B样条曲线面的性质及其生成算法的研究.doc

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 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