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

TL;DR

This paper introduces a unified approach using Fourier transforms and Gauss's law to analyze shapes in multiple dimensions, connecting classical physics and mathematical problems with explicit formulas for polygons and polyhedra.

Contribution

It presents a novel method combining Fourier analysis and Gauss's law to simplify shape analysis and derive classical results in physics and mathematics.

Findings

Explicit Fourier transform formulas for polygons and polyhedra.

Connections established between shape analysis and classical physics problems.

Alternative derivation of Davis's extension of the Motzkin-Schoenberg formula.

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.