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Fourier, Gauss, Fraunhofer, Porod and the Shape from Moments Problem

Published

Author(s)

Gregg M. Gallatin

Abstract

We show how the Fourier transform of a shape in any number of dimensions can be simplified using Gauss' 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 Dido's problem. We also use this approach to provide an alternative derivation of Davis' extension of the Motzkin-Schoenberg formula to polygons in the complex plane.
Citation
Journal of Mathematical Physics
Volume
53
Issue
1

Keywords

Gauss law, Fourier transforms, Fraunhofer diffraction, Porods law, isoperimetric inequality, Hopfs Umlaufsatz, shape from moments

Citation

Gallatin, G. (2012), Fourier, Gauss, Fraunhofer, Porod and the Shape from Moments Problem, Journal of Mathematical Physics, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=909668 (Accessed February 27, 2024)
Created January 17, 2012, Updated February 19, 2017