一类满足G^2连续的三角Bézier曲线曲面

A class of trigonometric Bézier curve and surface which satisfy G^2 continuity

为了在相对简单的条件下满足相对较高的光滑融合,同时在不改变控制顶点的情况下也可以修改曲线曲面的形状,构造了一组低阶的带有两个形状参数的三角Bézier基函数。基于该组基函数,通过三角函数的组合方式定义了任意阶三角Bézier曲线曲面,并详细讨论曲线的基本性质,同时也讨论了曲线、曲面的光滑融合所满足的条件。根据融合条件,可构造分段光滑的组合曲线曲面。这种融合的曲线曲面可以通过修改控制顶点和参数的方法来调节曲线曲面的形状,但不会改变曲线曲面的连续性并且在一定条件下能自动保证组合曲线、曲面的G2连续且计算简单。数值实例结果显示了该方法的有效性。

In order to meet the relatively high smooth jointing in relatively simple conditions,at the same time we can also modify the shape of curve and surface under without changing the control points,and then a group of low order trigonometric Bézier basis function with two shape parameters was constructed. Based on the group of basic functions,a class of curve and surface of arbitrary order trigonometric Bézier is defined by trigonometric function. The basic properties of the curves are discussed,and the conditions of the smooth blending of curves and surfaces are also discussed.According to the blending condition,the combination curves and surfaces of the piecewise smooth curves can be constructed.The blending curves and surfaces will not be changed the continuity of curve and surface by modifying the control points and parameters of the method and automatically meet with the G^2 continuity and simple calculation. The results of numerical examples show the effectiveness of this method.