B样条曲线最小二乘降阶方法

Least Squares Degree Reduction of B-spline Curves

提出一种新的B样条曲线降阶方法.该方法利用B样条基转换矩阵建立B样条曲线降阶的数学模型,将B样条曲线的降阶问题转化为求线性方程组的最小二乘解问题.该方法基于整体考虑不必对B样条曲线分段处理,步骤简单易实现;可一次降多阶,避免了重复一次降一阶运算引起的误差累积,而当仅降一阶时与基于控制顶点扰动的约束优化降阶方法的逼近效果一致;在降阶的同时可满足各种给定的端点约束条件,以满足实际应用中的各种要求.

New degree reduction method for B-spline curves is proposed, With this method, the mathematical mode for the degree reduction of B-spline curves is established by the transformation matrix of B-spline bases, and then the problem of degree reduction of B-spline curves is turned into the problem of getting the least squares solution of a linear equation system. The proposed method can be easily realized with simple steps without the piecewise processing to B-spline curves in the procedure of the degree reduction. It can reduce multi degrees at one time to avoid the error accumulation caused by the repetition of the one-degree reduction calculation. And when reducing only one degree, the resulting B-spline curves has the same approximate effect with those produced by the constrained optimization degree reduction method based on control point perturbations. The resulting B-spline curves produced by this method can satisfy various given constrained conditions at two ends of the curves to meet different demands in applications.