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基于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)

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(Quasi Extened Chebyshev space)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 本文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 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,不仅能够灵活的进行形状调整,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
Construction of Trigonometric Polynomail Curves and Surfaces Based on Two Shape Parameter

Wang Kai,Zhang Guicang,Gong Jinhui(Northwest normal university)

Abstract
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, high-order continuity and wide applicability, this article makes use of the florescent nature 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 firstly construct a set of optimal normalized totally positive basis, and give the corner cutting algorithm of the cubic TC-Bézier curves generated by the base.Secondly, it makes 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, and then further gets its formula. Then 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 proposed cubic uniform TC-B spline are defined, proving the nature of the curve and surface.Furthermore, the expressive method of using the curve to show the full circle and rotating surface and the direct generation method of spherical surface and rotating surface are provided. Result Experiments show that curve and surface with totally positivity constructed in Quasi Extended Chebyshev space can not only adjust the shape flexibly but also share high-order continuity and shape preserving, which is applicable to the design of curve and surface. Conclusion In this paper, we use two shape parameters in trigonometric function space to curve, surface construction, a lot of analysis and case indicate that the curves and surfaces constructed in this paper not only retains the good properties of the traditional Bézier method and B-spline method, but also has conformal, shape-adjustable, high order continuity and wide application, Suitable for curved
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