$G^0$条件下最接近给定平面B\'ezier曲线的PH曲线

本文讨论了分别利用Gauss-Legendre多边形和Gauss-Lobatto多边形, 在$G^0$条件下找到最接近给定平面B\''ezier曲线的五次PH曲线,无论是否指定弧长.通过计算给定B\''ezier曲线的Gauss-Legendre或Gauss-Lobatto多边形的顶点与PH曲线的顶点之间的平方差之和,可以将此问题表述为带有两个或三个二次约束的多项式优化问题,并且此问题由拉格朗日乘子法和牛顿-拉弗森迭代法有效地解决.文中给出了几个计算实例来说明优化方法的实现.计算结果表明,与B\''ezier控制多边形相比,使用Gauss-Legendre和Gauss-Lobatto多边形的方法可以在$G^0$条件下产生更接近给定B\''ezier曲线的PH曲线,且弧长更接近.此外,还可以实现具有预定弧长的良好近似.

$G^{0}$ Pythagorean-Hodograph Curves Closest to Prescribed Planar B\'{e}zier Curves

The task of identifying the quintic PH curve $G^{0}$ ``closest'' to a given planar B\''{e}zier curve with or without prescribed arc length is discussed here using Gauss-Legendre polygon and Gauss-Lobatto polygon respectively. By expressing the sum of squared differences between the vertices of Gauss-Legendre or Gauss-Lobatto polygon of a given B\''{e}zier and those of a PH curve, it is shown that this problem can be formulated as a constrained polynomial optimization problem in certain real variables, subject to two or three quadratic constraints, which can be efficiently solved by Lagrange multiplier method and Newton-Raphson iteration. Several computed examples are used to illustrate implementations of the optimization methodology. The results demonstrate that compared with B\''{e}zier control polygon, the method with Gauss-Legendre and Gauss-Lobatto polygon can produce the $G^{0}$ PH curve closer to the given B\''{e}zier curve with close arc length. Moreover, good approximations with prescribed arc length can also be achieved.