路的257E子u以ed 。-w2-3-63号 是pd (e.)-F(e. 4 Extraordinary Patch 4.1 Stam'Paramctrization Vig.Pteparameder domin }, Bounding the Distance between a Loop Surface and Its Limit Mesh 7 By symmetry, the maximum of B(u, v),(u, v) ∈ Ω must occur on the diagonal B(t, t) = 2t − 3t 2 , 0 ≤ t ≤ 1 2 . Since max 0≤t≤1/2 2t − 3t 2 = 1 3 = B( 1 3 , 1 3 ) , we have a bound on the maximal distance between S(u, v) and F(u, v) as stated in the following theorem: Theorem 2. The distance between a regular Loop patch S and the corresponding limit triangle F is bounded by max (v,w)∈Ω kS(v, w) − F(v, w)k ≤ 1 3 M . 4 Extraordinary Patch For an extraordinary patch, we first partition it into regular triangular sub￾patches with Stam’s parametrization [9], then derive a distance bound function for each regular subpatch with the technique developed in Section 3. 4.1 Stam’s Parametrization Ω 1 1 Ω 1 2 Ω 1 3 Ω 2 1 Ω 2 2 Ω 2 3 Ω3 1 Ω3 2 Ω3 3 (0,0) (1,0) (0,1) Fig. 4. Partition of the parameter domain Ω. Through subdivision an extraordinary Loop patch S of valence n can be partitioned into an infinite sequence of regular triangular patches {S k m}, k ≥ 1, m = 1, 2, 3. If we partition the unit triangle Ω into an infinite set of tiles {Ωk m}, k ≥ 1, m = 1, 2, 3 (see Figure 4), accordingly, with Ω k 1 = {(v, w)| v ∈ [2−k , 2 −k+1] and w ∈ [0, 2 −k+1 − v]} , Ω k 2 = {(v, w)| v ∈ [0, 2 −k ] and w ∈ [2−k − v, 2 −k ]} , Ω k 3 = {(v, w)| v ∈ [0, 2 −k ] and w ∈ [2−k , 2 −k+1 − v]}