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带两个参数的三角多项式曲线曲面构造

汪凯,张贵仓,龚进慧(西北师范大学, 兰州 730070)

摘 要
目的 为了使扩展的曲线曲面保留传统Bézier方法以及B样条方法良好性质的同时,具备保形性、形状可调性、高阶连续性以及广泛的应用性,本文在拟扩展切比雪夫空间利用开花的性质构造了一组最优规范全正基,并利用该基进行曲线曲面构造。方法 首先构造一组最优规范全正基,并给出该基生成的拟三次TC-Bézier曲线的割角算法;接着利用最优规范全正基的线性组合构造拟三次均匀TC-B样条基,根据曲线的性质假设拟三次均匀B样条基函数具有规范性和C2连续性,进而得到其表达式;然后证明拟三次均匀TC-B样条基具有全正性和高阶连续性;最后定义拟三次均匀TC-B样条曲线曲面,并证明曲线曲面的性质,给出曲线表示整圆和旋转曲面的表示方法,设计出球面和旋转曲面的直接生成方法。结果 实验表明,本文在拟扩展切比雪夫空间构造的具有全正性曲线曲面,不仅能够灵活地进行形状调整,而且具有高阶连续性、保形性。结论 本文在三角函数空间利用两个形状参数进行曲线曲面构造,大量的分析以及案例说明本文构造的曲线曲面不仅保留了传统的Bézier方法以及B样条方法的良好性质,而且具备保形性、形状可调性、高阶连续性以及广泛的应用性,适合用于曲线曲面设计。
关键词
Constructing trigonometric polynomial curves and surfaces with two parameters

Wang Kai,Zhang Guicang,Gong Jinhui(Northwest Normal University, Lanzhou 730070, China)

Abstract
Objective The Bézier and B-spline curves play an important role in traditional geometric design. With the development of the geometric industry over the recent years, the traditional Bézier and B-spline curves cannot meet people's needs due to defects. At the same time, many rational forms of Bézier curves are proposed, which solve the problems faced by traditional methods. However, rational methods have not only progressive problems, but also employ the improper use of weight factors, which can be destructived to the curve and surface design. In view of the abovementioned problems, a large number of Bernstein-like and B-spline-like basis functions with shape parameters are proposed. These methods are mainly constructed in trigonometric, hyperbolic and exponential function spaces, a combination of said spaces, and polynomial space. Although many improved methods are available, these methods are rarely applied in solving practical problems. In the final analysis, these methods increase the flexibility of the curve by adding shape parameters, compared with the traditional Bézier and B-spline methods. However, the method itself does not have the ability to replace the traditional method. Several aspects still need improvement. For example, the majority of these methods only discuss basic properties, such as non-negativity, partition of unity, symmetry, and linear independence. Shape preservation, total positivity, and variation diminishing are often overlooked, which are important properties for curve design. However, the basis function, which has total positivity, will ensure that the related curve contains variation diminishing and shape preservation. Therefore, possessing total positivity is highly important for basis function. In addition, constructing cubic curves and surfaces remains the main method among the improved methods. In general, these improved methods have C2 continuity, which largely meets engineering requirements. However, in many practical applications, C2 continuity cannot meet current needs. In summary, this study aims enable the extended curve and surface to maintain the good nature of traditional Bézier and B-spline methods, while maintaining shape preservation and adjustability, high-order continuity, and wide applicability. Thus, this article 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, we first construct a set of optimal normalized totally positive basis and then present a corner cutting algorithm of the cubic TC-Bézier curves generated by the base. Second, it renders use of the linear combination of optimal normalized totally positive basis construct the proposed cubic uniform TC-B spline basis. It assumes that the proposed cubic uniform B-spline basis function is characterized by normalization and continuity according to the nature of the curve, then further obtains its formula. The article proves that the proposed cubic uniform TC-B spline basis features total positivity and high-order continuity. Finally, the curve and surface of the proposed cubic uniform TC-B spline are defined, which proves the nature of the curve and surface. Furthermore, an expressive method of using the curve to show the full circle and rotating surface and direct generation method of spherical and rotating surfaces are provided. Result Analysis show that the TC-Bézier basis function in the ECC space possesse totally positivity, and the cubic uniform TC-B spline basis function in relation to the TC-Bézier basis function also possesses totally positivity. Therefore, the cubic TC-Bézier and cubic uniform B-spline curves that are generated by the corresponding basis functions also have important properties, such as variation diminishing and shape preservation, which further means that the proposed method is suitable for curve design. In addition, this paper also provides a large number of application cases to further show that the method is suitable for curve design. Conclusion We use two-shape parameters in the trigonometric function space for curve design. Surface construction, numerous analyses, and cases indicate that the curves and surfaces constructed retain not only the good properties of the traditional Bézier and B-spline methods, such as affine invariance, convex hull and variation diminishing, which are crucial in curve design but also shape preservation and adjustability, high-order continuity, and wide application. In summary, this paper proposes the trigonometric polynomial curve and surface method with two shape parameters, which solves the shortcomings of the traditional improvement methods in terms of property and high-order continuity. Therefore, the proposed method is more suitable for curve design compared with the traditional improved method.
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