Numerical polynomial reparametrization of rational curves

Abstract

Given a real rational parametrization P(t) of a plane curve C, we present an algorithm to compute polynomial curves to approximate C for the whole parameter domain. In this case, the denominators often have real roots in the whole interval. We decompose the interval as the union of finitely many intervals according to the real roots of the denominators. The key technique of the paper is to approximate the given curve by their asymptotes and error analysis at each interval is also presented. The asymptotes are associated with the infinity points corresponding to the real roots of the denominators. Numeric algorithms and examples are proposed to illustrate our results.