On geometric interpolation by planar parametric polynomial curves

In this paper the problem of geometric interpolation of planar data by parametric polynomial curves is revisited. The conjecture that a parametric polynomial curve of degree $\le n$ can interpolate $2 n$ given points in $\mathbb{R}^2$ is confirmed for $n \le 5$ under certain natural restrictions. This conclusion also implies the optimal asymptotic approximation order. More generally, the optimal order $2 n$ can be achieved as soon as the interpolating curve exists.