发布时间: 2019-11-16 摘要点击次数: 全文下载次数: DOI: 10.11834/jig.190003 2019 | Volume 24 | Number 11 计算机图形学

 收稿日期: 2019-01-09; 修回日期: 2019-04-17; 预印本日期: 2019-04-22 基金项目: 国家自然科学基金项目（61772163，61761136010，61472111） 第一作者简介: 朱雨凡, 1994年生, 女, 硕士研究生, 主要研究方向为计算机辅助设计。E-mail:zhuyufan_hdqx@163.com;凌成南, 男, 硕士研究生, 主要研究方向为计算机辅助设计。E-mail:522703208@qq.com;李博剑, 男, 硕士研究生, 主要研究方向为计算机辅助设计。E-mail:1062736023@qq.com;许金兰, 女, 博士, 讲师, 主要研究方向为计算机辅助设计、等几何分析。E-mail:jlxu@hdu.edu.cn. 中图法分类号: TP301.6 文献标识码: A 文章编号: 1006-8961(2019)11-1998-11

# 关键词

Construction of energy-minimizing Bézier surfaces interpolating given diagonal curves
Zhu Yufan, Xu Gang, Ling Chengnan, Li Bojian, Xu Jinlan
College of Computer Science and Technology, Hangzhou Dianzi University, Hangzhou Zhejiang 310018, China
Supported by: National Natural Science Foundation of China (61772163, 61761136010, 61472111)

# Abstract

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 distances 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 multiplier method. 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. The linear relationship between the control points of the 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 and boundary curves, we determine the extremum of the internal energy function and 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, they must have an intersection. According to this condition, the correction of the control points of the diagonal curve is added. The sum of the distances of the control points is taken as the objective function to ensure that the deviation between the modified diagonal curve and the user-defined diagonal curve is minimized. 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 design three- and four-order surface modeling examples to satisfy the requirements of different minimal internal energy and verify 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. The examples of surface modeling with the same boundary and different degrees are also designed. These examples show that the higher the order of the surface is, the closer the corrected boundary and diagonal curves are to the boundary and diagonal curves given by the user and the smaller the deviation will be. Compared with other surface modeling methods, the proposed method considers 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 proposed method can be widely used in practical engineering. Conclusion The surface constructed not only interpolates the modified diagonal curves and boundary curves but also has minimal internal energy. The proposed surface construction method is simple and practical and satisfies the relevant requirements of surface modeling.

# Key words

surface modeling; diagonal curve; Lagrangian multiplier method; bending energy; quasi-harmonic energy; Dirichlet energy

# 1 Bézier曲面的对角曲线

$\mathit{\boldsymbol{b}}_{ij}$为控制顶点，则Bézier曲面可定义为控制顶点与Bernstein基函数$B_i^n$($u$)和$B_j^n$($v$)的线性组合，即

 $\mathit{\boldsymbol{x}}\left( {u,v} \right) = \sum\limits_{i = 0}^n {\sum\limits_{j = 0}^n {{\mathit{\boldsymbol{b}}_{ij}}B_i^n\left( u \right)B_j^n\left( v \right)} }$ (1)

 ${\mathit{\boldsymbol{s}}_1}\left( u \right) = \mathit{\boldsymbol{x}}\left( {u,u} \right)$

 ${\mathit{\boldsymbol{s}}_1}\left( u \right) = \sum\limits_{l = 0}^{2n} {B_l^{2n}\left( u \right)\left[ {\frac{1}{{C_{2n}^l}}\sum\limits_{i + j = l} {{\mathit{\boldsymbol{b}}_{ij}}C_n^iC_n^j} } \right]}$ (2)

 ${\mathit{\boldsymbol{s}}_2}\left( u \right) = \mathit{\boldsymbol{x}}\left( {u,1 - u} \right)$

 ${\mathit{\boldsymbol{s}}_2}\left( u \right) = \sum\limits_{l = 0}^{2n} {B_l^{2n}\left( u \right)\left[ {\frac{1}{{C_{2n}^l}}\sum\limits_{i + j = l} {{\mathit{\boldsymbol{b}}_{i\left( {n - j} \right)}}C_n^iC_n^{n - j}} } \right]}$ (3)

 ${\mathit{\boldsymbol{a}}_l} = \frac{1}{{C_{2n}^l}}\sum\limits_{i + j = l} {{\mathit{\boldsymbol{b}}_{ij}}C_n^iC_n^j}$ (4)

 ${\mathit{\boldsymbol{c}}_l} = \frac{1}{{C_{2n}^l}}\sum\limits_{i + j = l} {{\mathit{\boldsymbol{b}}_{i\left( {n - j} \right)}}C_n^iC_n^{n - j}}$ (5)

# 2.2 基于单条对角曲线约束的输入曲线调整

 ${\mathit{\Phi }_5}\left( {x,y,z} \right) = \sum\limits_{l = 0}^{2n} {B_l^{2n}\left( {0.5} \right){\mathit{\boldsymbol{a}}_l}} - \sum\limits_{l = 0}^{2n} {B_l^{2n}\left( {0.5} \right){\mathit{\boldsymbol{c}}_l}}$

 $\begin{array}{*{20}{c}} {{f_5}\left( {x,y,z} \right) = {{\left( {{\mathit{\boldsymbol{a}}_2} - {{\mathit{\boldsymbol{a'}}}_2}} \right)}^2} + {{\left( {{\mathit{\boldsymbol{a}}_3} - {{\mathit{\boldsymbol{a'}}}_3}} \right)}^2} + \cdots + }\\ {{{\left( {{\mathit{\boldsymbol{a}}_{2n - 2}} - {{\mathit{\boldsymbol{a'}}}_{2n - 2}}} \right)}^2} + }\\ {{{\left( {{\mathit{\boldsymbol{c}}_2} - {{\mathit{\boldsymbol{c'}}}_2}} \right)}^2} + {{\left( {{\mathit{\boldsymbol{c}}_3} - {{\mathit{\boldsymbol{c'}}}_3}} \right)}^2} + \cdots + {{\left( {{\mathit{\boldsymbol{c}}_{2n - 2}} - {{\mathit{\boldsymbol{c'}}}_{2n - 2}}} \right)}^2}} \end{array}$

# 4.1.2 在4次情况下的Bézier曲面构造实例

$n$=4时，根据曲面控制顶点和对角曲线控制顶点之间的关系，可以得到5个关于待定内部控制顶点与边界控制顶点和对角曲线控制顶点的表达式。

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