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. 2020 Jun 26;10:10489. doi: 10.1038/s41598-020-66793-1

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© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply 2020

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Schematic of the theory of 3D/2D reconstructions from two projections. A 3D object can be treated as a stack of 2D objects, while 3D map projections can be treated as a stack of 1D projection lines from 2D objects along the same projection angle. Given that one 2D object is composited with a total of N_k atoms, in the coordinates of (x_i, y_i) or (r_i, θ_i), two projection lines could be sufficient to determine the coordinates of each atom. There is no requirement regarding how large the angle between two projection angles needs to be.