  The Fourier transforms of polyhedral cones can be used, via Brion's theorem,
to compute various geometric quantities of polytopes, such as volumes, moments,
and lattice-point counts. We present a novel method of computing these conic
Fourier transforms by polynomial interpolation. Given the fact that computing
volumes of polytopes is #P-hard (Dyer--Frieze [DF88]), we cannot hope for fast
algorithms in the general case. However, with extra assumptions on the
combinatorics of the cone, we demonstrate it is possible to compute its Fourier
transform efficiently.
