C¹-rational cubic fractal interpolation surface using functional values

Fractal interpolation is a modern technique for fitting of smooth/nonsmooth data. In the present article, we develop the C1-rational cubic fractal interpolation surface (FIS) as a fixed point of the Read-Bajraktarević (RB) operator defined on a suitable function space. Our C1-rational cubic FIS is effective tool to stich surface data arranged on a rectangular grid. Our construction needs only the functional values at the grids being interpolated, therefore implementation is an easy task. We first construct the x-direction rational cubic FIFs (x-direction fractal boundary curves) to approximate the data generating function along the grid lines parallel to x-axis. Then we form a rational cubic FIS as a blending of these fractal boundary curves. An upper bound of the uniform distance between the rational cubic FIS and an original function is estimated for the convergence results. A numerical illustration is provided to explain the visual quality of our rational cubic FIS. An extra feature of this fractal surface scheme is that it allows subsequent interactive alteration of the shape of the surface by changing the scaling factors and shape parameters.