基于Bernstein基表示的变次数样条曲线的精确降次算法  

Exact Degree Reduction Algorithm for Multi-Degree Spline Curves Based on Bernstein Basis Representation

作　　者：肖祖尚 沈莞蔷[1] Xiao Zushang;Shen Wanqiang(School of Science,Jiangnan University,Wuxi 214122)

机构地区：[1]江南大学理学院,无锡214122

出　　处：《计算机辅助设计与图形学学报》2025年第1期40-50,共11页Journal of Computer-Aided Design & Computer Graphics

基　　金：国家自然科学基金(61772013).

摘　　要：为完善变次数样条的精确降次理论、提升降次的速度,提出基于Bernstein基表示的变次数样条曲线的精确降次算法.首先根据在不做参数变换的情况下曲线降次前后的值不变的规则,使用单点处的高阶右导数公式得到每段曲线可精确地降低次数的充要条件;然后基于Bernstein基的线性表示,给出降次后控制顶点关于降次前控制顶点的显式表达;最后给出精确降次算法,用于先验判断一条变次数样条曲线每段能精确降至的最低次数,并且在可降次的情况下能选择每段需要降低的次数,求解降次后的控制顶点.通过自拟的2个简单示例与对偶基算法的精确降次相比的结果表明,在相同的环境下,所提算法的计算时间大约是对偶基算法的30%.To enhance the exact degree reduction theory for multi-degree splines and expedite the reduction process,an exact degree reduction algorithm for multi-degree spline curves is proposed based on the Bernstein basis representation.Initially,adhering to the rule that the values of the curve remain invariant before and after degree reduction without parameter transformation,a necessary and sufficient condition for exact degree reduction of a segment is established using high-order right derivatives at a single point.Subsequently,leveraging the linear representation of the Bernstein basis,explicit expressions of the control points after degree reduction in relation to those before degree reduction are provided.Finally,an exact degree reduction algorithm is presented,enabling a priori determination for the minimum degree that each segment of a multi-degree spline curve can be precisely reduced to.Within the range of degree,the reduced degree for each segment can be chosen,and the control points after degree reduction can be calculated.Compared with dual basis method for exact degree reduction by two self-constructed simple examples,the results indicate that,in the same environment,the computation time of the proposed algorithm is about 30% of that using the dual basis method.

关 键 词：B样条 变次数样条 精确降次 基函数 对偶基 

分 类 号：TP391.72[自动化与计算机技术—计算机应用技术]