Trilateration using unlabeled path or loop lengths

Ioannis Gkioulekas, Steven J. Gortler, Todd Zickler, Louis S. Theran*

*Corresponding author for this work

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Abstract

Let p be a configuration of n points in Rd for some n and some d≥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 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 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 p is generic.
Original languageEnglish
Number of pages43
JournalDiscrete & Computational Geometry
VolumeFirst Online
Early online date25 Nov 2023
DOIs
Publication statusE-pub ahead of print - 25 Nov 2023

Keywords

  • Distance geometry
  • Rigidty theory
  • Global rigidity

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