Trilateration using unlabeled path or loop lengths
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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.
Gkioulekas , I , Gortler , S J , Zickler , T & Theran , L S 2023 , ' Trilateration using unlabeled path or loop lengths ' , Discrete & Computational Geometry , vol. First Online . https://doi.org/10.1007/s00454-023-00605-x
Discrete & Computational Geometry
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DescriptionFunding: Steven Gortler was partially supported by NSF grant DMS-1564473. Ioannis Gkioulekas and Todd Zickler received support from the DARPA REVEAL program under contract no. HR0011-16-C-0028.
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