  Using techniques from geometry and complex analysis in their simplest form,
we present a derivation of electric fields on surfaces with non-trivial
topology. A byproduct of this analysis is an intuitive visualization of
elliptic functions when their argument is complex-valued. The underlying
connections between these techniques and the theory of Riemann surfaces are
also explained. Our goal is to provide students and instructors a quick
reference article for an extraordinary topic that is not included in the
standard books.
