Bounded Property and Point Control of a Bivariate Rational Interpolating Surface

A bivariate rational interpolation method with parameters was created in an earlier work which was based on function values only. This paper will deal with the bounded property and the point control method of the interpolating surface. It is proved that the values of the interpolating function in the interpolation region are bounded no matter what the parameters might be; this is called the bounded property of the interpolation. Also, the approximation expressions of the interpolation are derived; they do not depend on the parameters. More important is that the value of the interpolating function at any point in the interpolating region can be modified under the condition that the interpolating data are not changed by selecting suitable parameters, so the interpolation surface may be modified for the given interpolation data when needed in practical design. In the special case, the “Central Point-Mean Value” control is studied, and an example is given to show the control.