Fourier, Gauss, Fraunhofer, Porod and the Shape from Moments Problem
Mathematical Physics
2015-05-30 v1 math.MP
Data Analysis, Statistics and Probability
Optics
Abstract
We show how the Fourier transform of a shape in any number of dimensions can be simplified using Gauss's law and evaluated explicitly for polygons in two dimensions, polyhedra three dimensions, etc. We also show how this combination of Fourier and Gauss can be related to numerous classical problems in physics and mathematics. Examples include Fraunhofer diffraction patterns, Porods law, Hopfs Umlaufsatz, the isoperimetric inequality and Didos problem. We also use this approach to provide an alternative derivation of Davis's extension of the Motzkin-Schoenberg formula to polygons in the complex plane.
Keywords
Cite
@article{arxiv.1109.5262,
title = {Fourier, Gauss, Fraunhofer, Porod and the Shape from Moments Problem},
author = {Gregg M. Gallatin},
journal= {arXiv preprint arXiv:1109.5262},
year = {2015}
}
Comments
21 pages, no figures