摘要
本文的主题是对CAGD中可展Bézier曲面和可展Poisson曲面的设计以及球域Bézier曲线的边界曲面进行研究。
第一章综述了Bézier曲线曲面的理论发展过程,引申出可展曲面设计和球域Bézier曲线设计的必要性。
第二章介绍了以任意一条空间Bézier曲线为准线的一张可展Bézier曲面设计方法。该方法应用有关曲面可展充要条件的微分几何理论,Bézier曲线的升阶公式以及Bernstein基函数的线性无关性,不用解非线性特征方程,就可以直接决定可展曲面的直母线、可展锥面的顶点和切线面的脊线,完成可展曲面的设计。本章的结果对于工程外形设计具有良好的应用前景。
第三章是空间可展Poisson曲面的设计。利用Poisson基函数的线性无关性,Poisson曲线的升阶公式,以及微分几何理论中可展曲面的充要条件,设计出超越可展曲面。本章结果可广泛应用于旋转切割加工中频繁遇见的超越曲面的设计以及螺旋形管道曲面的外钣展开。
第四章是球域Bézier曲线的边界曲面的求解。利用微分几何中空间曲面族的包络算法和变量替换方法,求得球域Bézier曲线的精确边界表示;进一步利用函数逼近论中Legendre多项式的最佳一致平方逼近方法,把球域Bézier曲线的边界曲面近似地表示为一张Bézier曲面或分片Bézier曲面的组合。球域Bézier曲线是表达方式简洁、存储空间节省、运算速度较快的误差分析和误差控制工具。
In this paper, some researches have been done on design of developable Bezier surfaces and Poisson surfaces, the boundary surfaces of ball Bezier curve in Computer Aided Geometric Design (CAGD).In Chapter 1, an overview of development of Bezier curves and surfaces is presented, and hence developable surfaces and ball Bezier curve are introduced.In Chapter 2, an algorithm is presented that generates a developable Bezier surface through a Bezier curve called a directrix. The algorithm is based upon the differential geometry theory about the necessary and sufficient conditions for a surface which is developable, the degree elevation formula for parameter curves and the linear independency for Bernstein basis. No nonlinear characteristic equations have to be solved. Moreover the vertex for a cone and the edge of regression for a tangent surface can be given easily.In Chapter 3, design of developable Poisson surface is presented. Based on the necessary and sufficient conditions for a surface which is developable, the degree evaluation formula for Poisson curves and the linear independency for Poisson basis functions, an algorithm that generates developable Poisson surfaces through a Poisson curve of arbitrary degree and shape is given. These results could be applied to design of surfaces frequently met in revolution cutting, and plate expansion of spiral-like pipe surfaces.In Chapter 4, how to solve boundary surface of a ball Bezier curve is presented. With the envelope algorithm of the family of space surfaces in differential geometric and variable transformation, an accurate representation for the boundary of a ball Bezier curve was gained;and furthermore, it was approximately represented as a Bezier surface or an union of Bezier patches by using Legendre polynomial best square uniform approximation. Ball Bezier is a kind of error control and error analysis with simple expression, less storage and fast computation.
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