摘要：为了增强细分曲面的造型功能，讨论了C-C细分曲面的交互形状修改算法。通过实时建立局部坐标系定义C-C细分曲面上点、法向量和局部等参数线等约束并将其转化为对控制顶点的约束，得到全局线性系统，从而可以在满足不同类型的几何约束时修改曲面的形状。基于最小二乘法和能量优化法给出两种修改算法，前者可以保持控制顶点扰动量的总和最小，运行速度快，适合于局部、精确调整；后者利用罚函数法给出了能量极小意义下的最优解，适合于保持光顺性要求的全局修改。两种方法都可以利用广义逆矩阵求得显式解，具有可逆性、可交换性、结合性等优点，提高了曲面形状修改的效率和可控性。

Abstract：To improve modeling ability of subdivision surfaces, interactive shape modification algorithms for C-C subdivision surfaces are discussed. The constraints of points, normal vectors and local isoparametric curves on C-C surfaces, which can be converted into those on control vertices, are specified via setting up local coordinate systems in real-time operation. A global linear system is obtained and the shape of C-C subdivision surfaces can be modified with the various geometric constraints. Two methods based on least-square and energy optimization are presented. The former minimizes the total variation of control vertices. It is fast and suits for local, precise modification. The latter gives the optimal solution to the energy equation with penalty function method and it is a global modification with good fairness. Both methods can be solved explicitly by a pseudo-inverse matrix and are of invertible, commutative, and associative properties, which enhance the efficiency and controllability of the manipulation.