  We prove an Euler-Maclaurin formula for double polygonal sums and, as a
corollary, we obtain approximate quadrature formulas for integrals of smooth
functions over polygons with integer vertices. Our Euler-Maclaurin formula is
in the spirit of Pick's theorem on the number of integer points in an integer
polygon and involves weighted Riemann sums, using tools from Harmonic analysis.
Finally, we also exhibit a classical trick, dating back to Huygens and Newton,
to accelerate convergence of these Riemann sums.
