Hermite interpolation by piecewise rational surface

In this paper a bivariate rational Hermite interpolation is developed to create a space surface using both function values and the first-order partial derivatives of the function being interpolated as the interpolation data. The interpolation function has a simple and explicit rational mathematical representation with parameters. There are two schemes to construct the interpolation functions, an example shows that both of them approximate the function being interpolated very well. The basis of this interpolation is derived and the interpolating function can be C1 in all the interpolation region when the parameters satisfy a simple condition. When a patch of the interpolating surface does not satisfy the need of design, say it is too high or too low at a point and its neighbourhood, the values of the interpolating function in these points need to be decreased or increased under the condition that the interpolating data are not changed. For this, sufficient conditions are derived, and it is shown that this can be done just by selecting suitable parameters in the interpolation.