1(=t1 First derivative continuity: 2(C-A (6.12) t+2-t 2(B-C 2B (6.13 t+2-t hence, t2+1-t;t+2-t+1 2-t1+1 (6.14 1 4+2-1+ (6.15) t2+3-t where A, B= functions(C) Need one more condition(normalization) Nik(a)d We obtain A(t1+2-t)+B(t+3-t+1)+C(t+2-t+1)=t+3-t (6.16 From Equation 6.14 to 6.16, we obtain 4B1 t+1-t +3-1+2 t2+3-t1+1 Finall N23(u)=v1(u)N21()+y()M+11(x)+y()N+2(u) t t2+3 (6.18)
at s = 0, s = 1(u = ti+1, u = ti+2). First derivative continuity: 2(C − A) 1 ti+2 − ti+1 = 2A 1 ti+1 − ti (6.12) 2(B − C) 1 ti+2 − ti+1 = −2B 1 ti+3 − ti+2 (6.13) hence, A " 1 ti+1 − ti + 1 ti+2 − ti+1 # = C 1 ti+2 − ti+1 (6.14) B " 1 ti+2 − ti+1 + 1 ti+3 − ti+2 # = C 1 ti+2 − ti+1 (6.15) where A, B = functions(C) Need one more condition (normalization): Z ti+k or +∞ ti or −∞ Ni,k(u)du = 1 k (ti+k − ti) We obtain A(ti+2 − ti) + B(ti+3 − ti+1) + C(ti+2 − ti+1) = ti+3 − ti (6.16) From Equation 6.14 to 6.16, we obtain A = ti+1 − ti ti+2 − ti , B = ti+3 − ti+2 ti+3 − ti+1 , C = 1 (6.17) Finally, Ni,3(u) = y1(u)Ni,1(u) + y2(u)Ni+1,1(u) + y3(u)Ni+2,1(u) = u − ti ti+2 − ti Ni,2(u) + ti+3 − u ti+3 − ti+1 Ni+1,2(u) (6.18) 6
6.2.5 Example: 4 order basis function (Cubic B-spline case-K= 4 see Figure 6.3 n=6→7 control points m+k+l=ll knots 2,4 七 s,4 Figure 6.3: Cubic B-spline functions From property 1 N04(to)+N1,4(to)=0 (6.19) erefore P(to)=(P1-Po)N.(to) Similarly P(t10)=(P6-P5)N64(to) (6.21)
6.2.5 Example: 4 th order basis function (Cubic B-spline case– K = 4 see Figure 6.3) n = 6 → 7 control points n + k + 1 = 11 knots t 1 o t1 t2 t3 t4 t7 t8 t9 t10 t5 t6 N0,4 N1,4 N2,4 N3,4 N4,4 N5,4 N6,4 N0,4 is C -1 N0,4 is C 2 N1,4 is C 2 N5,4 is C 2 N2,4 is C 2 N6,4 is C 0 N1,4 is C 2 N4,4 is C 2 N1,4 is C 0 N2,4 is C 1 N3,4 is C 2 N3,4 is C 2 N4,4 is C 1 N5,4 is C 0 N6,4 is C -1 Figure 6.3: Cubic B-spline functions. From property 1: N˙ 0,4(t0) + N˙ 1,4(t0) = 0 (6.19) Therefore, P˙ (t0) = (P1 − P0)N˙ 1,4(t0) (6.20) Similarly, P˙ (t10) = (P6 − P5)N˙ 6,4(t10) (6.21) 7
P span 2 affected b span 3 af ●--- ar i affected by t span 4 affected by u=t,=t。=t。=t igure 6.4: Example of local convex hull property of B-spline curve
u=t0=t1=t2=t3 t4 t5 t6 P0 P1 P2 P3 P4 P5 u=t7=t8=t9=t10 P6 span 1 affected by P0,P1,P2,P3 span 2 affected by span 3 affected by span 4 affected by P1,P2,P3,P4 P2,P3,P4,P5 P3,P4,P5,P6 Figure 6.4: Example of local convex hull property of B-spline curve 8