Finite piecewise polynomial parametrization of plane rational algebraic curves

Abstract

We present an algorithm with the following characteristics: given a real non-polynomial rational parametrization ${\mathcal{P}(t)}$ of a plane curve and a tolerance ${\epsilon > 0}$ , ${\mathbb{R}}$ is decomposed as union of finitely many intervals, and for each interval I of the partition, with the exception of some isolating intervals, the algorithm generates a polynomial parametrization ${\mathcal{P}_{I}(t)}$ . Moreover, as an option, one may also input a natural number N and then the algorithm returns polynomial parametrizations with degrees smaller or equal to N. In addition, we present an error analysis where we prove that the curve piece ${{\cal C}_{I}=\{\mathcal{P}(t)\,|\,t\in I\}}$ is in the offset region of ${{\cal C}_{I}^{\ast}=\{\mathcal{P}_{I}(t)\,|\,t\in I\}}$ at distance at most ${\sqrt{2}\epsilon}$ , and conversely.