Rational Cubic Spline Fractal Interpolation Based On Function Values

Fractal methodology provides a good deterministic way for the under-standing of real-world phenomena. It can be regarded as a generalization of a polynomial interpolation and spline interpolation. It not only provides us one of the very few methods of non-differentiable interpolation, but also provides a new method to construct smooth or piecewise smooth interpolation curves and surfaces. On the basis of existing research literature, this paper mainly studies a class of new system of fractal interpolation, namely rational frac-tal interpolation with shape constraint parameters. The main results are as follows:1、 We propose a new Cl continuous rational cubic spline fractal interpo-lation function ψs{x) based on function values, by using the classical rational cubic spline interpolation function ψi(x)=pi(x)/qi(x) with two parameters αi,βi, where pi(x) are suitable cubic polynomials and qi(x) are linear polyno-mials. We investigate the uniform boundedness of the error for the rational fractal interpolation function ψs(x) when·∈C1as well as f∈C2, and obtain the error estimate formula of rational fractal interpolation function;Also, we derive that the values of the error constant C and C*are bounded, it means that the constructing rational cubic spline fractal interpolation function yψs(x) is stable for the shape parameters αi,βi.2、 We also studied the sensitivity of the constructed rational cubic spline fractal interpolation function ψs(x) for the scale factor Si. The results show that the perturbation error of the rational cubic spline fractal interpolation function ψs (x) is convergent for the scale factors Si.3、 The shape-preserving properties of the curve and surface are an impor-tant research topic in curve and surface modeling, it has a wide applications in practical design.For the given shape data, this paper discusses the shape-preserving properties of the rational cubic spline fractal interpolation function ψs(x), including preserving monotonicity and preserving convexity.The suffi-cient conditions, which the scale factor Si and the shape parameters αi,βi satisfy, for the rational cubic spline fractal interpolation function ψs(x) to be monotonic or convex are derived. Thus, the problems of preserving mono-tonicity and preserving convexity for the fractal interpolation curve ψs(x) can be transformed to that of constrained control for the scale factor Si and the shape parameters αi,βi, that is to say, it is alternated to solve the algebraic inequalities on the scale factor s, and the shape parameters αi,βi.4、 The shape constraint and control of curve and surface is a fundamen-tal task, and also is an important subject that we must face in curve and surface modeling. And yet, it is not well explored hitherto in references on the fractal interpolation. Here, we are concerned with the constrained control of the shape for the rational spline fractal curves when the constraint functions g(x) are piecewise linear function and piecewise quadratic function,including above constraint, below constraint and bilateral constraints. And the sufficient conditions for constraining the interpolating curves to be in a given region are obtained. One may constrain the interpolating curve to lie above, below or between two given function curves by selecting suitable the scale factor s, and the shape parameters αi,βi. Thus, the constrained control of the shape for the rational fractal interpolating curve ψs(x) can be alternated that of the scale factor Si and the shape parameters αi,βi.