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

TL;DR

This paper introduces rational Hausdorff divisors and their associated linear systems to address the approximate parametrization of algebraic curves, providing new algorithms and theoretical insights into rational parametrizations within a Hausdorff distance framework.

Contribution

It defines rational Hausdorff divisors, analyzes their linear systems, and develops algorithms for approximate curve parametrization, offering a new approach to the problem.

Findings

All curves in the linear system are rational and at finite Hausdorff distance.

A projective linear subspace solution for approximate parametrization is identified.

Every irreducible Hausdorff curve admits a generic rational parametrization.

Abstract

In this paper we introduce the notion of rational Hausdorff 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 finite Hausdorff 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. Furthermore, we identify the linear system with a plane curve that is shown to be rational and we develop algorithms to parametrize it analyzing its fields of parametrization. Therefore, we present a generic answer to the approximate parametrization problem. In addition, we introduce the notion of Hausdorff curve, and we prove that every irreducible Hausdorff curve can always be parametrized with a generic rational parametrization having coefficients depending on as many parameters as the degree of the input curve.