C2 positivity-preserving rational interpolation splines in one and two dimensions

Abstract

A class of rational quartic/cubic interpolation spline with two local control parameters is presented, which can be C2 continuous without solving a linear system of consistency equations for the derivative values at the knots. The effects of the local control parameters on generating interpolation curves are illustrated. For C2 interpolation, the given interpolant can locally reproduce quadratic polynomials and has O(h2) or O(h3) convergence. Simple schemes for the C2 interpolant to preserve the shape of 2D positive data are developed. Moreover, based on the Boolean sum of quintic interpolating operators, a class of bi-quintic partially blended rational quartic/cubic interpolation surfaces is also constructed. The given interpolation surface provides four local control parameters and can be C2 continuous without using the second or higher mixed partial derivatives on a rectangular grid. Simple sufficient data dependent constraints are also derived on the local control parameters to preserve the shape of a 3D positive data set arranged over a rectangular grid.