Surface Grid Point Fractal Interpolation

In this paper,we discusses the fractal interpolating surface problem(S-FIP). At first,itclearly provides the strict definition of S-FIP,introduces the fractal interpolating space,and points out the basic properties on the space. Secondly,using the above mathematical language,it make a summary of the tensor-prodct IFS fractal interpolating method. At last,it puts forward the concepts of holographic self-similar surface and inter-graphic self-similar surface, and points out the properties of them.the holographic self-similarity means that the local is similar to the global; the intergraphic self-similarity means that the local is similar to the local.this paper gives two kinds of new methods on S-FIP-the holographic method and the intergraphic method,And gives out when the fractal interpolating surface is contnuous.The main results are as follows.1. By the product process of the fractal surface, obtains the if and only if condition of the general fractal interpolating surface, and obtains the following general results.Theorem A: Let {wij}ni=1,1mj=1 be a IFS of S-FIP. Assums {(xlt yjt ZlJ)}ni=0,f=0 is the Interpolating data group, D = I x J = {(x,y) : a < x < 6, c < y < d} is the interpolating area,we make grids as follows:a = XQ < x\ << xn = b , c = y0 < yi ?< ym = d K ?D x R is the interpolating space Let ^\y=yo is on the boundary y = yo, of the shape surface \I>, ?Lo is the linterpolating data group, D = IxJ = {(x,y) : a < x < b, c < y < d} is the interpolating area, K ?D x R is the interpolating space. and under the condition of Theorem A. Let {a;}"=1 denote the horizontal contraction ratios, and let i denote the vertical contraction ratios, let {dt.,}"i7,j_i denote the whole contraction ratios. Suppose Mj., is the Lipschitz constant of the shape surface $,L is the the Lipschitz constant of the base surface to, Let M = max {Mi,}, a = max {a;} , l is the lifting map of(3)There exists / e F[D]such that >4 = (7[/].That is to say there exists / 6 C"[Z)],such that /(xj, ?/_,? = Zij, And j4is the function graph ofIntroduces the inetrgraphic IFS of S-FIP ,proves that the inter-graphic IFS is topologically conjugate to a holographic IFS,and obtains the following general results.Theorem C: Let {wy }^=^=1 be a IFS of S-FIP. Assums {(a;., yj} zi;-)}=0 ,0 is the macro Interpolating data group,and {(xi, yj, hij)}?0,?0 is the micro Interpolating data group. Let D = I x J = {(x,y] : a < x < b, c < y < d} is the interpolating area, K = D x R is the interpolat...