A new weighted rational cubic interpolation and its approximation

https://doi.org/10.1016/j.amc.2004.09.041Get rights and content

Abstract

A weighted rational cubic spline interpolation has been constructed using two kinds of rational cubic spline with quadratic denominator. The degree of smoothness of this spline is C2 in the interpolating interval when the parameters satisfy a continuous system. The sufficient and necessary conditions that constrain the interpolant curves to be convex in the interpolating interval or subinterval are derived. Also, the error estimate formulas of this interpolation are obtained.

Introduction

Many authors have studied several kinds of spline [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] for curve and surface design and control. In general, the common spline interpolation is the fixed interpolation which means that the shape of the interpolating curve or surface is fixed for the given interpolating data, since the interpolating function is unique for the given interpolating data. If one wishes to modify the shape of the interpolating curve, the interpolating data need to be changed. An important question is how can the shape of the curve be modified under the condition that the given data are not changed? That is a big problem in computer aided geometric design. Theoretically speaking, it is contradictory to the uniqueness of the interpolating function for the given interpolating data. In recent years, the rational spline with parameters has received attention in the literature [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. For the given interpolating data, the change of the parameters causes the change of the interpolating curve, so that the interpolating curve may be modified to be the needed shape if suitable parameters exists. That is, the uniqueness of the interpolating function for the given data is replaced by the uniqueness of the interpolating curve for the given data and the selected parameters.

A C2 rational cubic spline with quadratic denominator was given by Gregory [17], which depends on the function values and derivatives of the function being interpolated. Another C2 rational cubic spline with quadratic denominator which was based on the function values was given by Duan [23]. Both these two kinds of spline can be applied to constrain the shape of the interpolating curves in some cases [23], [24]. 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.

The paper is arranged as the follows. In Section 2 the weighted rational cubic spline will be constructed using two kinds of rational cubic spline with quadratic denominator, there are the interpolant parameters αi, βi and the weight coefficient λ in the new weighted spline. Section 3 is about the convex control of the interpolating curves, and the sufficient and necessary conditions that constrain the interpolating curves to be convex (or concave) in the interpolating interval or subinterval are derived. Some numerical examples are given in Section 4. The approximation properties of the weighted rational interpolation are studied in Section 5.

Section snippets

Weighted rational cubic spline interpolation

A rational cubic spline with quadratic denominator based on function values and derivatives was given in [17]. Given a data set {(ti, fi, di), i = 0, 1,  , n, n + 1}, where fi and di are the function values and the derivative values defined at the knots, respectively, and t0 < t1 <  < tn < tn+1 are the knots. Let hi = ti+1  ti, θ = (t  ti)/hi, and let αi and βi be positive parameters. DenoteP(t)|[ti,ti+1]=pi(t)qi(t),i=0,1,n,wherepi(t)=(1-θ)3αifi+θ(1-θ)2Vi+θ2(1-θ)Wi+θ3βifi+1,qi(t)=(1-θ)2αi+2θ(1-θ)+θ2βiandVi=(2

Convexity control of the weighted rational interpolating curves

Engineering practice usually requires that the interpolating function retains the shape of the given data. In order to get the condition for the interpolation to keep convexity in the interpolating interval, consider the condition for the second-order derivative to remain positive or negative in the interpolating interval. For this rational cubic interpolation, this task can be carried out simply by selecting suitable values of the parameters αi, βi and λ to satisfy the linear inequalities. In

Numerical examples

In this section two examples are given. Example 1 shows that the weighted interpolation defined by (3) can approach the values of the function being interpolated better than that of the interpolation defined by (1). Example 2 is about the interpolating curve’s convexity control.

Example 1

For the circle (x  1)2 + (y  1)2 = 1, 0  x  2, even if writing it as y=1±1-(x-1)2, the works could not be completed for the weighted rational interpolation in [0, 2], since the derivative dydx at x = 0 and x = 2 are infinite. Rewrite

Approximation properties of the weighted interpolation

For the error estimation of the weighted piecewise rational cubic interpolating function defined by (3), since the interpolation is local, without loss of generality it is necessary only to consider the error in the subinterval [ti, ti+1]. When f(t)  C2[t0, tn] and P(t) is the rational cubic interpolating function of f(t) in [ti, ti+1], it is easy to see that this kind of interpolation is exact for f(t), the polynomial, being interpolated, in which the degree is no more than 1. Consider the case

Acknowledgments

The supports of the National Nature Science Foundation of China, the Nature Science Foundation of Shandong Province of China and the Education Foundation of China are gratefully acknowledged.

References (25)

  • L.L. Schumaker

    On shape preserving quadratic spline interpolation

    SIAM J. Numer. Anal.

    (1983)
  • F.N. Fritsch et al.

    A method for constructing local monotone piecewise cubic interpolation

    SIAM J. Sci. Stat. Comput.

    (1984)
  • Cited by (0)

    View full text