$$\varvec{C}^2$$ C 2 Rational Quartic/Cubic Spline Interpolant with Shape Constraints

<i>C</i>¹ convexity-preserving piecewise variable degree rational interpolation spline

“…Various convexity-preserving interpolation spline methods have been proposed, such as the piecewise rational interpolation splines developed in (Clements, 1990;Sarfraz, 2002;Abbas et al, 2014;Han, 2008;Merrien and Sablonnière, 2013;Zhu andHan, 2015a, 2015b;Zhu, 2018), the piecewise cubic interpolation splines constructed in (Costantini, 1984;Brodlie, 1991), the piecewise weighted polynomial interpolation splines given in (Kvasov, 2014a;Kvasov, 2014b), the piecewise exponential interpolation splines presented in (Heß and Schmidt, 1986) and the piecewise polynomial interpolation splines of nonuniform degree proposed in (Kaklis, 1990). A common point of these methods is that in order to preserve convexity property, it is requested sufficient data dependent constraints on the shape parameters.…”

“…However, the resulting rational quadratic/linear interpolant is unique to the given convex data, thus users can not interactively adjust the shape of the obtained convexity-preserving interpolation curves without changing the given convex data. Recently, some improvements on this rational quadratic/linear interpolation spline have been proposed, such as the shape preserving quartic rational splines developed in (Han, 2008;Zhu andHan, 2015a, 2015b;Zhu, 2018), which include the rational quadratic/linear interpolation spline as a special or a limit case. These improved quartic rational splines have extra local shape parameters for interactively modifying the shape of spline curves.…”