10 Wang n,the v "-宫城 aa s=血 (13) g=:1st≤+明atiamle be n n创- -P红 rn1(n=c1(n以.n>3 +≤rn.j1 thn th ve nhem 10 Z. Huang and G. Wang where rj (n) is a constant which depends on n, the valence of the extraordinary vertex, and r0(n) ≡ 1. We call rj (n) the j-step convergence rate of second order norm. The convergence rate reflects how fast the control mesh converges to the limit surface. The smaller the convergence rate is, the faster the control mesh converges. It is obvious that rj+k ≤ rj rk. Let α k i , i = 1, 2, . . . , n + 9 be the MSDs of Πk 0 , k ≥ 0 defined as in Equation (1). For each l = 1, 2, . . . , n + 9, we express α k+1 l as a linear combination of α k i : α k+1 l = nX +9 i=1 x l iα k i , where x l i , i = 1, 2, . . . , n + 9 are undetermined real coefficients. Then we can bound kα k+1 l k by cl(n)Mk 0 , where cl(n) is the solution of the following con￾strained minimization problem cl(n) = min nX +9 i=1 |x l i | , s.t. nX +9 i=1 x l iα k i = α k+1 l . (13) Since Mk+1 0 = max{kα k+1 l k : 1 ≤ l ≤ n + 9}, we get an estimate for r1(n) as follows r1(n) = max 1≤l≤n+9 cl(n) . By symmetry, we only need to solve at most four constrained minimization problems corresponding to α k+1 1 = P k+1 1 + P k+1 2 − P k+1 n+1 − P k+1 3 , α k+1 n+1 = P2 + Pn+1 − P1 − Pn+2, α k+1 n+2 = P2 + Pn+2 − Pn+1 − Pn+3, and α k+1 n+3 = P2 + Pn+3 − Pn+2 − Pn+4, respectively. Since P k+1 1 is the extraordinary vertex, it is not surprising to find out that c1(n) is the maximum for n > 3. The special case is r1(3) = c4(3) = 0.4375 > c1(3) = 0.3125. Then it follows that r1(n) = c1(n), n > 3 . Similarly, we can estimate rj (n), n ≥ 3, j > 1 by solving only one constrained minimization problem (13) with α k+1 1 replaced by α k+j 1 . Then we have Lemma 1. If Mk 0 represents the second order norm of the level-k extraordinary subpatch S k 0 , k ≥ 0 of valence n, then it follows that Mk+j 0 ≤ rj (n)Mk 0 , j ≥ 1 . The above lemma works in a more general sense, that is, if Mk 0 is replaced with Mk , the second order norm of the level-k control mesh Πk , the estimates for rj (n) still work. Though r2(n) can be roughly estimated as r1(n) 2 , in practice r2(n) may derive better results than r1(n) 2 as shown in the next subsection. Table 1 shows