Computing the topology of the image of a parametric planar curve under a birational transformation

Abstract

We provide a method to compute the topology of the image of a parametric curve under a birational mapping of the plane. The method proceeds by exploiting as much as possible the initial parametrization in order to reduce the computational cost. The self intersections of the image curve are derived from points in the image where the inverse of the birational mapping is not defined. In order to compute these points, we prove a result characterizing birational planar mappings, together with an algorithm to compute the inverse of a birational mapping. We apply the method when the original curve is rational, in which case the image of the curve is also rational but with a higher degree, and when the original curve is an exp-log-arctan function. In this last case the image is a non-rational curve admitting an analytic parametrization, a problem not treated in the literature so far