推导了单位圆域和单位方域内的双变量正交多项式曲面的数学模型，详细分析了将不同正交多项式曲面应用于自由曲面拟合的精度问题。采用均匀随机、阵列分布和环状辐射三种采样方式，并选择具有代表性的普通非球面、自由曲面以及Peaks自由曲面进行了大量的拟合实验。实验结果表明：三种采样方法中，阵列采样的拟合适应度最高；XY多项式和正交XY多项式的拟合适应度最高；方域和圆域内正交的泽尼克多项式在曲面拟合中优势显著；双变量正交切比雪夫多项式在方域内、阵列采样的情况下曲面拟合优势明显。

The orthogonal polynomials of two variables are generated on the unit circle and unit square, and a detailed analysis of the free-form fitting precision is carried out using the orthogonal polynomials with three different sampling grids, which are uniformly pseudo-random grid, array grid and circular grid. To ensure the universality of the fitting analysis, many experiments are conducted on rotationally symmetric aspheric surfaces, free-form surfaces and Peaks free-form surfaces. According to the experiments, among the three sampling grids, the array sampling grid is suitable for most fitting situations. XY-polynomial and orthogonal XY-polynomial give better fitting precision than other surface types in most cases on the wave-front fitting, the orthogonal Zernike polynomial has advantage in circle or square domain and orthogonal Chebyshev is the best polynomial when fitting is required on a square domain using the array sampling grid.