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.