插值给定对角曲线的能量极小Bézier曲面造型  

Construction of energy-minimizing Bézier surfaces interpolating given diagonal curves

作　　者：朱雨凡 徐岗[1] 凌成南 李博剑 许金兰[1] Zhu Yufan;Xu Gang;Ling Chengnan;Li Bojian;Xu Jinlan(College of Computer Science and Technology,Hangzhou Dianzi University,Hangzhou Zhejiang 310018,China)

机构地区：[1]杭州电子科技大学计算机学院

出　　处：《中国图象图形学报》2019年第11期1998-2008,共11页Journal of Image and Graphics

基　　金：国家自然科学基金项目(61772163,61761136010,61472111)~~

摘　　要：目的曲面造型是计算机辅助几何设计中的重要研究内容,张量积类型曲面的对角曲线是衡量曲面性质的重要度量,与曲面的几何形状密切相关。基于输入对角曲线的曲面设计方法在实际应用中具有一定的价值,因此提出一种插值给定对角曲线的能量极小Bézier曲面造型的方法。方法给定一条对角曲线时,修正用户输入的对角曲线及边界曲线的几何信息,然后运用拉格朗日乘数法,结合曲面内部能量函数,求解待定的内部控制顶点,构造曲面。给定两条对角曲线时,在上述内容基础上加入了两条对角曲线必有交点的考量,增加对对角曲线控制顶点的修正。结果增加了对角曲线这一约束条件,从对比实验曲线图可以看出,随着横坐标曲面阶数升高,纵坐标修正曲线和用户曲线间的差值越来越小,结果表明曲面阶数越高,修正曲线与用户曲线偏差越小,造型效果越好。结论该曲面造型方法简单,基于修正后的对角曲线和边界曲线构造的曲面具有极小内部能量,可满足曲面造型方面的相关需求。Objective Surface modeling is an important research content of computer-aided geometric design,architectural geometry,and computer graphics. The diagonal curve of the tensor product surface is an important tool to measure surface properties. In the aspect of modeling design,people have various requirements for the diagonal curves and boundary curves of a surface. People want to optimize the boundary of the entire surface through the special boundary curves and determine the overall shape of the surface by designing one or two diagonal curves. Therefore,constructing a surface based on the boundary and diagonal curves given by the user is important. The diagonal curve of the Bézier surface is related to its geometry. The method of surface design based on the input diagonal curve will have certain value in practical applications.Bézier surface modeling based on diagonal curve has been rarely published. Method In this paper,the Bézier surface construction method is investigated for given diagonal and boundary curves. The method is mainly divided into the case of a diagonal curve and the case of two diagonal curves. The information of the curves needs to be corrected to achieve an ideal shape. The Lagrange multiplier method is mainly used in the correction. In the case of a given diagonal curve,first,the users input the diagonal and boundary curves of the surface according to their personal requirement. The sum of the dis-tances of the control points is taken as the objective function to ensure the minimum deviation between the modified diagonal curve and the boundary curves and the curves given by the user. The relationship between the diagonal curve and the boundary curve is used as the constraint condition,and the geometric information of the diagonal curve and the boundary curve input by the user is corrected. We then use the modified curve as the diagonal and boundary curves in subsequent surface construction. The internal control points to be determined are set as the independent variable by using Lagrangian

关 键 词：曲面造型 对角曲线 拉格朗日乘数法 弯曲能量 拟调和能量 Dirichlet能量 

分 类 号：TP301.6[自动化与计算机技术—计算机系统结构]