【作者】 于涛；

【导师】 韩力文；

【作者基本信息】 河北师范大学 ， 计算机应用技术， 2010， 硕士

【摘要】 细分方法是曲线曲面几何造型中的常用方法,它可以克服参数表示方法的局限性,并具有从离散到离散、规则简单、易于修改和极限曲线曲面良好的光滑性等优点,广泛应用于三维几何造型、计算机动画和多分辨率分析等领域。NURBS方法可以用统一的数学形式精确地表示二次曲线曲面,是定义工业产品几何形状的唯一数学描述方法,因此解决NURBS曲线曲面细分问题对几何造型有着重要的意义。为了满足实际几何造型中对于精度的要求,分析NURBS细分曲线曲面误差也显得极为重要。本文在讨论NURBS曲线系数矩阵基础上,研究了NURBS曲线细分和二次曲面细分,并应用NURBS曲线点投影的裁剪圆算法分析了NURBS细分曲线的误差。取得了以下主要研究成果:(1)利用NURBS曲线矩阵形式中的系数矩阵,得到了NURBS曲线的细分表示。并利用该细分表示,实现了具有代表性的二次NURBS曲线2-尺度细分、二次NURBS曲线3-尺度细分、三次NURBS曲线2-尺度细分,进一步推广至高次NURBS曲线的多尺度细分。(2)讨论了NURBS曲线点投影问题,研究了B样条曲线乘积和点投影的裁剪圆算法,应用裁剪圆算法分析了NURBS细分曲线的误差,并给出了NURBS曲线每次细分后控制多边形与极限曲线误差的计算实例。(3)通过圆和椭圆的九顶点NURBS表示,结合旋转曲面特征,将NURBS曲线细分应用到二次曲面细分,实现了单叶双曲面、双叶双曲面、椭圆抛物面和三轴椭球等曲面的细分。并采用MATLAB R2009a搭建了“NURBS曲线曲面细分”软件平台,在文中所用到的图例都由该平台计算生成。更多还原

【Abstract】 Subdivision schemes have important applications in curves and surfaces modeling and engineering designing. Applying subdivision schemes to geometric modeling may overcome the limitations of parametric schemes. Subdivision is efficient to design, represent and approach arbitrarily topological curves and surfaces. Because subdivision schemes have the advantages of discrete-to-discrete, simple rules, efficiency and convenient modification, which have already been widely used in 3D geometric modeling, computer animation and multiresolution analysis. NURBS may represent curves and surfaces accurately by uniform algebraic expression. Therefore, it is important to solve the problem of NURBS curves and surfaces subdivision for geometric modeling. For the accuracy of engineering, analyzing the error of NURBS curves subdivision is needed. This thesis studies subdivision schemes and applications of NURBS curves based on coefficient matrix, and point projection clipping circular algorithm is used to analyze the error of NURBS curves subdivision. The main results are as follow:(1) It studies the subdivision expression of NURBS curves based on NURBS coefficient matrix, with which 2-band subdivision scheme of quadratic NURBS curves, 3-band subdivision scheme of quadratic NURBS curves and 2-band subdivision scheme of cubic NURBS curves are obtained. Consequently, it can subdivide higher degree NURBS curves with multi-band.(2) It discusses NURBS point projection, and researches B-spline curves product and clipping circular algorithm in detail. Analyzing the error of NURBS curves subdivision with clipping circular algorithm of point projection and examples are given.(3) Through the nine vertex representation of circle and ellipse, combined with the characteristics of rotary surfaces, it obtains quadratic NURBS surfaces subdivision based on NURBS curves subdivision, and achieves subdivision of hyperboloid of one sheet, hyperboloid of two sheets, elliptic paraboloid and triaxial ellipsoid and so on.‘NURBS curves and surfaces subdivision’software platform is constructed by MATLAB R2009a, examples used in the thesis are generated by this software platform.更多还原