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
Pub Type: Journals
Gauss law, Fourier transforms, Fraunhofer diffraction, Porods law, isoperimetric inequality, Hopfs Umlaufsatz, shape from moments