8 .Hunng and G.Wang s,=s5=， P-r+s+mr,,ep。 rleFa be ptidtrdefnd F,时l选-t点c,》 oc晋他2T Bi-F(e.)-F(ea,wa).=F(rs.) 4.Douding the Distanc s6国)-F6,=s低，-偏训≤2-1以6，.9 Natir that is not the limit E( a-五引≤4M,1≤f≤15 m Equa (9)tha s,o-t训s5酷M,化，叫en 8 Z. Huang and G. Wang Each tile Ωk m corresponds to a box spline patch S k m. And S k m(v, w) is defined over the unit triangle with the form as Equation (2). Therefore, the parametrization for S(v, w) is constructed as follows [9]: S(v, w) |Ωkm = S k m(˜v, w˜) = S k m(t k m(v, w)) , where the transformation t k m maps the tile Ωk m onto the unit triangle Ω. The center triangle of S’s control mesh is F = {P1, P2, Pn+1} (see Figure 1), and the corresponding limit triangle is F = {P1, P2, Pn+1}, with Pi being the limit point of Pi , i = 1, 2, n + 1. Let F(u, v) be the linear parametrization of F: F(v, w) = uP1 + vP2 + wPn+1, (v, w) ∈ Ω . (7) The limit triangle F can be partitioned into sub-triangles defined over Ωk m as follows: F(v, w) |Ωkm = Fbk m(t k m(v, w))) . Here Fbk m is the linear patch defined as Fbk m(v, w) = ub1 + vb11 + wb15, (v, w) ∈ Ω . (8) If the three corners of Ωk m are (v1, w1),(v2, w2) and (v3, w3), which correspond to (0, 0),(1, 0) and (0, 1) in Ω via the transformation t k m, respectively. Then b1 = F(v1, w1) , b11 = F(v2, w2) , b15 = F(v3, w3) . 4.2 Bounding the Distance Similar to the analysis in Section 3, we can rewrite S k m(v, w) and Fbk m(v, w) into the quartic B´ezier forms as Equations (3) and (4), respectively. Thus for (v, w) ∈ Ωk m, we have kS(v, w) − F(v, w)k = kS k m(˜v, w˜) − Fbk m(˜v, w˜)k ≤ X 15 i=1 kbi − bikBi(˜v, w˜). (9) Notice that Fbk m is not the limit triangle of the triangular patch S k m but one portion of the extraordinary patch S’s limit triangle F. So we can not use the results for kbi − bik derived in Section 3 directly. By solving 15 constrained minimization problems with the form similar to Equation (6), we have kbi − bik ≤ δiM, 1 ≤ i ≤ 15 . Consequently, it follows from Equation (9) that kS k m(v, w) − Fbk m(v, w)k ≤ Bk m(v, w)M, (v, w) ∈ Ω