带形状参数的三角λ-Bézier曲线

为了克服传统 Bézier 曲线缺乏局部调整性并且不能精确表达圆锥曲线的缺点，构造了一个带形状参数的 n（n≥2）次三角λ-Bézier曲线，为了降低工作难度，各阶曲线的形状参数取值范围保持不变。首先将基函数设置在三角多项式空间中，利用递推性构造了λ-Bernstein基函数，进而讨论了该基函数的端点性和对称性等重要性质，并由该基函数定义了n（n≥2）次λ-Bézier曲线。另外，讨论了形状参数取不同值时对曲线形状的影响以及曲线的拼接条件：在一定的条件下，该曲线可达到G2拼接；最后，给出了张量积形式的λ-Bézier曲面以及性质。实例表明，该曲线克服了传统Bézier曲线缺乏局部调整性的缺点且能近似表达圆弧和抛物线等圆锥曲线。

Trigonometric λ-Bézier Curves with Shape Parameter

To handle the lack of local adjustability and low accuracy in expressing conic curves of traditional Bézier curves , a trigonometric λ-Bézier curve with n(n≥2) order shape parameter was constructed. The value range of shape parameters of each order curve remains unchanged to reduce the difficulty. A λ-Bernstein basis function was constructed in trigonometric polynomial space by means of recursion, and the important properties of endpoint and symmetry of this basis function were discussed. Then, the n(n≥2)λ-Bézier curve was defined by this basis function. In addition, the influence of different shape parameters on the shape of the curve and the splicing conditions of the curve were discussed. The curve can achieve G2 splicing under certain conditions. Finally, λ-Bézier surfaces in tensor product form and their properties were given. The example showed that this curve overcame the lack of local adjustability of traditional Bézier curve, and can accurately express conic curves such as circular arc and parabola.