复杂网格曲面上自由形状特征内蕴对称设计方法

Surface-based intrinsic symmetrical design has been significantly applied in a variety of fields including shape defects repairing, rapid modeling and industrial designs. Based on the ideal of design reuse, the explicit symmetry in the Euclidean space will be expanded into the curved surface space, based on which an intrinsic symmetrical design framework on surface space will be established. Through using the coarse positioning, fine correction strategy, the local coordinate framework transfer mechanism on discrete surfaces will be studied and the relationship of position symmetry will be built. Meanwhile, by applying differential geometry information contained in Laplacian coordinates, e.g., curvature, normal vector, etc., a differential coordinate-based parameterized method of the boundary loop of features will be fully studied, and the mapping relationship between an object and its mirror image will also be established during the symmetrical design for boundary of features, which can reduce the distortion of the boundary loop of features when symmetrical mapping. Then, an update strategy combing the vector field with the scalar field will be used to compute the geodesic distance, which will do great help for the subsequent research of the field quantities propagation mechanism on triangular mesh. In addition, a local parameterized method of triangular mesh based on its corresponding dual mesh will also be put forward, which will be more robust and not depend on the assessment method for normal vector of mesh vertex. Finally, after proposing and defining the shape-matching concept on parameter space, a characterization methods and criteria of the intrinsic shape similarity before and after freeform features symmetry will be studied. What’s more, a new reusable shape features retention quantitative evaluation framework will be constructed too. Based on the above research, this project will enrich the research topic of the shape design and reuse on surface space in theory, and provide the basis for intrinsic symmetrical design of engineering applications.

曲面上的内蕴对称设计在形状缺损修复、快速建模和工业图案设计等领域具有重要应用。基于设计重用思想，将欧氏空间中的显式对称映射拓展到曲面空间，建立曲面空间中内蕴对称设计框架。采用粗定位、精修正策略，研究局部坐标框架在离散曲面上的迁移机制，构建位置对称映射关系；借助拉普拉斯坐标蕴含的曲率、法矢等微分几何信息，研究基于微分坐标的重用特征边界环参数化方法，建立特征边界对称设计时的物-像映射关系，减小特征边界环对称映射时的形变；采用矢量和标量相结合的测地距离更新策略，研究三角网格曲面上的场量传播机制，提出基于三角网格对偶图的曲面局部参数化方法，使其不依赖于网格顶点的法矢评估，更加鲁棒；提出参数空间内的形状匹配概念，研究自由形状特征对称前后内蕴形状保持性的表征方法与评判准则，形成一种新的重用特征形状保持性定量评价框架。项目将在理论上丰富曲面空间形状设计与重用研究的内容，为内蕴对称设计的工程应用奠定基础。

曲面上的内蕴对称设计在形状缺损修复、快速建模和工业图案设计等领域具有重要应用。基于设计重用思想，将欧氏空间中的外蕴对称映射拓展到曲面空间，建立曲面空间中内蕴对称设计框架。采用粗定位、精修正策略，研究局部坐标框架在离散曲面上的迁移机制，构建位置对称映射关系。借助拉普拉斯坐标蕴含的曲率、法矢等微分几何信息，提出一种基于微分坐标的重用特征边界环参数化方法，该方法保特征、形变小，将其用于建立特征边界对称设计时的物-像映射关系，大大减小特征边界环对称映射时的形变。提出一种新的曲面上的场量传播机制，以便协调解决计算精度与效率之间的矛盾问题。以三角面重心点代替网格顶点作为传播“介质”，使场量的传播不依赖于顶点的法矢；采用标量与矢量相融合的传播更新机制，综合利用二者的优势，最佳化计算精度与效率的平衡。构建一种非完备模型对称面提取算法，并将其应用于模型修复、个性化眼镜定制之中，其中个性化眼镜定制部分，已申请软件著作权，所设计的眼镜正在中试，将于年内实现市场化。提出一种基于样本块的带纹理特征匹配与迁移方法，并将其应用于含纹理特征模型的修复。提出参数空间内的形状匹配概念，研究自由形状特征对称前后内蕴形状保持性的表征方法与评判准则，形成一种新的重用特征形状保持性定量评价框架。项目成果在理论上丰富了曲面空间形状设计与重用研究的内容，；在实践上，开拓了内蕴对称设计的工程化应用范畴，将其应用于形状修复和个性化眼镜设计中，具有较好的市场前景。