A Kind Of New Rational Fractal Oplines And Visualization Of Shaped Data

On the basis of existing research literature, this paper mainly studies a new type of fractal interpolation system of curves and surfaces, The primary strategies are as following:In section one, we introduce the iterated function system and give the common formats of fractal curves and surfaces.In section two, we propose a family of C1 rational spline fractal interpo-lation functions (RSFIFs) with the help of classical rational cubic splines, and obtain the error estimate formula of RSFIF. Further, a constrained rational fractal interpolating scheme and a monotonicity-preserving rational fractal in-terpolating scheme are developed to visualize constrained data and monotonic data in the view of constrained curves and monotone curves by using con-straints on scaling factors and shape parameters in the description of RSFIF, respectively.In section three, a family of bivariate rational fractal interpolation func-tions (BRFIFs)Φ(x, y) based on the function values and derivative values of a original function is proposed with the help of the classical bivariate rational interpolation function Pi,j(x, y)=pij(x,y)/qi,j(y). In order to show the effectiveness which the BRFIF approximates the original function f(x,y), we consider the error of the BRFIFΦ(x, y), and derive the error estimate formula for f ∈ C1 and f E C2. In addition, for the given monotone data, suitable values of the rational IFS parameters are identified so that the property of monotonicity carries from the data set to the BRFIFs.The fractal curves and surfaces developed in this paper have some char-acteristics comparing with the present fractal interpolants. First of all, the shape parameters can be embed within the structure of differentiable fractal functions. Next, each RSFIF or BRFIF is identified uniquely by the values of vertical scaling factors and shape parameters. Finally, the fractal curves and surfaces can be varied as scaling factors and shape parameters vary, so that the shape of the interpolating curves and surfaces can be modified by selecting suitable scaling factors and shape parameters for the unchanged interpolating data.