五阶与六阶三角样条曲线

Five-and Six-Order Trigonometric Spline Curves

利用三角函数构造了两个含参数的函数组,它们分别由6个、7个函数组成,分析了这两个函数组的性质。由这两组函数定义了两种新的样条曲线,它们分别具有与五次、六次B样条曲线相同的结构。新曲线在继承B样条曲线基本性质的同时,又具备了一些新的优点。例如,在等距节点下,新曲线在节点处均可以达到C5连续,而且在不改变控制顶点的情况下,新曲线的形状均可以通过改变形状参数的值进行调整。另外,给出了使新曲线插值于控制多边形首末端点的方法,以及构造闭曲线的方法等,文中的图例说明了新方法的正确性和可行性。

Using trigonometric functions, two groups of functions with parameters are constructed, which consist of six and seven functions respectively. The properties of the two groups of functions are analyzed. Based on them, two kinds of new spline curves are defined, which have the same structure with quintic and sextic B-spline curves respectively. The new curves not only inherit the basic properties of B-spline curve, but also have some new advantages. For example, when the knot points are equidistant, the new curves can reach C5 continuous at the knot points, and the shape of the new curves can be adjusted by changing the value of the shape parameter with the control points unchanged. In addition, the methods of making the new curves interpolating the first and end points of the control polygon, and the methods of constructing closed curves, etc, are given. The examples in the paper show the new method is correct and feasible.