Cubic polynomial and cubic rational C1 sign, monotonicity and convexity preserving Hermite interpolation

The subject of this paper is C 1 sign, monotonicity and convexity preserving spline interpolation to a set of ordered points from a real function of one real variable. Two solutions are proposed constructing, respectively, a Hermite parametric polynomial Cubic Spline (CS), and a Hermite Cubic Rational polynomial Spline (CRS). Both curves are based on the shape preserving Hermite Variable Degree Spline (VDS) Gabrielides and Sapidis (2018) (first introduced in Kaklis and Pandelis (1990)) and they use the Bézier representation of polynomials. Since the CS curve is parametric, the present problem also requires calculation of the y-component of CS for any specific x-value; a robust solution to this problem is discussed in detail. The CRS is non-parametric and it does solve the given interpolation-problem with its weights (which play the role of tension parameters) being directly computed using the properties of the VDS segments.