目的 为了使构造的曲线拥有传统Bézier曲线的良好性质,同时还具备形状可调性、逼近性、保形性以及实用性。方法 本文首先在拟扩展切比雪夫空间的框架下,构造了一类具有全正性的拟三次三角Bernstein基函数,并给出了该基函数的性质；基于此基函数,构造了相应的拟三次三角Bézier曲线,分析了其曲线的性质,得到了生成曲线的割角算法以及 , 光滑拼接条件,同时还提出了一种估计曲线逼近控制多边形程度的三角Bernstein算子；接着在拟三次三角Bernstein基函数的基础上提出一种三角域上带三个指数参数的拟三次三角Bernstein-Bézier基,基于此基生成了一种三角域上的拟三次三角Bernstein- Bézier曲面,该曲面可以构建边界为椭圆弧、抛物线弧以及圆弧的曲面。在此过程中,为了能够高效稳定地计算所提出的三角域曲面,提出了一种实用的de-Casteljau-type 算法,同时还给出了连接两个曲面的 连续条件。结果 实验表明,本文在拟扩展切比雪夫空间中构造的具有全正性的曲线曲面,能够灵活地进行形状调整,而且具有良好的逼近性以及适用性。结论 本文在拟扩展切比雪夫空间的框架下构造了一类具有全正性的基函数,并以此基函数进行曲线曲面构造。实验表明本文构造的曲线具备传统三次Bézier曲线的所有优良性质,而且具有灵活的形状可调性。随着参数的增大,所生成的曲线能够更加逼近控制多边形,模拟控制多边形的行为。此外,本文在三角域上构造的曲面能够生成边界为椭圆弧的曲面,还给出了一种实用的计算曲面de-Casteljau-type算法。综上,本文提出的基函数满足几何工业的需要,是一种实用的方法。
A new trigonometric basis with exponential parameters
Wang Kai,Zhang Guicang,Tuo Mingxiu(Northwest normal university)
Objective To enable the extended curve and surface to maintain the good nature of traditional Bézier method and B-spline method, while shape preserving, shape adjustability and practicability, this paper makes use of the blossom property in Quasi Extended Chebyshev space to construct a group of optimal normalized totally positive basis for curve and surface construction. Method In this paper, a class of cubic trigonometric Quasi Bernstein basis functions with totally positivity is constructed under the framework of the Extended Chebyshev space, and the properties of the basis functions are given. Based on this basis function, the corresponding curve is given. The properties of the curve are analyzed. The cutting algorithm of the curve and the smooth connecting conditions are given. A trigonometric Quasi Bernstein operator for estimating the degree of the control polygon is also proposed. Then, based on the cubic trigonometric Quasi Bernstein basis function, a class of trigonometric polynomial basis functions with three shape parameters over triangular domain is proposed. Based on this basis functions, a kind of triangular polynomial patch over triangular domain is proposed. This patch can be used to construct patches whose boundires are elliptical arcs, parabolic arcs and arcs. In order to calculate the proposed triangular polynomial surface efficiently and stably, a practical de-Casteljau-type algorithm is proposed. In addition, G1 continuous conditions for joining two triangular polynomial patches are given. Result Experiments show that the proposed totally positivity patch in the frame of Chebyshev space can not only flexibly adjust the shape, but also has shape preserving and good approximation. Conclusion In this paper, we construct a class of basis functions with totally positivity under the framework of the Extended Chebyshev space, and construct the curve and surface with this basis function. Experiments show that the curve constructed in this paper has all the excellent properties of the traditional cubic Bezier curve, and has flexible shape adjustability. As the parameters increase, the generated curve can be closer to the control polygon, simulating the behavior of the control polygon. In addition, the surface constructed on the triangular domain can generate the surface whose boundries are elliptical arcs, and a de-Casteljau-type algorithm for calculating the surface is also given. In summary, the basis function proposed in this paper satisfies the needs of the geometric industry and is a practical method.