  Calculations of the Fourier transform of a constant quantity over an area or
volume defined by polygons (connected vertices) are often useful in modeling
wave scattering, or in fourier-space filtering of real-space vector-based
volumes and area projections. If the system is discretized onto a regular
array, Fast Fourier techniques can speed up the resulting calculations but if
high spatial resolution is required the initial step of discretization can
limit performance; at other times the discretized methods result in
unacceptable artifacts in the resulting transform. An alternative approach is
to calculate the full Fourier integral transform of a polygonal area as a sum
over the vertices, which has previously been derived in the literature using
the divergence theorem to reduce the problem from a 3-dimensional to line
integrals over the perimeter of the polygon surface elements, and converted to
a sum over the straight segments of that contour. We demonstrate a software
implementation of this algorithm and show that it can provide accurate
approximations of the Fourier transform of real shapes with faster convergence
than a block-based (voxel) discretization.
