一种带形状参数的三角样条曲线

A Trigonometric Spline Curve with Shape Parameters

本文针对三次B样条曲线相对于其控制多边形形状固定,不能描述除抛物线以外的圆锥曲线的不足进行改进。将形状参数与三角函数进行有机结合,构造了一组含参数的三角样条基,基于这组基定义了一种结构类似于三次B样条曲线的带形状参数的三角样条曲线。新曲线在继承B样条曲线主要优点的同时,既具有形状可调性,又能精确表示椭圆,而且其连续性和对控制多边形的逼近性也都优于三次B样条曲线。对于等距节点,在一般情况下该曲线整体C3连续,在特殊条件下可达C5连续。利用张量积方法,将曲线推广后所得到的曲面具有与曲线类似的性质,给出了用曲面表示椭球面的方法。

In order to improve the disadvantages of the cubic B-spline curve,its shape is fixed to its control polygon.It cannot describe the conic section except the parabola,and the shape parameters and the trigonometric functions are combined.A trigonometric curve with shape parameters is defined.Its structure is similar to the cubic B-spline curve.The new curve not only inherits the major advantages of the cubic B-spline curve,but also can express ellipse with shape adjustability.Besides,the continuity and approximability of the new curve is better than the cubic B-spline curve.For equidistant knots,the new curve is C3continuous,and it is C5 continuous under special conditions.Using the tensor product method,the curve can be extended to the surface.The surface has properties similar to the curve.The way to represent the ellipse surface using the new surface is given.