【摘要】 利用权的思想并结合奇异混合技术,对传统的拟Bézier曲线进行扩展,构造了一种带形状参数的奇异混合拟Bézier曲线。首先将奇异混合函数和三角多项式空间的拟三次Bézier基函数相结合得到奇异混合拟Bézier曲线的定义,进而根据奇异混合拟Bézier曲线的定义反推出奇异混合拟Bézier基函数;接着讨论了奇异混合拟Bézier基函数及其对应曲线的性质,并探究了奇异混合函数及参数对二者的影响;最后给出了奇异混合拟Bézier曲线曲面的设计实例。实验结果表明,与传统Bézier曲线相比,本文构造的曲线在具有传统Bézier曲线实用性质的同时还具有灵活的形状可调性,新曲线不仅能够精确表示二次曲线,并且在满足特定条件时曲线还能够达到G~1及G~2连续,将曲线运用张量积方法拓展到曲面还可以精确表示椭球面及球面。大量的分析以及实例表明,本文构造的曲线在几何造型设计中十分有效。更多还原

【Abstract】 Weighting idea and singular blending technology are used to extend the traditional quasi-Bézier curve, and a singular blending quasi-Bézier curve with shape parameters is constructed. Firstly, the singular blending function and the quasi-cubic Bézier basis function of the triangular polynomial space are combined to obtain the definition of the singular blending quasi-Bézier curve, and the singular blending quasi-Bézier basis function is deduced according to the definition of the singular blending quasi-Bézier curve. Secondly, we discuss the singular blending quasi-Bézier basis functions and the properties of their corresponding curves, and explore the influences of singular blending and parameters on them. Finally, an example of a singular blending quasi-Bézier curve and surface design is given. The experimental results show that the curve constructed in this paper has the flexible shape adjustability while having the practical properties of the traditional Bézier curve. The new curve can not only accurately represent conic curves such as elliptical arc, circular and parabola arc, but also achieve G~1 and G~2 continuity under certain conditions. Extending the curve to the surface using the tensor product method can also accurately represent the ellipsoid and the spherical surface. A large number of analysis and examples prove that the curves constructed in this paper are very effective in geometric design.更多还原