摘 要 ：目的 曲面造型是计算机辅助几何设计中重要研究内容，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。在造型设计方面，人们对曲面的对角曲线和边界曲线有各种各样的要求。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。但目前基于对角曲线的Bézier曲面造型方法鲜有工作发表。方法 本文研究了插值给定对角曲线和边界曲线的Bézier曲面构造方法。主要分为基于一条对角曲线和基于两条对角曲线情况，针对给定一条对角曲线情况，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。针对给定两条对角曲线情况，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 本文设计了几组曲面造型实例并进行比较，验证了该方法的有效性，并给出了具体应用实例和应用场景。与其他造型方法相比，本文方法加入了对角曲线这一约束条件，满足了用户在对角曲线方面的需求。对比实验表明曲面阶数越高，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，实用性强，构造的曲面不仅插值修正后的对角曲线和边界曲线，而且具有极小内部能量，可满足曲面造型方面的相关需求。
Construction of Energy-Minimizing Bézier Surfaces Interpolating Given Diagonal Curves
ZHU Yufan,XU Gang,LING Chengnan,LI Bojian(College of Computer Science and Technology,Hangzhou Dianzi University,Hangzhou Zhejiang 310018)
Objective Surface modeling is an important research content of computer-aided geometric design, architectural geometry and computer graphics. The diagonal curve of tensor product surface is an important measure to measure the 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, it is especially important to construct a surface based on the boundary and diagonal curves given by the user. The diagonal curve of the Bézier surface is closely related to the geometry of the surface. The method of surface design based on the input diagonal curve will have certain value in practical applications. But the method of Bézier surface modeling based on the diagonal curve is rarely published. Method In this paper, the Bézier surface construction method for given diagonal and boundary curves is studied. The method mainly divided into the case of a diagonal curve and the case of two diagonal curves. In order to achieve an ideal shape, the information of the curves needs to be corrected. The Lagrange multiplier method is mainly used in the correction. In the case of a given diagonal curve, firstly, the users input the diagonal and boundary curves of the surface according to their personal requirement. In order to ensure the minimum deviation between the modified diagonal curve and the boundary curves and the curves given by the user, the sum of the distances of the control points is taken as the objective function, and the relationship between the diagonal curve and the boundary curve is used as the constraint condition, the geometric information of the diagonal curve and the boundary curve input by the user is corrected. Then we use the modified curve as the diagonal and boundary curves in the subsequent surface construction; by using Lagrangian multiplier method, the internal control points to be determined are used as the independent variable, the three internal energy functions of the surface (bending energy function, quasi-harmonic energy function and Dirichlet energy function) are taken as the objective function, and the linear relationship between the control points of diagonal curve and the surface is taken as the constraint condition. We convert a conditionally restricted extreme value problem to an extreme value problem without conditions. According to the modified diagonal curves and boundary curves, by finding the extremum of the internal energy function, we find the relationship that the internal control points should satisfy and solve the internal control points. Finally, the surface is constructed from the modified boundary curves, the modified diagonal curves, and the obtained internal control points. In the case of two given diagonal curves, these two diagonal curves must have an intersection. According to this condition, the correction of the control points of diagonal curve is added. In order to ensure that the deviation between the modified diagonal curve and the user-defined diagonal curve is minimized, the sum of the distances of the control points is taken as the objective function, then we correct the diagonal curve given by the user. In a similar way as the previous case, we correct the geometric information of the two diagonal and boundary curves. Result We have designed three- and four-order surface modeling examples to satisfy the requirements of different minimal internal energy, and verified the effectiveness of the surface construction method. By giving a diagonal curve or two diagonal curves, we design modeling examples to verify the practicality of the method. At the same time, the examples of surface modeling with the same boundary and different degree are designed. These examples show that the higher the order of the surface, the closer the corrected boundary and diagonal curves are to the boundary and diagonal curves given by the user, and the smaller the deviation. Compared with other surface modeling methods, the proposed method considered the constraint condition of the diagonal curve of the surface, which satisfies the requirements of the user on the diagonal curve, and is closer to the user"s design intention. The method proposed in this paper can be widely used in practical engineering. Conclusion The surface constructed in this way not only interpolates the modified diagonal curves and boundary curves, but also has minimal internal energy. This surface construction method is simple and practical, and satisfies the relevant requirements of surface modeling.