有理样条的保凸性研究

Research on convexity-preserving rational spline

曲线曲面的形状控制问题是计算机辅助几何设计中一个重要的研究课题,有理样条插值方法是曲线曲面设计中强有力的工具,带参数的有理样条由于其形状控制的灵活性,近年来受到普遍关注.主要研究了带参数的保凸有理样条插值问题.首先基于带参数的有理三次样条曲线,构造了双三次混合有理样条曲面,这类带参数的有理样条曲面具有孔斯曲面的良好性质,并且由于含有参数,因而给形状控制带来了方便.进而研究了保凸性问题,给出了有理曲线保凸的充要条件,由边界曲线的保凸条件推导出了有理样条曲面的保凸条件.其优势就是在不改变给定数据信息的情况下,通过调整形状参数来达到保凸的目的.最后通过数值例子验证了方法的有效性.

The shape control problem of curves and surfaces is a main research of computer aided geometric design.Rational spline interpolation is a useful and powerful tool.In recent years,the rational spline with parameters has received attention for the flexibility in shape control.The conditions for interpolating surfaces on the free parameters to be convex are derived in this paper.First of all,a rational cubic spline function with free parameters is extended to rational bi-cubic partially blended function.These rational patches have inherent good features to coons patches.And it can be controlled easily for the parameters contained.The convexity-preserving problem is discussed,and the sufficient and necessary conditions of the rational spline function are given.The conditions of boundary curves to be convex are extended to the convexity conditions of the rational spline function.The convexity of the surfaces can be controlled by adjusting the shape parameters with the data not changeable.Examples are given to show the effectiveness of the methods.