  A local lattice point counting formula, and more generally a local
Euler-Maclaurin formula follow by comparing two natural families of meromorphic
functions on the dual of a rational vector space $V$, namely the family of
exponential sums (S) and the family of exponential integrals (I) parametrized
by the set of rational polytopes in $V$. The paper introduces the notion of an
interpolator between these two families of meromorphic functions. We prove that
every rigid complement map in $V$ gives rise to an effectively computable
\SI-interpolator (and a local Euler-MacLaurin formula), an \IS-interpolator
(and a reverse local Euler-MacLaurin formula) and an \ISo-interpolator. Rigid
complement maps can be constructed by choosing an inner product on $V$ or by
choosing a complete flag in $V$. The corresponding interpolators generalize and
unify the work of Berline-Vergne, Pommersheim-Thomas, and Morelli.
