An official website of the United States government
Here’s how you know
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
Secure .gov websites use HTTPS
A lock (
) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.
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.
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 November 14, 2024)