Fractal interpolation surfaces
by
Robert Malysz
Depatrment of Mathematics and Information Science, Warsaw University of Technology

We consider fractal interpolation surfaces (FIS) of the form z=f(x, y) which are continuous and interpolate given data (i/n, j/n, z_{i, j}), i, j=0, 1, ...  n such that f:D --> R where D=[0, 1]². FIS have been introduced by Geronimo, Hardin and Massopust. They constructed self-affine FIS using the triangulation of the region D. Our FIS is an attractor of the iterated function system ([0, 1]²×R, w) where w=(L_{i, j}, F_{i, j}) and L_{i, j}:[0, 1]² --> [(i-1)/n, i/n]×[(j-1)/n, j/n] , F_{i, j}(x, y, z)=dz+a_{i, j}x+b_{i, j}y+c_{i, j}xy+f_{i, j}. The Minkowski dimension of FIS generated by this new class is of the form dim_M Gr f=3+log| d| / logn (if 1 > | d| > 1/n ). Such surfaces can be applied in data analysis and computer landscapes.

Date received: February 18, 2000