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收稿：2019-01-09，
修回：2019-4-17，
录用：2019-4-22，
纸质出版：2019-11-16
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目的
2
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
2
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
2
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
2
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
2
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
2
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
2
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
2
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.
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