ACCEPTED MANUSCRIPT Finally we conclude the paper with discussion of future work 2 Deflnition and notation ● 2.1 Distances of the natch s (sre Fis. e Is-Fu训 2 Second order norm ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT Finally we conclude the paper with discussion of future work. 2 Definition and notation Without loss of generality, we assume the initial control mesh has been sub￾divided at least twice, isolating the extraordinary vertices so that each face is a quadrilateral and contains at most one extraordinary vertex. 2.1 Distances Given a control mesh and the corresponding limit mesh of a Catmull-Clark subdivision surface S˜, for each interior mesh face F in the control mesh, there is a corresponding limit face F in the limit mesh, and a corresponding surface patch S in the limit surface S˜. The limit face F is a quadrilateral formed by connecting the four corner points of the patch S. 2n+8 control points in the 1-neighborhood of F form S’s control mesh, where n is the valence of F’s only extraordinary vertex (if it has one and n = 4 if not) and called the valence of the patch S (see Fig. 2a). A CCSS patch S can be parameterized over the unit square Ω = [0, 1] × [0, 1] as S(u, v) (Stam, 1998). Let F(u, v) be the bilinear parameterization of the corresponding limit face F over Ω. For (u, v) ∈ Ω, we denote S(u, v) − F(u, v) as the distance between the points S(u, v) and F(u, v). The distance between a CCSS patch S and the corresponding limit face F is defined as the maximum distance between S(u, v) and F(u, v), that is, max (u,v)∈Ω S(u, v) − F(u, v) , which is also called the distance between the patch S and the limit mesh of the surface S˜. 2.2 Second order norms Let Π = {Pi : i = 1, 2,..., 2n + 8} be the control mesh of an extraordinary patch S = S0 0, with P1 being an extraordinary vertex of valence n. The control points are ordered following Stam’s method (Stam, 1998) (Fig. 2a). The second order norm of Π (or S), denoted M = M0 = M0 0 , is defined as the maximum norm of the following 2n + 10 second order differences (SODs) {αi : i = 4