Numerical proper reparametrization of parametric plane curves

• We introduce the problem of numerical improper parametrization of plane curves.
• The approximate improper index is firstly defined in this paper.
• A numerical algorithm is designed to find the approximate proper reparametrization.
• The relationship between the numerical curve and its reparametrization is well studied.

We present an algorithm for reparametrizing algebraic plane curves from a numerical point of view. More precisely, given a tolerance ϵ>0ϵ>0 and a rational parametrization PP of a plane curve CC with perturbed float coefficients, we present an algorithm that computes a parametrization QQ of a new plane curve DD such that QQ is an ϵϵ–proper reparametrization   of DD. In addition, the error bound is carefully discussed and we present a formula that measures the “closeness” between the input curve CC and the output curve DD.