Planar cubic G 1 interpolatory splines with small strain energy

Abstract In this paper, a classical problem of the construction of a cubic G 1 continuous interpolatory spline curve is considered. The only data prescribed are interpolation points, while tangent directions are unknown. They are constructed automatically in such a way that a particular minimization of the strain energy of the spline curve is applied. The resulting spline curve is constructed locally and is regular, cusp-, loop-and fold-free. Numerical examples demonstrate that it is satisfactory as far as the shape of the curve is concerned

On geometric interpolation by planar parametric polynomial curves

Abstract. 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 ≤ n can interpolate 2n given points in R 2 is confirmed for n ≤ 5 under certain natural restrictions. This conclusion also implies the optimal asymptotic approximation order. More generally, the optimal order 2n can be achieved as soon as the interpolating curve exists. 1