Convexity/Concavity and Stability Aspects of Rational Cubic Fractal Interpolation Surfaces

Fractal interpolation is more general than the classical piecewise interpolation due to the presence of the scaling factors that describe smooth or non-smooth shape of a fractal curve/surface. We develop the rational cubic fractal interpolation surfaces (FISs) by using the blending functions and rational cubic fractal interpolation functions (FIFs) with two shape parameters in each sub-interval along the grid lines of the interpolation domain. The properties of blending functions and C1-smoothness of rational cubic FIFs render C1-smoothness to our rational cubic FISs. We study the stability aspects of the rational cubic FIS with respect to its independent variables, dependent variable, and first order partial derivatives at the grids. The scaling factors and shape parameters seeded in the rational cubic FIFs are constrained so that these rational cubic FIFs are convex/concave whenever the univariate data sets along the grid lines are convex/concave. For these constrained scaling factors and shape parameters, our rational cubic FIS is convex/concave to given convex/concave surface data.