  In this paper we introduce geometric tools to study the families of rational
vector fields of a given degree over $\mathbb C\mathbb P^1$. To a generic
vector field of such a parametric family we associate several geometric
objects: a periodgon, a star domain and a translation surface. These objects
generalize objects with the same name introduced in previous works on
polynomial vector fields. They are used to describe the bifurcations inside the
families. We specialize to the case of rational vector fields of degree $4$.
