This is not a direct answer, but rather an historical, and somewhat tangential comment. Back in 1979, the general problem you posed was proved NP-hard by Saxe and by Yemini independently. There has been quite a rich literature on this topic in the last 35+ years, which you might trace via Google Scholar. Added. For example, see the 2014 survey below.
James B. Saxe. Embeddability of weighted graphs in k-space is strongly NP-hard. In Proceedings of the 17th Allerton Conference on Communications, Control, and Computing, pp. 480–489, 1979. Also in James B. Saxe: Two Papers on Graph Embedding Problems, Department of Computer Science, Carnegie-Mellon University, 1980. (PDF download.)
Yechiam Yemini. Some theoretical aspects of position-location problems. In 20th Annual Symposium on Foundations of Computer Science (FOCS), pp. 1–8, Oct. 1979. DOI: 10.1109/SFCS.1979.39 (ACM link.)
Liberti, Leo, Carlile Lavor, Nelson Maculan, and Antonio Mucherino. "Euclidean distance geometry and applications." Siam Review 56(1) (2014): 3-69. (Journal link.)
(Figure from "Untangling planar graphs from a specified vertex position—Hard cases" ([Elsevier link][3]).)