Constrained and Monotone Curves Derived from Rational Fractal Interpolation

Local shape constraint and adjustment of fractal curves and surfaces derived from traditional fractal polynomial interpolation is difficult.In order to make the fractal curves to give a good approximation for the irregular data and its shape to be modified,a rational spline fractal interpolation method is proposed.In view of this method,the traditional non-recursive shape modifiable interpolation can be generalized by fractal.Firstly,a type of C¹-continuous rational spline fractal interpolation functions is constructed with the help of classical rational cubic spline,which allows us to embed shape parameters within the structure of differentiable fractal functions,so that the shape of fractal curves can be adjusted by making constraints on scaling factors and shape parameters.And then,the analytical properties of the fractal interpolation functions are investigated,including convergence and stability.Finally,on the basis of the constructed rational fractal functions,constrained and monotone interpolating schemes are developed by making constraints on iterated function system（IFS） parameters,respectively.Experimental results show that the presented rational fractal functions have the good capacity of quasi-locality due to the embedded shape parameters,which provides an effective tool for the shape adjustment of fractal curves.