Let \(\textbf{p}\) 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 Euclidean length 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 \(\textbf{p}\) 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 \(\textbf{p}\) 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 \(\textbf{p}\) is generic.

令\(\textbf{p}\)为\(\mathbb R^d\)中n 个点的配置，其中某些n和某些\(d \ge 2\)。每对点定义一条边，该边在配置中具有欧几里德长度。路径是点的有序序列，循环是在同一点开始和结束的路径。作为边序列的路径或循环也具有欧几里德长度，它只是其欧几里德边长度的总和。我们感兴趣的是在给定一组边、路径和循环长度的情况下重建\(\textbf{p}\) 。特别是，我们考虑未标记的设置，其中长度简单地作为一组实数给出，并且没有用描述哪些路径或循环产生这些长度的组合数据进行标记。在本文中，我们研究了何时通过详尽的三边测量过程从某些给定的路径或循环长度集唯一确定\(\textbf{p}\) （直到不可知的欧几里得变换）的问题。这样的过程已经被用于使用未标记的边缘长度进行重建的更简单的问题。本文还提供了完整的证明，证明当给定足够丰富的边缘测量集并假设\(\textbf{p}\)是通用的时，该过程必须在该边缘设置中起作用。