Finite piecewise polynomial parametrization of plane algebraic curves

Abstract

We present an algorithm with the following characteristics: given a real non-polynomial rational parametrization P(t) of a plane curve and a tolerance ϵ>0 , 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 PI(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 CI={P(t)|t∈I} is in the offset region of C∗I={PI(t)|t∈I} at distance at most 2–√ϵ , and conversely.