  We study the symmetry groups and winding numbers of planar curves obtained by
taking the image of the complex unit circle under a Laurent polynomial, a class
that includes weighted sums of exponentials. We generalize various results on
such sums of exponentials and relate them to other classes of curves present in
the literature. Moreover, we consider the evolution under the wave equation of
such curves for the case of binomials. Interestingly, our methods provide a
unified and systematic way of constructing curves with prescribed properties,
such as number of cusps, number of intersection points or winding number.
