【作者】 李莉；

【导师】 唐月红；

【作者基本信息】 南京航空航天大学 ， 应用数学， 2014， 硕士

【摘要】 本文主要研究了一类有理插值样条曲面的构造及其形状控制问题.本文的研究是对目前基于函数值有理插值样条方法的改进和扩展.获得的主要结果为：讨论了一类(4,2)1阶带形状参数加权有理插值样条在给定区域内的形状控制问题，分别就约束于给定折线、二次曲线之上，之下或之间三种情形，通过将问题归之为形状参数的约束，推导出了相应的显示不等式，得到了有理约束插值的充分条件.由此，通过对参数和权系数的选取，实现对有理插值样条曲线的形状控制.数值例子表明了区域约束下的曲线形状控制.采用一类基于函数值(4,2)1阶带形状参数有理样条曲线，给出了一类(4,2)1阶有理插值样条曲面的构造，推导了这类新的有理插值样条曲面诸如边界插值、极限、解析和正则等性质，并证明了样条插值的精度为O (k3)(其中k是矩形网格的尺度)，而目前此类有理插值样条曲面的误差阶为O (k2).通过Gauss曲率的计算，引入双10次11阶矩阵表示的凸性判别函数，推导了这类有理插值样条曲面凸性判定的一个充要条件.并结合实系数多项式零点理论，具体给出了判定一类(4,2)1阶有理插值样条曲面凸性的几个充分条件，实现了有理样条曲面全局凸性的控制.解决了现有有理插值样条曲面凸性相对刚性的问题.数值例子也验证了方法的正确性和有效性.本文提出的一类(4,2)1阶有理插值样条曲面具有简单分片的显式表达式，良好的几何行为，作为插值工具逼近效果好，并在不改变插值条件的前提下，只要通过调整形状参数就可进行曲面的局部修改，特别是可先验地判定插值曲面的凸性，进行预期全局凸曲面设计，达到曲面局部修改和形状控制.更多还原

【Abstract】 This paper mainly studies the construction of a kind of rational interpolation spline surface andits shape control. The research in the paper is the extension and improvement of the method ofrational interpolation spline based on function values. The main results are as follows:The shape control of a kind of weighted rational interpolation with shape parameters of order(4,2)1is discussed in the given region. Considering the interpolating curves to be above, below orbetween the given broken lines or piecewise quadratic curves respectively, and by falling theproblems into constraints of shape parameters, the corresponding exhibit inequations are derived andthe sufficient conditions are got. Thus, by choosing parameters and weight coefficient shape control ofrational interpolation are realized.Using the method of a kind of rational interpolation with shape parameters of order (4,2)1basedonly on function values, constriction of a kind of rational interpolation of order (4,2)1is given, someproperties are studied for the new rational spline interpolating surfaces, for example, boundaries,limits, analysis and canonical etc..It is proved that the precision of the spline isO (k3)(k is thedimension of the rectangular net), while the precision of the kind of rational interpolation surface isO (k2)at present.By calculating the Gauss curvature, a matrix representation with11-order coefficient matrix and10-degree bibariate polynomial vector function is introduced as identification of the functionalconvexity. The necessary and sufficient condition for the rational interpolation surface to be convex isderived. Combining the theory of zero points for polynomial with real coefficient, several conditionsof judging the convexity of a kind of rational surface of order (4,2)1are deduced. Control of thesurface to local convex is come true. The relative stiffness of the convexity of the exsitinginterpolation surface are solved. Examples are given to verify the validity and effectiveness.The kind of order (4,2)1rational interpolation spline surface proposed in the article has a simpleand explict mathematical representation, good geometry behavior. As a tool of interpolation, it has agood approximation, and the interpolating surface can be modified by selecting suitable parametersunder the condition that the interpolating data are not changed. Especially, the convexity of thesurface can be judged prior, and it can be processed the expected global convexity preserving surfacedesign, achieving the local modification and shape control.更多还原