ACCEPTED MANUSCRIPT 2+8 2+ 1.+10)of the control points (Cheng006) 2P+P +P 2+7 ( spect y(Fig), ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT 2n + 1 2 3 2n + 8 8 1 4 2n + 7 7 6 5 2n + 6 2n + 5 2n +4 2n +3 2n + 2 9 S = S0 0 (a) 2n+1 2 3 2n+8 2n+17 9 8 1 4 2n+7 2n+16 7 6 5 2n+6 2n+15 2n+5 2n+4 2n+3 2n+2 2n+14 2n+13 2n+12 2n+11 2n+10 2n+9 S1 0 S1 3 S1 1 S1 2 (b) Fig. 2. (a) Ordering of the control points of an extraordinary patch. (b) Ordering of the new control points (solid dots) after a Catmull-Clark subdivision. 1,..., 2n + 10} of the control points (Cheng et al., 2006): M = max{{P2i − 2P1 + P2((i+1)%n+1) : 1 ≤ i ≤ n} ∪{P2i+1 − 2P2(i%n)+2 + P2(i%n)+3 : 1 ≤ i ≤ n} ∪{P2 − 2P3 + P2n+8, P1 − 2P4 + P2n+7, P6 − 2P5 + P2n+6, P4 − 2P5 + P2n+3, P1 − 2P6 + P2n+4, P8 − 2P7 + P2n+5, P2n+6 − 2P2n+7 + P2n+8, P2n+2 − 2P2n+6 + P2n+7, P2n+2 − 2P2n+3 + P2n+4, P2n+3 − 2P2n+4 + P2n+5}} = max{αi : i = 1,..., 2n + 10} . (1) For a regular patch (n = 4), there are only two second order differences with the form P2i − 2P1 + P2((i+1)%n+1). Thus, the second order norm of a regular patch is defined as the maximum norm of 16 second order differences. By performing a Catmull-Clark subdivision on Π, one gets 2n+17 new vertices P1 i , i = 1,..., 2n + 17 (see Fig. 2b), which are called the level-1 control points of S. All these level-1 control points compose the level-1 control mesh of S: Π1 = {P1 i : i = 1, 2,..., 2n + 17}. We use Pk i and Πk to represent the level-k control points and level-k control mesh of S, respectively, after applying k subdivision steps to Π. The level-1 control points form four control point sets Π1 0, Π1 1, Π1 2 and Π1 3, corresponding to the control meshes of the subpatches S1 0, S1 1, S1 2 and S1 3, re￾spectively (see Fig.2b), where Π1 0 = {P1 i : 1 ≤ i ≤ 2n + 8}, and the other three control point sets Π1 1, Π1 2 and Π1 3 are shown in Fig. 3. The subpatch S1 0 is an extraordinary patch, but S1 1, S1 2 and S1 3 are regular patches. Follow￾ing the notation in Eq. (1), one can define the second order norms M1 i for S1 i , i = 0, 1, 2, 3, respectively. M1 = max{M1 i : i = 0, 1, 2, 3} is defined as the 5