L_∞范数下使用基本曲线和修正曲线的带约束Bézier曲线降阶

Constrained Degree Reduction of Bézier Curve in L_∞ Norm Using Basic Curve and Correction Curve

为避免直接求解基于L∞距离的带约束逼近的非线性最优解引起的复杂性,提出了一种把降阶逼近曲线分解为基本曲线和修正曲线的降阶方法.基本曲线利用约束Legendre多项式可得到显式解,且保证降阶后曲线满足要求的边界插值条件;修正曲线的控制顶点由降阶逼近曲线和原曲线的差定义,能够在L∞范数意义下极小化降阶逼近曲线与原曲线的误差.文中方法以简单稳定的方式实现保端点插值的一次降多阶,并达到L∞范数意义下对原曲线的近似最佳逼近.最后通过实例说明了文中方法的有效性.

To avoid the complexity that arose from directly solving the constrained optimization in L∞ norm, we present a new method, which decomposes the approximation curve into two parts： the basic curve and the correction curve. The basic curve can be explicitly obtained by using constrained Legendre polynomials, and it satisfies the constrained conditions imposed on the approximation curve. The correction curve, whose control points are defined by the difference between the original curve and the approximation curve, is used to minimize the error in L∞ norm. The new method performs multi-degree reduction at one time in a steady and simple way, and achieves near optimal uniform approximation. Examples are included to show the performance of new method.