On shape preserving C2 Hermite interpolation

We propose a general parametric local approach for functional C(2) Hermite shape preserving interpolation. The constructed interpolant is a parametric curve which interpolates values, first and second derivatives of a given function and reproduces the behavior of the data. The method is detailed for parametric curves with piecewise cubic components. For the selected space necessary and sufficient conditions are derived to ensure the convexity of the constructed interpolant. Monotonicity is also studied. The approximation order is investigated for both cases. The use of a parametric curve to interpolate data from a function can be considered a disadvantage of the scheme. However, the simple structure of the used curves greatly reduces such a disadvantage.