On affine rigidity

TL;DR

This paper introduces the concept of affine rigidity for hypergraphs, providing fundamental results that link affine rigidity to graph connectivity and applications in point registration and manifold learning.

Contribution

It defines affine rigidity for hypergraphs, shows it is a generic property, and connects high vertex connectivity to universal rigidity in geometric frameworks.

Findings

Affine rigidity can be determined by the rank of a specific matrix.

Graphs with (d+1)-vertex connectivity are generically neighborhood affinely rigid in d-dimensional space.

Such rigidity implies universal rigidity of frameworks of their squared graphs.

Abstract

We define the notion of affine rigidity of a hypergraph and prove a variety of fundamental results for this notion. First, we show that affine rigidity can be determined by the rank of a specific matrix which implies that affine rigidity is a generic property of the hypergraph.Then we prove that if a graph is is $(d+1)$-vertex-connected, then it must be "generically neighborhood affinely rigid" in $d$-dimensional space. This implies that if a graph is $(d+1)$-vertex-connected then any generic framework of its squared graph must be universally rigid. Our results, and affine rigidity more generally, have natural applications in point registration and localization, as well as connections to manifold learning.