A Surface Modeling Method by Using $$C^2$$ C 2 Piecewise Rational Spline Interpolation

Abstract

A piecewise bivariate rational spline interpolation is presented only based on the values of the interpolated function. The interpolation has the following advantages: it is $$C^2$$C2 continuous in the whole interpolating region; the interpolation function has a explicit rational mathematical representation, and can be represented by the basis functions; more important, since there are three free parameters $$\alpha _{i,j}$$?i,j, $$\alpha _{i,j+1}$$?i,j+1 and $$\beta _{i,j}$$βi,j in this interpolant, the shape of the interpolating surfaces can be modified by selecting suitable parameters for the unchanged interpolating data. Also, the values of the interpolation function are bounded no matter what the parameters might be, and the approximation expressions of the interpolant are derived.