A Bivariate Rational Fractal Interpolation And The Properties

Fractal interpolation which is an alternative to traditional interpolation techniques, gives a broader set of interpolants, and provides a good deter-ministic way for the understanding of the real world phcnomena. Using this method, we can construct not only interpolants with non-integral dimension but also smooth interpolants. Fractal interpolation techniques are effective and applied widely to deal with a highly irregular data from many real phe-nomena. The most of the current fractal interpolation functions (FIFs) are generated by iterated function system (IFS) based on the polynomial function.In this paper, based on the existing research literature, we consider a type of new bivariate rational fractal interpolation with the help of the classical bivariate interpoant. The primary strategies arc as following:Firstly, a family of bivariate rational fractal interpolation function (BR-FIF) Φ(x,y) based on the function values and derivative values of original function is proposed with the help of the classical bivariate rational interpo-lation function Pi,j(x,y)= Pi,j(x,y)/qi,j(y).Each BRFIF of the family is identified uniquely by the values of scaling factors Si,j and shape parameters αi.j*,βi，j* αi,j and βi,j. There is a large body of literature in the fractal interpolation, however, there is few literature in the bivariatc rational fractal interpolation.Secondly, for many interpolation methods, the error on the boundary of interpolating region is usually bigger than internal. In order to show the ef-fectiveness which the BRFIF approximates the original function f(x,y), we consider the boundary error of the BRFIF Φ(x,y), and derive the error esti-mate formula for f∈C1 and f∈C2.Thirdly, the shape-preserving properties of the curve and surface are im-portant research topic in curve and surface modeling, it has a wide applications in practical design. For the given shaped data, this paper discusses the shape preserving properties of the bivariate rational fractal interpolation function Φ(x,y), including preserving monotonicity and preserving convexity. The suf-ficient conditions, which the scaling factors Si，j and shape parameters αi，j*, βi，j*,αi,j and,βi,j satisfy, for the bivariate rational fractal interpolation func-tion Φ(x,y) to be monotonic or convex are derived. Thus, the problem of preserving monotonicity and preserving convexity for the fractal interpolation surface Φ(x, y) can be transformed to that of constrained control for the scal-ing factors Si,j and shape parameters αi.j*,βi，j*,αi,j and βi,j, that is to say, it is alternated to solve the algebraic inequalities on the scaling factors Si,j and shape parameters αi.j*,βi，j*,αi,j and βi,jAt last, we investigate the sensitivity of the constructed bivariate rational fractal interpolation function Φ(x,y) for the scale factors Si,j. The results show that the perturbation error of the bivariate rational fractal interpolation function Φ(x,y) is convergent for the scale factors Si,j.