Properties Local support(eg. P6 affects span 4 only), see Figure 6.4 ti, ti+k](k Convex hull (stronger that Bezier) let u E ti, ti+1l, then Nik(u)+0 for j Ei-k+1 ∑Nk(n)=1,NJk(n)≥ Each span is in the convex hull of the k vertices contributing to its definition Consequence: k consecutive vertices are collinear - span is a straight line segment Variation diminishing property as for Bezier curves Exploit knot multiplicity to make complex curves 6.2.6 Special case n=k-1 The B-spline curve is also a Bezier curve in this case tk-1<tk=t k equal knots k equal knots 6.3 Derivatives P(u)=∑PNk(a) P′ (6.23 P-P (6.24) tProperties: • Local support (eg. P6 affects span 4 only), see Figure 6.4 i.e. Pi affects [ti ,ti+k] (k spans) • Convex hull (stronger that B´ezier) let u ∈ [ti ,ti+1], then Nj,k(u) 6= 0 for j ∈ i − k + 1, · · · , i (k values) X i j=i−k+1 Nj,k(u) = 1, Nj,k(u) ≥ 0 • Each span is in the convex hull of the k vertices contributing to its definition • Consequence: k consecutive vertices are collinear → span is a straight line segment • Variation diminishing property as for B´ezier curves • Exploit knot multiplicity to make complex curves 6.2.6 Special case n = k − 1 The B-spline curve is also a B´ezier curve in this case. T = {to = t1 = · · · = tk−1 | {z } k equal knots < tk = tk+1 = · · · = t2k−1 | {z } k equal knots } 6.3 Derivatives P(u) = Xn i=0 PiNi,k(u) (6.22) P 0 (u) = Xn i=1 diNi,k−1(u) (6.23) di = (k − 1) Pi − Pi−1 ti+k−1 − ti (6.24) 9