where the quartic Bezic function 院em-∑iae,e时en (an) 器款告 8el=.≥1.m=123 sEC m点置a而er5对aen Ise,o)-F化，训sM， in the domsin元-Pn,)l+w≥】 m照o4ea制 1 ble 2.F 5 Subdivision Depth Estimation 1+与≤，)A货，≥0 (12Bounding the Distance between a Loop Surface and Its Limit Mesh 9 where the quartic B´ezier function B k m(v, w) = X 15 i=1 δiBi(v, w), (v, w) ∈ Ω (10) is the distance bound function of S k m(v, w) with respect to Fbk m(v, w). Then the distance bound function of S(v, w) with respect to F(v, w), B(v, w),(v, w) ∈ Ω, can be defined as follows: B(v, w) |Ωkm = B k m(t k m(v, w)), k ≥ 1, m = 1, 2, 3 . It is obvious that B(0, 0) = B(1, 0) = B(0, 1) = 0, and B(v, w) is a piece￾wise quartic triangular B´ezier function over Ω away from (0, 0). Let β(n) = max(v,w)∈Ω B(v, w), we have the following theorem on the maximal distance be￾tween S(v, w) and F(v, w): Theorem 3. The distance between an extraordinary Loop patch S of valence n and the corresponding limit face F is bounded by max (v,w)∈Ω kS(v, w) − F(v, w)k ≤ β(n)M , (11) where β(n) is a constant that depends only on n, the valence of S. For 3 ≤ n ≤ 50, we investigate the maximums of the quartic B´ezier functions B k m(v, w), k ≥ 1, m = 1, 2, 3 over Ω assisted by plotting the graph of B(v, w). The following facts are found: 1. B(v, w) attains its maximum in the domain Ω = Ω ∩ {(v, w)| v + w ≥ 1 4 } (shaded region in Figure 4). 2. B(v, w) attains its maximum either on the diagonal B(t, t), t ∈ [0, 1 2 ] or on the borders B(t, 0) and B(0, t), t ∈ [0, 1]. By symmetry, to compute the value of β(n), at most four distance bound func￾tions corresponding to S 1 1 , S 1 2 , S 2 1 , S 2 2 are needed to be analyzed. Numerical results for β(n), 3 ≤ n ≤ 10 are given in Table 2. For regular Loop patches, the constant β(n) = 1 3 is optimum. But for extraordinary Loop patches, the constants can be improved through further subdividing the subpatches S k m. 5 Subdivision Depth Estimation Before estimating subdivision depth, we investigate the recurrence relation be￾tween the second order norms of the control meshes of S at different levels. 5.1 Convergence Rate of Second Order Norm If the second order norms Mk+j 0 and Mj 0 satisfy the following recurrence inequal￾ity Mk+j 0 ≤ rj (n)Mk 0 , j ≥ 0 , (12)