B样条曲线曲面的性质及其生成算法的研究

B样条曲线曲面的性质及其生成算法的研究

XXX

（XXX学院数学与信息科学学院05级信本（2）班）

摘要

从B样条曲线曲面的定义入手，阐述了B样条曲线曲面的性质，在生成算法中提出了一个扩展B样条曲线曲面的新方法.扩展B样条曲线曲面的关键是为新增加的点确定节点值.生成算法的基本思想是:首先，B样条曲线和扩展部分在连接点处满足2

GC连续，用能量极小化方法确定扩展部分的曲线形状，通过对曲线重新参数化使两部分曲线满足2

C连续，进而确定新增加点的节点值.文章还讨论了运用该方法进行B样条曲面扩展，且以实例对新方法与其它方法进行了比较.

关键词：B样条曲线; B样条曲面;参数化;曲线扩展

Research On The Nature Of B-spline Curves And Surfaces And

Their Generation Algorithm

Yongning Zhang

College of Mathematics and Information Science , Xianyang Normal University , Information

Class 05(2)

Abstract

First, the definitions of B-spline curves and surfaces are introduced, and then the nature of B-spline curves and surfaces are studied. On the final, a new generation algorithm on expansion of B-spline curves and surfaces is proposed. The key thought of expansion of B-spline curves and surfaces is to determine the value of the new points. The basic idea of generation algorithm is: First of all, B-spline curve and the extension of the connecting points in a row to meet with the energy minimization method to determine the extension of the curve shape of the curve through re-parameterized so that the two parts meet the continuous curve, and then determine the new value of the node points. The article also discussed the use of the method of B-spline surfaces expansion, and compared an example of the new method with other methods.

Keywords: B-spline curve; B-spline surfaces; parameter; curve extension

目录

引言........................................................... - 3 - 1．B样条曲线.................................................. - 3 -

1.1 B样条基函数的定义...................................... - 3 -

1.2 B样条曲线的定义........................................ - 3 -

1.3 B样条曲线的性质....................................... - 4 -

1.3.1 严格的凸包性...................................... - 4 -

1.3.2 分段参数多项式.................................... - 4 -

1.3.3 可微性或连续性.................................... - 4 -

1.3.4 几何不变性........................................ - 4 -

1.3.5 局部可调性........................................ - 4 -

1.3.6近似性............................................ - 5 -

1.3.7变差缩减性........................................ - 5 -

1.4 B样条曲线的分类....................................... - 5 -

1.4.1均匀B样条曲线.................................... - 5 -

1.4.2 非均匀B样条曲线.................................. - 6 -

1.5 B样条曲线的生成算法.................................... - 7 -

1.5.1B样条曲线的扩展................................... - 7 -

1.5.2 2

GC连续条件...................................... - 8 -

1.5.3确定 的值........................................ - 8 -

1.5.4确定节点u的值.................................... - 9 -

1.5.5求控制顶点........................................ - 9 -

2.1 B样条曲面的定义....................................... - 12 -

2.2 B样条曲面的性质...................................... - 12 -

2.3 B样条曲面的分类...................................... - 12 -

2.3.1 均匀B样条曲面................................... - 12 -

2.3.2 非均匀B样条曲面................................. - 13 -

2.4 B样条曲面的生成算法.................................. - 14 -

3.应用........................................................ - 15 - 4结论........................................................ - 16 - 谢辞.......................................................... - 17 - 参考文献...................................................... - 17 -