in 12uw Caaa创 su=∑bae.een 3) 是 o程 F(c.w)=uba hearpa F,=∑5成c回 0 ,tL-F子，》，，k之0,0≤j+≤4 are the Rzir points. sic.)-Fc.)1-)s-1(c.).8 Bounding the Distance between a Loop Surface and Its Limit Mesh 5 b b b b b b b b b b b b b b b 1 3 6 10 15 5 9 14 8 13 12 11 7 4 2 (a) u 4 4u 3 v 4u 3w 6u 2 v 2 12u 2 vw 6u 2w 2 4uv3 12uv2w 12uvw2 4uw3 v 4 4v 3w 6v 2w 2 4vw3 w 4 (b) Fig. 3. (a) Ordering of the B´ezier points of a quartic triangular B´ezier patch. (b) Quartic Bernstein polynomials corresponding to the B´ezier points. We introduce the third parameter u = 1 − v − w such that (u, v, w) forms a barycentric system of coordinates for the unit triangle. S is a quartic triangular B´ezier patch, thus S(u, v) can be written in terms of Bernstein polynomials [11]: S(u, v) = X 15 i=1 biBi(v, w), (v, w) ∈ Ω , (3) where bi , 1 ≤ i ≤ 15 are the B´ezier points of S, and Bi(v, w), 1 ≤ i ≤ 15 are the quartic Bernstein polynomials (see Figure 3). The correspondence between the standard representation B4 ijk(u, v, w) = 4! i!j!k! u iv jw k , i, j, k ≥ 0, i+j +k = 4 and Bi(v, w), 1 ≤ i ≤ 15 is (i, j, k) 7→ (j+k)(j+k+1) 2 + k + 1. With the algorithm developed in [10], we can get the 15 × 12 matrix T which converts from the 12 control vertices (pi) to the 15 B´ezier points (bi) (see Appendix 1). Note that F = {b1, b11, b15} is the limit triangle corresponding to the center triangle F = {p4, p7, p8} (see Figures 2 and 3). The linear parametrization of F is F(v, w) = ub1 + vb11 + wb15 , where u = 1−v−w. By the linear precision property of the Bernstein polynomials [11], we can express F(v, w) as the following quartic B´ezier form: F(v, w) = X 15 i=1 biBi(v, w) , (4) where b(j+k)(j+k+1) 2 +k+1 = F( j 4 , k 4 ), j, k ≥ 0, 0 ≤ j + k ≤ 4 are the B´ezier points. 3.2 Bounding the Distance According to the previous analysis, for (v, w) ∈ Ω, it follows that kS(v, w) − F(v, w)k = X 15 i=1 (bi − bi)Bi(v, w) ≤ X 15 i=1 kbi − bikBi(v, w) . (5)