g =(w)Ic0 and wc0,2 he distan Is,四)-re,lss(c,j-上5（，o +E-时e1-时-四侧 Lemma3.2f(e,）∈疏，m=l,23,hea 时s{m 【nmmn.3f(,)e哈：then for any0≤i≤k-2 ur haxe g卡2学end时 8Ω 1 1 Ω 1 2 Ω 1 3 Ω 2 1 Ω 2 2 Ω 2 3 Figure 4: Partition of the parameter domain Ω. Each tile Ωk m corresponds to a box spline patch S k m. And we denote the parameter space corresponding to the extraordinary subpatch S k 0 by Ω k 0 = {(v, w)| v ∈ [0, 2 −k ] and w ∈ [0, 2 −k − v]} . Let F(v, w), F k m(v, w) and F k 0 (v, w) be the linear parametrization of the center faces of the control meshes of S, S k m and S k 0 , respectively. Using the triangle inequality, for (v, w) ∈ Ω k m, m = 1, 2, 3, the distance between an extraordinary Loop patch S(v, w) and the corresponding triangle F(v, w) can be bounded as kS(v, w) − F(v, w)k ≤ kS k m(v, w) − F k m(v, w)k + kF k m(v, w) − F k−1 0 (v, w)k + X k−2 i=0 kF i+1 0 (v, w) − F i 0 (v, w)k . (8) With a proof analogous to the one of Lemma 3 in [9], we have the following two lemmas. Lemma 3.2 If (v, w) ∈ Ω k m, m = 1, 2, 3, then F k m(v, w) − F k−1 0 (v, w) ≤  3 8Mk−1 0 , m=1,3; 1 8Mk−1 0 , m=2. , where Mk−1 0 is the second order norm of S k−1 0 . Lemma 3.3 If (v, w) ∈ Ω k m, then for any 0 ≤ i ≤ k − 2 we have F i+1 0 (v, w) − F i 0 (v, w) ≤ ω(n)Mi 0 , where ω(n) = 5 8 − (3+2 cos(2π/n))2 64 , and Mi 0 is the second order norm of S i 0 and F 0 0 = F. 8