Abstract Segmented primary mirror is commonly adopted to construct the modern extremely large telescopes. For the sake of facilitating wavefront analysis for this complex aperture, we generalize the theory frame of obtaining orthonormal polynomials corresponding to the segmented pupil based on the non-recursive Gram-Schmidt process. The analytical expressions for Keck and GMT are presented as two typical examples. Moreover, the variances of residual error for first 11 orthogonal modes, with unit amplitude, after sub-aperture rigid correction are computed digitally. This work explores a path to analyzing global discontinuous wavefront.