Finite piecewise polynomial parametrization of plane rational algebraic curves

摘要

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 ${{cal C}_{I}={mathcal{P}(t),|,tin I}}$ is in the offset region of ${{cal C}_{I}^{ast}={mathcal{P}_{I}(t),|,tin I}}$ at distance at most ${sqrt{2}epsilon}$ , and conversely.