5. Recursive definition(Cox-de Boor algorithm u∈[t2,t+1) 0ugt1,t+1) a-t (6.6) 0 above when it occur Properties 1-4 or property 5 by itself(given a known knot vector T) define the B-spline 6.2.3 Eacample: 2 order B-spline basis function Piecewise linear case- see Figure 6.1) k=2,C t-1t;t+1t2+2t2+3 Figure 6. 1: Plot of 2nd order B-spline basis functions Ni,2() consists of two piecewise linear polynomials t;≤u≤t N2(u) t+1≤u≤t5. Recursive definition (Cox-de Boor algorithm) Ni,1(u) = ( 1 u ∈ [ti ,ti+1) 0 u 6∈ [ti ,ti+1) (6.5) Ni,k(u) = u − ti ti+k−1 − ti Ni,k−1(u) + ti+k − u ti+k − ti+1 Ni+1,k−1(u) (6.6) (set 0 0 = 0 above when it occurs) Properties 1-4 or property 5 by itself (given a known knot vector T) define the B-spline basis. 6.2.3 Example: 2 nd order B-spline basis function (Piecewise linear case– see Figure 6.1) k = 2, C k−2 = C 2−2 = C 0 ti−1 ti ti+1 ti+2 ti+3 Ni,2(u) Figure 6.1: Plot of 2 nd order B-spline basis functions. Ni,2(u) consists of two piecewise linear polynomials: Ni,2(u) = ( u−ti ti+1−ti ti ≤ u ≤ ti+1 ti+2−u ti+2−ti+1 ti+1 ≤ u ≤ ti+2 4