Determining generic point configurations from unlabeled path or loop lengths

In this paper we study the problem of reconstructing a configuration of points in \(d\ge 2\) dimensions from an unlabeled sequence of Euclidean lengths arising under an ensemble of paths or loops. We provide a sufficient trilateration-based condition for the reconstruction to be uniquely determined and a numerical procedure for performing this reconstruction.

Our results are obtained by completely characterizing the linear automrophisms of the “unsquared measurement variety” of \(n\) points in \(d\) dimensions for all \(n\) and \(d\). The special case of \(n=4\) and \(d=2\) corresponds to the well known Regge symmetries of the tetrahedron.