A novel approach to the convexity control of interpolant curves

Abstract

SUMMARYA method is presented for controlling the convexity of interpolant curves based on a rational cubic interpolating function with quadratic denominator. The key idea is that the uniqueness of the interpolating function for the given data is replaced by the uniqueness of the interpolating function for the given data and the parameters, so that for the given data the shape of the interpolating curve can be modiÿed by selecting suitable parameters. Necessary and su cient conditions are given for adjusting the convexity of the interpolating curve for given data. Examples are given and the optimal error estimation is given.

Region control and approximation of a weighted rational interpolating curves

“…When the knots are equally spaced, for the positive parameters i ; ÿ i and the weight coe cient ∈ R, then (14).…”

“…Another C 2 rational cubic spline with quadratic denominator which just bases on the function values was given by Duan et al [15]. Both these kinds of spline can be applied to constrain the shape of the interpolating curves in some cases [14,15]. Based on the idea of adding more parameters in the interpolating spline to enhance the constraining ability, the weighted rational spline will be constructed in this paper by using these two kinds of rational cubic spline with quadratic denominator.…”