Characterizing generic global rigidity

A configuration of a graph is a map from its vertices to Euclidean space. A configuration is globally rigid if its edge lengths determine it up to rigid motion. For which graphs is a generic configuration globally rigid? We answer this question by proving a conjecture by Connelly, that his sufficient condition is also necessary. The condition comes from considering the geometry of the length-squared mapping l; essentially, the graph is generically locally rigid iff the rank of l is maximal, and it is generically globally rigid iff the rank of the Gauss map on the image of l is maximal.

This is joint work with Steven Gortler and Alex Healy.