Numerical Reparametrization of Rational Parametric Plane Curves

TL;DR

This paper introduces a numerical algorithm for reparametrizing algebraic plane curves with approximate data, ensuring the new parametrization closely matches the original curve within a specified tolerance.

Contribution

It provides a novel numerical method for reparametrizing rational plane curves with error bounds, handling approximate input data.

Findings

Algorithm computes an $\e$-proper reparametrization of the curve.

Error bounds are explicitly formulated and analyzed.

The method effectively manages perturbed float coefficients in parametrizations.

Abstract

In this paper, we present an algorithm for reparametrizing algebraic plane curves from a numerical point of view. That is, we deal with mathematical objects that are assumed to be given approximately. More precisely, given a tolerance $\epsilon>0$ and a rational parametrization $\cal P$ with perturbed float coefficients of a plane curve $\cal C$, we present an algorithm that computes a parametrization $\cal Q$ of a new plane curve $\cal D$ such that ${\cal Q}$ is an {\it $\epsilon$--proper reparametrization} of $\cal D$. In addition, the error bound is carefully discussed and we present a formula that measures the "closeness" between the input curve $\cal C$ and the output curve $\cal D$.