一种二元有理插值样条函数的凸性

Convexity of a Bivatiate Rational Interpolating Spline Funciton

通过研究一种基于函数值的(3,2)1阶二元有理插值样条函数中诸如边界插值、极限、解析和正则等性质,指出极限曲面是双曲抛物面,揭示了参数对这种插值曲面的影响.首先引入双8次矩阵表示的凸性判别函数,推导了判定插值曲面凸性的充要条件;然后根据该条件给出数值实例,展示如何适当选取参数实现有理插值样条曲面的局部保凸性.特别发现了这种插值曲面凸性在某些点处即使型值是凸的数据也是相对刚性的,并提出了插值曲面局部保凸的必要条件.最后还讨论了文献(Zhang Y,Duan Q,Twizell E H.Convexity control of a bivariate rational interpolating spline surfaces.Computers&Graphics,2007,31(5):679-687)中存在的部分计算问题.

The properties of a bivariate rational interpolating spline function of order （3,2）1, which based on values of the function, including boundaries, limits, analysis, regularity and so forth are t key subject studied in this paper. First of all, the paper indicates that the limit surface is hyperbo lS he 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.