汪凯,张贵仓,拓明秀(西北师范大学, 兰州 730070)
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 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C1，C2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3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，此外，还提出一种实用的de-Casteljau-type算法，同时还给出了连接两个曲面的G1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。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。
New trigonometric basis with exponential parameters
Wang Kai,Zhang Guicang,Tuo Mingxiu(Northwest Normal University, Lanzhou 730070, China)
Objective The construction of basis functions has consistently been a difficult point of computer-aided geometric design (CAGD). The construction of a class of practical basis functions often plays a decisive role in the development of the geometric industry. The traditional Bézier curves and B-splines have been widely used in CAGD. However, when the control points are determined, the generated curve is relatively fixed with respect to the control points and has a certain rigidity. Although the proposed rational B-spline curve can adjust the curve by adjusting the weight factor, the rational methods have difficulty in predicting the influence of the weight factor on the curve due to its own shortcomings. Researchers have exerted efforts in the past two decades to solve this problem. However, most of the improved methods have the basic properties of the traditional Bézier method and the B-spline method, such as affine invariance, convex hull, non-negativity, geometric invariance and flexible shape adjustability. Moreover, the proposed curve can accurately represent special curves used in engineering, such as conic and hyperbolic curves. However, most of the literature does not discuss the variation diminishing the generated curve. The curve with vanishing variation must have convexity. The curve with the total positivity must have diminishing variation. Therefore, the total positivity of the basis functions indicates that these functions are suitable for geometric design. We can easily obtain rectangular patches with shape parameters through these new curves. However, the Bernstein-Bézier patch over the triangular domain is not a tensor product patch exactly. Therefore, we cannot obtain triangular surfaces with an adjustable shape using the method of tensor product. Surface modeling over triangular domain is important for many applications. Thus, the practical methods for generating surfaces over a triangular domain must be explored. The blossom property in quasi extended Chebyshev space is used to construct a group of optimal normalized entirely positive basis for curve and surface construction. This method enables the extended curve and surface, thereby maintaining the good nature of the traditional Bézier and B-spline methods while preserving shape, shape adjustability, and practicability. Method A class of cubic trigonometric quasi Bernstein basis functions with total positivity is constructed under the framework of the extended Chebyshev space, and the properties of the basis functions are provided. The corresponding curve is presented based on this basis function. The properties of the curve are analyzed. The cutting algorithm of the curve and the smooth connecting conditions are obtained. 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 the triangular domain is proposed. A type of triangular polynomial patch over the triangular domain is proposed based on this basis functions. This patch can be used to construct patches whose boundaries are elliptical arcs, parabolic arcs, and arcs. A practical de-Casteljau-type algorithm is proposed to calculate the proposed triangular polynomial surface efficiently and stably. In addition, G1 continuous conditions for joining two triangular polynomial patches are provided. Result Experimental results show that the proposed total positivity patch in the frame of Chebyshev space not only can adjust the shape flexibly but also has shape preservation and good approximation. Conclusion We construct a class of basis functions with total positivity under the framework of the extended Chebysh ev space, and construct the curve and surface with this basis function. Experimental results show that the curve constructed in this study has all the excellent properties of a traditional cubic Bézier curve and has flexible shape adjustability. As the parameters increase, the generated curve can be closer to the control polygon, thereby simulating its behavior. In addition, the surface constructed on the triangular domain can generate the surface whose boundaries are elliptical arcs. A de Casteljau-type algorithm for calculating the surface is also provided. In summary, the proposed basis function satisfies the requirements of the geometric industry and is a practical method.