6.6 Evaluation and subdivision of B-splines 6.6.1 De Boor algorithm for B-spline curve evaluation P P P P P P Figure 6.5: Diagram of de Boor subdivision algorithm over a cubic spline segment, where Evaluation of B-spline curves can be performed as follows(de boor's algorithm) P(u)=∑PNk(u) to≤u≤tn+k 6.2 Letu∈[t,t+1) be a particular span.N2(u)≠0fort=l-k+1,…,l Let P:=Pi De boor's recursive formula P!=(1-c)P-1+aP1-,i≥l-k+2 126.6 Evaluation and subdivision of B-splines 6.6.1 De Boor algorithm for B-spline curve evaluation P1 2 P2 2 P3 2 t4−t t4−t1 P1 1 P0 P1 P2 P3 P1 3 t−t1 t4−t1 t5−t t5−t3 t−t3 t5−t3 t4−t t4−t3 t−t3 t4−t3 t−t2 t4−t2 t5−t t5−t2 t−t2 t5−t2 t6−t t6−t3 t−t3 t6−t3 t4−t t4−t2 P3 3 Figure 6.5: Diagram of de Boor subdivision algorithm over a cubic spline segment, where t ∈ [t3,t4]. Evaluation of B-spline curves can be performed as follows (de Boor’s algorithm): P(u) = Xn i=0 PiNi,k(u) t0 ≤ u ≤ tn+k (6.27) Let u ∈ [tl ,tl+1) be a particular span. Ni,k(u) 6= 0 for i = l − k + 1, · · · , l Let P 0 i = Pi (6.28) De Boor’s recursive formula: P j i = (1 − α j i )P j−1 i−1 + α j iP j−1 i , i ≥ l − k + 2 (6.29) 12