Convexity of a Bivatiate Rational Interpolating Spline Funciton

The properties of a bivariate rational interpolating spline function of order(3,2)1,which is based on values of the function,including boundaries,limits,analysis,regularity and so forth are the key subject studied in this paper.First of all,the paper indicates that the limit surface is hyperbolic paraboloid and illuminates the influences of parameters on the rational interpolating spline surfaces.Secondly,the convex discriminant function expressed by dual 8 matrix has been introduced into the paper.Thirdly,the necessary and sufficient conditions for identifying the convexity of rational interpolation surfaces have been derived,while in accordance with which the examples,explaining how to choose appropriate parameters resulting in local convexity,are put forward to prove the validity of all above.Particularly,it is found that the convexity of the interpolation surfaces is relatively rigid at certain points,although the interpolated data is convex.Considering such situation,this paper has raised a necessary condition ensuring the local convexity of interpolation surfaces.What's more,this paper points out that some error results in literature 7 need to be discussed.