On Geometric Interpolation by Polynomial Curves

Abstract

In this paper, geometric interpolation by parametric polynomial curves is considered. Discussion is focused on the case where the number of interpolated points is equal to r + 2, and n=r denotes the degree of the interpolating polynomial curve. The interpolation takes place in $\mathbb R^d$ with d=n. Even though the problem is nonlinear, simple necessary and sufficient conditions for existence of the solution are stated. These conditions are entirely geometric and do not depend on the asymptotic analysis. Furthermore, they provide an efficient and stable way to the numeric solution of the problem.