ACCEPTED MANUSCRIPT P时P时PP吗 P吗PaPPinppin p以gaga+Paa中 FigControl pots of the bte ≤(a),k≥0 M*≤rm)ir,k≥0 derived by Cheu 3 Regular patches Isreguar CCSS patch.then S()can be expressed asafoACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT P1 2n+13P1 2n+12P1 2n+11P1 2n+10P1 2n+9 P1 2n+5P1 2n+4P1 2n+3P1 2n+2P1 2n+14 P1 7 P1 6 P1 5 P1 2n+6P1 2n+15 P1 8 P1 1 P1 4 P1 2n+7P1 2n+16 P1 2 P1 3 P1 2n+8P1 Π1 2n+17 1 Π1 3 Π1 2 v u Fig. 3. Control points of the subpatches S1 1, S1 2 and S1 3. second order norm of the level-1 control mesh Π1. After k steps of subdivision on Π, one gets 4k control point sets Πk i : i = 0, 1,..., 4k − 1 corresponding to the 4k subpatches Sk i : i = 0, 1,..., 4k − 1 of S, with Sk 0 being the only level-k extraordinary patch (if n = 4). We denote the second order norms of Πk i and Πk as Mk i and Mk, respectively. The second order norms Mk 0 and M0 satisfy the following inequality (Cheng et al., 2006; Chen and Cheng, 2006; Huang and Wang, 2007): Mk 0 ≤ rk(n)M0 , k ≥ 0 , (2) where rk(n) is called the k-step convergence rate of second order norm, which depends on n, the valence of the extraordinary vertex, and r0(n) ≡ 1. Fur￾thermore, it follows that Mk ≤ rk(n)M0 , k ≥ 0 . An expression for the one-step convergence rate r1(n) was derived by Cheng et al. (2006) with a direct decomposition method. The multi-step convergence rate rk(n) was introduced and estimated by Chen and Cheng (2006) with a matrix based technique, then improved by Huang and Wang (2007) with an optimization based approach. 3 Regular patches In this section, we first express a regular CCSS patch S and its corresponding limit face F in bicubic B´ezier form. Then we bound the distance between S and F by bounding the distances between their corresponding B´ezier points. If S is a regular CCSS patch, then S(u, v) can be expressed as a uniform 6