|
| B样条基函数向Bernstein基函数的转换 |
| Conversion from B-spline Basis Function to Bernstein Basis Function |
| 投稿时间：2023-09-02 修订日期：2024-01-18 |
| DOI： |
| 中文关键词: B样条基函数 Bernstein基函数 显式表示 转换矩阵 |
| 英文关键词: B-spline basis function Bernstein basis function explicit representation conversion matrix |
| 基金项目: |
|
| 摘要点击次数: 73 |
| 全文下载次数: 0 |
| 中文摘要: |
| 为了将B样条曲线曲面转换为Bézier曲线曲面, 讨论了B样条基函数向Bernstein基函数的转换算法. 从B样条基函数的de Boor-Cox递推定义出发, 给出了在任意一个长度非零的节点区间上, 相邻次数的B样条基函数之间的递推关系. 并进一步给出了在局部参数下, 在单位长度的定义区间上, 高次与低次B样条基函数之间的递推公式. 基于此, 并结合Bernstein基函数的递推公式, 导出了从B样条基函数到Bernstein基函数的转换矩阵中, 高次与低次结果中矩阵元素之间的递推关系. 从而可以从0次B样条基函数的Bernstein基函数表示结果开始, 递推得到任意次的表示结果. 所得递推关系形式简洁, 转换算法具有普适性, 为运用Bézier方法的理论成果与成熟算法解决B样条方法的相关问题提供了理论基础. 作为应用, 给出了用该算法实现从B样条曲面向分片Bézier曲面表示形式的转换流程以及数值实例, 验证了算法的正确性和实用性. |
| 英文摘要: |
| In order to convert the curves and surfaces from B-spline to Bézier representation, the conversion algorithm from B-spline basis to Bernstein basis is discussed. Starting from the de Boor-Cox definition of B-spline, the recursive relation between B-spline basis of adjacent degrees on any knot interval of non-zero length is given. Furthermore, the recursive formula between B-spline basis of higher degree and lower degree under local parameter and on the unit length of definition interval is given. Based on this and combining with the recursion formula of Bernstein basis, the recursion relationship between the matrix elements in the results of higher degree and lower degree in the transformation matrix from B-spline basis to Bernstein basis is derived. In this way, starting from the relationship between B-spline basis and Bernstein basis of degree 0, we can obtain the representation result of arbitrary degree by recursion. The recursion relation is simple and the conversion algorithm is universal, which provides a theoretical basis for applying the theoretical results and mature algorithm of Bézier method to solve the problems related to B-spline method. As an application, the conversion procedure from B-spline surface to piecewise Bézier surface and numerical examples are given to verify the correctness and practicability of the algorithm. |
|
View Fulltext
查看/发表评论 下载PDF阅读器 |
| 关闭 |
|
|
|
