C1 Shape-Preserving Rational Quadratic/Linear Interpolation Splines with Necessary and Sufficient Conditions

In this work, we introduce a novel class of C1 rational quadratic interpolation splines defined by two symmetric parameters. This generalization encompasses the rational quadratic interpolation schemes given by Schmidt in 1987 as a special case. For data sets with convexity, monotonicity, or positivity constraints, we derive the necessary and sufficient conditions, ensuring that the interpolant preserves these properties. Furthermore, we propose an algorithm for selecting visually appealing and shape-preserving spline curves by minimizing a particular approximated curvature functional.