Abstract: Based on the unorganized sample data points from the module，we can get a complete framework of subdivision surface，which is a nonlinear at least squard problem.Through the three geometrically approximate optimation method of the subdivision surface，point distance minimization，tangent distance minimization，and squared distance minimization，which reveals the intrinsic relation on contrianed nonline optimization，stability and convergence.For the point distance minimization，which is a variant of the grandient decent，and thus has only linear convergence.To the tangent distance minimization method，it’s quadratic convergence for zero residual problems may not convergence at all.Squared distance minimization method is the relative optimization of the three method，and is gotten from the Newton-formula.Through the research of the three methods，makes sure the subdivision surface reach to the convergence stability，and resolve the dispute of the optimization method.

摘要： 基于模型上采集的非组织的样本点数据，给出了一个非线性至少平方的表面细分的完整框架。通过研究和分析三个几何上最优的表面细分的方法：点距离最小、切线距离最小、平方距离最小，来揭示它们的相交性与稳定性，以及非线性约束最优化的内部联系。对于点距离最小方法的分析，它是切线下降的变体，因此，只有线性相交。对于切线距离最小方法，它是接近二次相交零余留的问题，也有可能没有相交。平方距离最小方法，可以通过牛顿公式得到，并且是三种方法中相对最优的。通过对这三种方法的研究，来保证表面细分稳定地相交，且解决了对最优方法的争议。