Hermite and Lagrange interpolation in R-d by G(1) cubic splines with small strain energy

In this paper, Hermite and Lagrange interpolation by G(1) smooth cubic spatial splines with small strain energy are considered. A parametric interpolation scheme, based on minimisation of an approximate strain energy, is introduced. As its particular cases, it reproduces three known planar schemes and one spatial. The resulting interpolants are shape preserving, locally without loops, cusps or folds. A construction of optimal tangent directions is analysed. This yields a solution of the Lagrange interpolation problem. The obtained curves are compared with the results of two classic schemes and are applied for the construction of a cubic Hermite spline surface interpolant with small Willmore energy.