6 h:他置o llet poiblec1..5 Ib,-5≤6M hdon where-1,2.....15 are undetermined real coefficients.It follows that Ib-e2Ials24anls上hiw anonniwrra 香-mm工时 6) 4∑-h- -6--3 6-6--是 It follows that 5四-4De,四-+w-2-m- Thcorem e ance bet)and w 1S,四y-r,sB(M6 Z. Huang and G. Wang In the following, we compute the smallest possible constants δi , i = 1, 2, . . . , 15 such that kbi − bik is bounded by kbi − bik ≤ δiM , where M is the second order norm of S. It is obvious that δ1 = δ11 = δ15 = 0. bi and bi , 1 ≤ i ≤ 15 are the convex combinations of the control vertices pi , i = 1, 2, . . . , 12, and bi−bi can be expressed as a linear combination of the 15 MSDs αl , l = 1, 2, . . . 15 defined in Equation (1) as follows: bi − bi = X 15 l=1 x i lαl , where x i l , l = 1, 2, . . . , 15 are undetermined real coefficients. It follows that kbi − bik ≤ X 15 l=1 kx i lαlk ≤ X 15 l=1 |x i l |kαlk ≤ X 15 l=1 |x i l |M . Therefore, to get a tight upper bound for kbi − bik, we solve the following constrained minimization problem: δi = minX 15 l=1 |x i l | s.t. X 15 l=1 x i lαl = bi − bi . (6) By symmetry, we only need to solve three constrained minimization problems. With the help of the symbolic computation of Mathematica, we have δ1 = δ11 = δ15 = 0 , δ2 = δ3 = δ7 = δ10 = δ12 = δ14 = 1 4 , δ4 = δ6 = δ13 = 1 3 , δ5 = δ8 = δ9 = 5 12 . It follows that B(v, w) = X 15 i=1 δiBi(v, w) = v + w − v 2 − vw − w 2 . We obtain a bound on the pointwise distance between S(v, w) and F(v, w): Theorem 1. For (v, w) ∈ Ω, we have kS(v, w) − F(v, w)k ≤ B(v, w)M , where B(v, w) = v + w − v 2 − vw − w 2 is called the distance bound function of S(v, w) with respect to F(v, w)