frameworks ruled on a single quadric.

Super stability is invariant under projective transforms, coning, and slicing.

Unified previous results on the Strong Arnold Property of matrices.

Abstract

We prove that if a framework of a graph is neighborhood affine rigid in $d$-dimensions (or has the stronger property of having an equilibrium stress matrix of rank $n-d-1$) then it has an affine flex (an affine, but non Euclidean, transform of space that preserves all of the edge lengths) if and only if the framework is ruled on a single quadric. This strengthens and also simplifies a related result by Alfakih. It also allows us to prove that the property of super stability is invariant with respect to projective transforms and also to the coning and slicing operations. Finally this allows us to unify some previous results on the Strong Arnold Property of matrices.