High Order Polynomial Interpolation of Parameterized Curves

Abstract

Interpolation of parameterized curves differs from classical interpolation in that we interpolate each spatial variable separately. A difficult challenge arises from the option of reparameterization: a presumably good interpolation (e.g., at the Gauss points) of a given parameterization does not necessarily give the best approximation of the curve, as there may exist a reparameterization better suited for polynomial interpolation. The reparameterization can be done implicitly by choosing different sets of interpolation points along the exact curve. We present common interpolation methods, and propose a new method, based on choosing the interpolation points in such a way that the interpolant is tangential to the exact (reparameterized) curve at these points. The new method is compared to the traditional ones in a series of numerical examples, and results show that classical interpolation is sometimes far from optimal in the sense of the Kolmogorov n-width, i.e., the best approximation using n degrees-of-freedom.