之 k一驶，地+子 专-e(a世引05sA-1A2 6 Comparison en1S-F≤CaM,A21, C(n)n Tsble2.Cumpuriaomn of Caf)mln.3≤a≤l0 10 (u612 Z. Huang and G. Wang then it follows that lj ≥ l log 1 rλ(n)  rj(n) max{β(n), 1 3 }M0 ǫ m. Consequently, we have the following subdivision depth estimation theorem for Loop patches. Theorem 4. Given a Loop patch S of valence n and an error tolerance ǫ > 0, after k = min 0≤j≤λ−1 λlj + j (14) steps of subdivision on the control mesh of S, the distance between S and its level-k limit mesh is smaller than ǫ. Here, lj =  log 1 rλ(n)  rj (n) max{β(n), 1 3 }M ǫ  , 0 ≤ j ≤ λ − 1, λ ≥ 1 . In particular, for regular Loop patches, k =  log4 ￾M 3ǫ
. 6 Comparison Both a control mesh and its corresponding limit mesh can be employed to ap￾proximate a Loop surface in practical applications. This section compares these two approximation representations within the framework of the second order difference techniques. The distance between a Loop patch S of valence n and its control mesh can be bounded in terms of the second order norm M as [6]: max (v,w)∈Ω kS(v, w) − F(v, w)k ≤ Cλ(n)M, λ ≥ 1 , (15) where Cλ(n) = β(n) Pλ−1 i=0 ri(n) 1 − rλ(n) . Table 2 illustrates the comparison results of the constants C3(n) and β(n) for 3 ≤ n ≤ 10. It can be seen that C3(6) = 1 2 is the smallest of C3(n), n ≥ 3. β(n) < C3(6), n ≥ 3 means that the limit mesh approximates a Loop surface better than the corresponding control mesh in general. Table 2. Comparison of C3(n) and β(n), 3 ≤ n ≤ 10 . n 3 4 5 6 7 8 9 10 C3(n) 0.872850 0.780397 0.858623 0.500000 0.875407 0.866176 0.856245 0.847476 β(n) 0.358813 0.350722 0.342499 0.333333 0.329252 0.332001 0.333880 0.335299 Given a Loop patch S of valence n and an error tolerance ǫ > 0, the subdi￾vision depth estimation formula for the control mesh approximation is [6]: k = min 0≤j≤a−1 alj + j , (16)