3次Bézier曲线的新扩展

A New Extension of the Cubic Bézier Curve

为了使扩展的3次Bézier曲线在C^(2)光滑拼接条件下也能实现FC^(3)连续,并保证单段曲线和组合曲线均可以在不改变控制顶点的前提下自由调整形状,构造了一种结构与3次Bézier曲线相同的新曲线.首先,给出调配函数初步表达式,其中含待定系数;然后,根据预设的曲线在拼接时的性质,反推出调配函数应具有的端点性质,以此建立调配函数中待定系数应满足的方程组;通过解方程组,得到了一组含4个参数的5次多项式调配函数,取特殊参数时其次数可降为4次;最后,将调配函数与控制顶点作线性组合,定义了一种由4个控制顶点确定的带4个形状参数的新曲线,其具有Bézier曲线的凸包性、几何不变性和仿射不变性等基本性质.得益于多个形状参数的引入,由相同控制多边形可以定义形状各异的曲线,每个形状参数的改变都会带动曲线上的点沿固定方向线性移动.构造组合曲线时,相邻曲线段之间的FC^(3)连续条件与C^(2)连续条件一致,组合曲线可以在保持连续性、控制顶点和参数分割均不变的前提下,自由调整形状.该方法能扩展CAD系统对工业产品的处理能力,提高产品设计、修改和优化的速度,节约设计中人力和物力.

An extended cubic Bézier curve is constructed, which can realize FC^(3) continuity under the condition of C^(2) continuity, and ensure that the shape of both single-segment curve and composite curve can be freely adjusted without changing the control points. For this reason, a new curve with the same structure as the cubic Bézier curve is constructed. Firstly, the preliminary expression of the blending function is given, including undetermined coefficients. Then, according to the preset properties of the curves during joining, the endpoint properties of the blending function are deduced in reverse, so as to establish the equations that the undetermined coefficients in the blending function should satisfy. By solving the equations, a set of quintic polynomial blending functions with four parameters is obtained, which can be reduced to quartic when special parameters are taken. At last, a new curve with four shape parameters determined by four control points is defined by linear combination of the blending function and the control points. It has the basic properties of Bézier curve, such as convex hull property, geometric invariance and affine invariance. Thanks to the introduction of multiple shape parameters,curves of different shapes can be defined by the same control polygon, and the change of each shape parameter will drive the points on the curve to move linearly along a fixed direction. When constructing composite curve,the FC^(3 )continuity conditions between adjacent curve segments are the same as the C^(2) continuity conditions, and the shape of composite curve can be adjusted freely on the premise of maintaining continuity, control points and parameter segmentation unchanged. The new method can expand the processing ability of CAD system to industrial products, improve the speed of product design, modification and optimization, and save the investment of manpower and material resources in design.