基于几何约束的Doo-Sabin细分曲面修改算法

给出了Doo-Sabin细分曲面在奇异面的极限位置和法矢计算公式，定义了正则网格带的局部等参数线。通过建立局部坐标系对曲面上所有点进行局部参数化，把曲面上点的位置、法向量及局部等参数线等约束转化为所有待调整控制顶点的约束，得到线性系统，从而可以在满足上述多种不同类型的几何约束时修改曲面的形状。从控制网格扰动量最小和能量优化的角度给出两种修改算法，并利用广义逆矩阵求得显式解。约束的线性关系表明，两种方法都存在逆过程，修改的结果与过程无关，便于实际操作与控制。

Shape Modification of Doo-Sabin Subdivision Surfaces with Geometric Constraints

The constraints of points, normal vectors and local isoparametric curves on Doo-Sabin subdivision surfaces, which were converted into those on control vertices to be adjusted, were specified via setting up local coordinate systems. A linear system was obtained and the shape of Doo-Sabin subdivision surfaces can be modified with various geometric constraints. By minimizing the variation of control net and optimizing the membrane energy, two methods were presented, which can be solved explicitly with pseudo-inverse matrices. Both methods are invertible, commutative and associative, which facilitates practical manipulation and control.