Approximate Parametrization of Plane Algebraic Curves by Linear Systems of Curves

Abstract

It is well known that an irreducible algebraic curve is rational (i.e. parametric) if and only if its genus is zero. In this paper, given a tolerance $\epsilon >0$ and an $\epsilon$-irreducible algebraic affine plane curve $\mathcal{C}$ of proper degree $d$, we introduce the notion of $\epsilon$-rationality, and we provide an algorithm to parametrize approximately affine $\epsilon$-rational plane curves, without exact singularities at infinity, by means of linear systems of $(d-2)$-degree curves. The algorithm outputs a rational parametrization of a rational curve $\bar{\mathcal{C}}$ of degree at most $d$ which has the same points at infinity as $\mathcal{C}$. Moreover, although we do not provide a theoretical analysis, our empirical analysis shows that $\bar{\mathcal{C}}$ and $\mathcal{C}$ are close in practice.