带两个参数的非均匀三次三角B样条曲线

为了使构造的三次三角非均匀 B-样条曲线在具备形状可调性、高阶连续性、精确 表示椭圆等性质的同时还具有变差缩减性，构造了一类具有全正性的带 2 个参数的非均匀三次 三角 B-样条基函数，进而进行曲线构造。首先假设待构造的非均匀三次三角 B-样条基在每一个 节点处具有 C2连续且具有单位性，进而确定基函数的表达式；然后给出了基函数具有全正性等 重要性质；最后给出了非均匀三次三角 B-样条曲线的定义，并证明了其具有变差缩减性等重要 性质，还证明了曲线在取特殊参数值时具有 C(2n–1)阶连续。实例表明，本文构造的曲线有效解 决了传统方法存在的问题，适合于几何设计。

Non-Uniform Cubic Trigonometric B-Spline Curve with Two Shape Parameters

To make the extended cubic trigonometric non-uniform B-spline curves possess not only shape adjustability, high order continuity, and exact representation of ellipse, but also variation diminishing, a class of non-uniform cubic trigonometric B-spline basis functions based on totally positivity is constructed. Firstly, we assume that the non-uniform cubic trigonometric B-spline basis functions to be constructed have C2 continuity and partition of unity at each knot, and accordingly the expressions of the basis functions are determined. Then it is proved that the basis functions have total positivity and other important properties. The definition of non-uniform cubic trigonometric B-spline curves are given, and its important properties such as variation diminishing are proved. It is also proved that the curve has C(2n–1) order continuity when taking special parameter values. The example shows that the curve constructed in this paper effectively solves the problems existing in the traditional method and is suitable for geometric design.