Rational Hausdorff Divisors: a New approach to the Approximate Parametrization of Curves

Abstract

In this paper we introduce the notion of rational Hausdor divisor, we analyze the dimension and irreducibility of its associated linear system of curves, and we prove that all irreducible real curves belonging to the linear system are rational and are at nite Hausdor distance among them. As a consequence, we provide a projective linear subspace where all (irreducible) elements are solutions to the approximate parametrization problem for a given algebraic plane curve. Further- more, we identify the linear system with a plane curve that is shown to be rational and we develop algorithms to parametrize it analyzing its elds of parametriza- tion. Therefore, we present a generic answer to the approximate parametrization problem. In addition, we introduce the notion of Hausdor curve, and we prove that every irreducible Hausdor curve can always be parametrized with a generic rational parametrization having coe cients depending on as many parameters as the degree of the input curve.