Affine rigidity and conics at infinity

Authors: Robert Connelly, Steven J. Gortler, and Louis Theran
Journal: International Mathematics Research Notices, 2018134084--4102, 2017.
Full text: arXiv DOI

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 its edge directions lie on a conic at infinity 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.