文档介绍：南京航空航天大学
硕士学位论文
一类有理样条插值曲线及其形状控制
姓名:刘琳
申请学位级别:硕士
专业:应用数学
指导教师:唐月红
20090301
南京航空航天大学硕士学位论文
摘要
n )2,( 阶有
理参数样条曲线的构造,,
尤其是有理参数样条曲线特别适用于大挠度插值场合,,比多项式样
条更加灵活、有效,还能刻画被插函数的奇性等固有特性.
首先,以 Peano-kernel 定理为工具,研究了)2,3( 1 阶有理插值样条在被插函数为 C 2 连
续及 C1 连续时的逼近误差和)2,4( 1 阶有理插值样条在被插函数为 C 2 连续时的逼近误差,估
2
计了逼近误差中系数 ci 的界,得到了被插函数为 C 连续的有理插值函数的逼近误差阶为
O(h2),而被插函数为 C1 连续的逼近误差阶为 O(h) .
然后,讨论了插值曲线在给定区域的形状控制问题以及保正问题,提出了插值曲线约束
于给定的折线、二次曲线之上、
数的约束,推导出了相应的显式约束不等式,,通过适当选择形状
参数,.
最后,构造一类包含极点和控制参数的 k kn = )2,1(,)2,( 阶有理参数插值样条曲线. 推
. 以)2,4( 1
阶有理参数段作为零件拼接成了二阶视觉连续的有理样条插值曲线,并给出了大挠度有理样
条插值曲线的许多实例.

关键词:样条曲线,有理插值,误差估计,形状控制,视觉连续
i
一类有理样条插值曲线及其形状控制
Abstract
In the paper, the approximation properties and shape control problems of a class of rational
interpolation splines are studied. Furthermore, a class of rational spline interpolation curves with
k
both control and connection parameters of order ()(nk,2= 1,2 ) are presented. The
parameters increase the degree of freedom for geometric design. Especially, the rational
interpolation spline curves, which have geometric invariability, are quite suitable for the occasion
of large deflection interpolation. Therefore, As a tool for geometric modeling they are more
flexible and effective than polynomial splines while they could describe the inherent characters of
the interpolation functions, such as the singularity.
Firstly, With Peano-Kernel theory, the approximation errors of a rational interpolation spline
function of order (3,2)1 when the exact function has continuous first or second derivative are
estimated and similarly, the estimate of a rational interpolation spline function of order (4,2)1
when interpolating to a function with c 2 continuity is given as well . Meanwhile, we prove the
optimal e