subdivision.Then M=S M,where the subdivisionrix s amatrix [+学-导号+学 警十好3+婴一4 +j++专+u+ + =2+nX+-+-2+ -3+2a+(+s,(m+s1-2+) 与-3-4+m0o+,-1,1- m2ae0R装hMoa 3.2 Irregular faces 瓦-r Z. Huang et al. /Journal of Information & Computational Science 7: 1 (2010) 1–6 5 subdivision. Then M = S4M, where the subdivision matrix S4 is a 4 × 4 matrix: S4 =         1 2 + w0 2 − w2 4 w1 2 + w2 4 w2 4 w3 2 + w2 4 w0 2 + w2 4 1 2 + w1 2 − w3 4 w2 2 + w3 4 w3 4 w0 4 w1 2 + w0 4 1 2 + w2 2 − w0 4 w3 2 + w0 4 w0 2 + w1 4 w1 4 w2 2 + w1 4 1 2 + w3 2 − w1 4         =       l0 l1 l2 l3       −1       1 0 0 0 0 1 2 0 0 0 0 1 2 0 0 0 0 1 4             l0 l1 l2 l3       Here l0, l1, l2 and l3 are the left eigenvectors of S4 corresponding to eigenvalues λ0 = 1, λ1 = λ2 = 1 2 , and λ3 = 1 4 respectively: l0 =[w0 3 + 2 3 (w0 + w3)(w0 + w1), w1 3 + 2 3 (w0 + w1)(w1 + w2), w2 3 + 2 3 (w1 + w2)(w2 + w3), w3 3 + 2 3 (w2 + w3)(w0 + w3)] , l1 =[(w0 + w1)(1 − 2(w0 + w3)), −2(w0 + w1)(w1 + w2), (w1 + w2)(1 − 2(w2 + w3)), w0 − w2 + 2(w1 + w2)(w2 + w3)] , l2 =[−2(w0 + w1)(w0 + w3), −(w0 + w1)(1 − 2(w0 + w3)), w1 − w3 + 2(w0 + w3)(w2 + w3),(w0 + w3)(1 − 2(w2 + w3))] , l3 = 1 3 (w1 − 4(w0 + w1)(w1 + w2))[1, −1, 1, −1] . (3) Using the approach described in [7], one can get: Proposition 1 For a regular face with vertices M = [P0, P1, P2, P3] T , its center will converge to the point l0 · M on the limit surface. Proposition 2 The normal vector to an EDS at the limit point l0 ·M corresponding to a regular face with vertices M is the vector (l1 · M) × (l2 · M), here ”×” denotes vector cross product. 3.2 Irregular faces To solve the divergence problem of quadratic NURSSes, EDSes employ the same refinement rules for irregular faces as Doo-Sabin surfaces [1]. For an irregular face with n vertices P0, . . . , Pn−1 (see Fig. 4), the new vertex Pi is computed as Pi = Xn−1 j=0 ai,jPj , (4)