Trilateration using Unlabeled Path or Loop Lengths

TL;DR

This paper investigates the unique reconstruction of point configurations in Euclidean space from unlabeled path or loop lengths, extending trilateration methods to more complex length data.

Contribution

It extends trilateration techniques to reconstruct point configurations from unlabeled path and loop lengths, providing conditions for unique determination.

Findings

Reconstruction is possible with sufficiently rich unlabeled path/loop lengths.

Complete proof of trilateration process effectiveness for edge lengths.

Conditions for generic configurations ensuring uniqueness.

Abstract

Let $\mathbf{p}$ be a configuration of $n$ points in $\mathbb{R}^d$ for some $n$ and some $d \ge 2$. Each pair of points defines an edge, which has a Euclideanlength in the configuration. A path is an ordered sequence of the points, and a loop is a path that begins and ends at the same point. A path or loop, as a sequence of edges, also has a Euclidean length, which is simply the sum of its Euclidean edge lengths. We are interested in reconstructing $\mathbf{p}$ given a set of edge, path and loop lengths. In particular, we consider the unlabeled setting where the lengths are given simply as a set of real numbers, and are not labeled with the combinatorial data describing which paths or loops gave rise to these lengths. In this paper, we study the question of when $\mathbf{p}$ will be uniquely determined (up to an unknowable Euclidean transform) from some given set of path or loop lengths through an exhaustive trilateration process. Such a process has already been used for the simpler problem of reconstruction using unlabeled edge lengths. This paper also provides a complete proof that this process must work in that edge-setting when given a sufficiently rich set of edge measurements and assuming that $\mathbf{p}$ is generic.