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 asafo
ACCEPTED 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. Furthermore, 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
ACCEPTED MANUSCRIPT deglrCsprhnihtscoatralpoiatsaddot,theBaicrpo pounts P.follo s.)=广b,o回 (4 Po.Po,1 P0.2 P0. P20P2,P2,2P2 bio bi.:bi.z bi.s Pao Pi Pa2 Pa3 7
ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT p3,0 F _ b3,0 p2,1 b0,0 p2,2 S p b 1,1 0,3 p0,0 F p1,2 p0,3 p0,0 p0,3 p3,0 p3,3 p1,1 p1,2 p2,1 p2,2 b0,0 b3,0 b0,3 b3,3 Fig. 4. A regular CCSS patch with its control points (solid dots), the B´ezier points (hollow dots) and the limit face. bicubic B-spline surface patch defined over the unit square Ω with control points pi,j , 0 ≤ i, j ≤ 3 as follows: S(u, v) = 3 i=0 3 j=0 pi,jN3 i (u)N3 j (v) , (3) where N3 i (u), 0 ≤ i ≤ 3 are the uniform cubic B-spline basis functions. S(u, v) can be converted into the following bicubic B´ezier form (Farin, 2002): S(u, v) = 3 i=0 3 j=0 bi,jB3 i (u)B3 j (v) , (4) where bi,j , 0 ≤ i, j ≤ 3 are the B´ezier points of S (see Fig. 4), and B3 i (u), 0 ≤ i ≤ 3 are the cubic Bernstein polynomials. The relationship between (bi,j ) and (pi,j ) is as follows: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ b0,0 b0,1 b0,2 b0,3 b1,0 b1,1 b1,2 b1,3 b2,0 b2,1 b2,2 b2,3 b3,0 b3,1 b3,2 b3,3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = T ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ p0,0 p0,1 p0,2 p0,3 p1,0 p1,1 p1,2 p1,3 p2,0 p2,1 p2,2 p2,3 p3,0 p3,1 p3,2 p3,3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Tt , (5) 7