B3() 2 0.0 0.0 1.0 Figure 4.8: Plot of the cubic bernstein basis functions 2. Partition of unity Eio Bi n(u)=[1-u+un=l(by the binomial theorem) Bi n(u) (1-u)B1,n-1()+uB2-1,n-1() ith Bi n(u)=0 for i<0. and Boo(u)=1 4. Linear Precision Property ∑=B,n(u) onversion of explicit curves to parametric Bezier curves Parametrization of straight lines y(a) 5. Degree elevation: The basis functions of degree n can be expressed in terms of those degree n+1 as +1 B1+1m+1(x),i=0,1 +1 6. Symmetry Bi, n(u)= Bn-i,n(1-u 7. Derivative B(u) =n[Bi-In-1(u)-Bin-Iu) where B-1n-1(u)=B,, n-1(u)=0 8. Basis conversion (for each of the y, a, coordinates Ferguson or monomial form f 2 u 1- a f1 f0.0 0.0 1.0 1.0 u i=3 i=2 i=1 i=0 Bi,3(u) Figure 4.8: Plot of the cubic Bernstein basis functions. 2. Partition of unity Pn i=0 Bi,n(u) = [1 − u + u] n = 1 (by the binomial theorem) 3. Recursion Bi,n(u) = (1 − u)Bi,n−1(u) + uBi−1,n−1(u) with Bi,n(u) = 0 for i < 0, i > n and B0,0(u) = 1 4. Linear Precision Property u = Xn i=0 i n Bi,n(u) • Conversion of explicit curves to parametric B´ezier curves. • Parametrization of straight lines y = y(x) → x = u; y = y(u) 5. Degree elevation: The basis functions of degree n can be expressed in terms of those of degree n + 1 as: Bi,n(u) =  1 − i n + 1  Bi,n+1(u) + i + 1 n + 1 Bi+1,n+1(u), i = 0, 1, · · · , n 6. Symmetry Bi,n(u) = Bn−i,n(1 − u) 7. Derivative B0 i,n(u) = n[Bi−1,n−1(u) − Bi,n−1(u)] where B−1,n−1(u) = Bn,n−1(u) = 0 8. Basis conversion (for each of the x, y, z, coordinates) • Ferguson or monomial form: f(u) = X 3 i=0 fiu i = h u 3 u 2 u 1 i      f3 f2 f1 f0      = U · FM 9