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# Curves and surfaces in computer aided geometric design - Yamaguchi F.

Yamaguchi F. Curves and surfaces in computer aided geometric design - Tokyo, 1988. - 390 p.
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t4 12
Therefore:
Nii3(0) = (l-0)-l = l
Λγ1;3(0.25) = (1 -0.25)  0.75 = 0.5625 iVx 3(0.5) = (1-0.5) -0.5 = 0.25 ^1,3 (0-75) = (1 0.75)  0.25 = 0.0625
N2.3 (t) = ^7- W2,2 (t) + JV3,2 (t)
Γ4 t2
= tN2<2(t)+j(2-t)N3a(t)
Therefore:
N2>3(0) = 0- 1+0.5-(2-0)-0 = 0
N2^ (0.25) = 0.25  0.75 + 0.5  (2 -0.25)  0.25 = 0.40625
δγ2 η (0.5) = 0.5  0.5 + 0.5  (2- 0.5)  0.5 = 0.625
iV2 η (0.75) = 0.75  0.25 + 0.5  (2 -0.75)  0.75 = 0.65625
N3,3 (0 = N3,2 (t) = 0.5 (JV3,2 (t)
5 f3
Therefore:
iV3 3(0) = 0.5 0-0 = 0 N3 3(0.25) = 0.5 -0.25 -0.25 = 0.03125 δγ3 3 (0.5) = 0.5 -0.5 -0.5 = 0.125 δγ3 3 (0.75) = 0.5  0.75  0.75 = 0.28125
*o,4(0 = *i,3(t) = (1 - t)NU3(t)
4 ri
Therefore:
NOf4(0) = (l-0)-l = l
Noa(0.25) = (1 -0.25) ¦ 0.5625 = 0.422
Noa(0.5) = (1-0.5)  0.25 = 0.125
6.10 Α-Spline Curve Type (3)
307
N0A(0.75) = (1 -0.75)  0.0625 = 0.0156 NlAt) = ^~N1<3(t)+-^-N2>3(t)
Χ Γ1 l5~l2
= tNli3(t)+^(2-t) N2i3(t)
Therefore:
δγ14(0) = 0- 1+0.5 -(2-0)-0 = 0 N, A (0.25) = 0.25  0.5625 + 0.5  (2 - 0.25)  0.40625 = 0.496 N, A(0.5) = 0.5  0.25 + 0.5  (2 - 0.5)  0.625 = 0.594 N1A(0.75) = 0.75  0.0625 + 0.5  (2 - 0.75)  0.65625 = 0.457
N2A(t) = ~^N2Jt) + -^~N(t)
l5~ l2 l6~h
= \tN2t3(t)+j(3-t)N3'3(t)
N2a{0) = 0.5-0-0 + 0.333-(3-0)-0 = 0
N2A(0.25) = 0.5  0.25  0.40625 + 0.333  (3 - 0.25)  0.03125 = 0.0794
N2A(0.5) = 0.5  0.5  0.625 + 0.333  (3 -0.5) - 0.125 = 0.260
N2,4(0.75) = 0.5  0.75  0.65625 + 0.333  (3-0.75)  0.28125 = 0.457
N3At) = ^~N3>3(t)=\tN3>3(t)
l6 γη J
Therefore:
δγ3 4(0) = 0.333-0-0 = 0 N3A(0.25) = 0.333  0.25  0.03125 = 0.0026 δγ3 4(0.5) = 0.333  0.5-0.125 = 0.0208 ΔΓ3 4(0.75) = 0.333 -0.75 -0.28125 = 0.0703
The curve can be generated in the range ?3( = 0)^?<?4( = 1) by substituting the above values in:
Π(*)=1ΧλΉσ
]= ξ
The curve in the range f4( = 1)^?<?5( = 2) is found from the values of N1A(t), N2A(t), N3A(t) and iV44(f). The relations among the lower order B-spline functions that are relevant to these function values are shown in Fig. 6.46(b). By the same method as before we determine:
308
6. The Β-Spline Approximation
N1>4(1), N1i4(1.25), *lf4(1.5), *M(1.75)
*2,4(1), iV2;4(1.25), N2>4(1.5), *2>4(1.75)
*3,4(1), *3.4(1-25), *η,4(1-5), iV3,4(1.75)
*4,4(1), ΛΓ4>4(1.25), ΛΓ4>4(1.5), *4s4(1.75)
and thus generate the curve in f4( = l)^t<t5( = 2).
Repeating a similar procedure, we determine the curve throughout 0 ^ t < 6. Since the Β-spline function values at t = 6 cannot be determined directly by
using the recurrence formula (6.109), it is approximated by the value at a
nearby point such as t = 5.999. Since 5.999<f1 + 1, i = 8, so that it is nec^sary to find Nsa(5.999), N64(5.999), iV7,4(5.999) and NSA{5.999). The relations among the lower order B-spline functions that are relevant to these B-spline function values are shown in Fig. 6.46(c).
N7,2(t) = -~- NsJt)
tg  t8
6-5.999
.-. N1 2 (5.999) =---------------1 = 0.001
6  5
JV8.2(t) = -^LJV8,1(t)
tg t8
5.999-5
... *8 2(5.999) = -------------1 = 0.999
6  5
JV6,3(t)=-^-JV7i2(o
tg tj
6-5.999 N6,3 (5.999)= 6 4  0.001 N0
tg t7 t10 tg
5.999-4 6-5.999
... δγ η(5.999) =--------------- 0.001 + --  0.999 = 0.002
6-4 6-5
*8,3 = *8,2 (0
Γ10 f8
5.999-5
... ΔΓ8 η(5.999) =---------------0.999 = 0.998
6  5
*5,4(0 = ~ 7^ *α,η(0
9 f6
6-5.999
 *5,4(5.999) = - 6_3 ' 0 = 0
6.11 Differentiation of Β-Spline Curves
^6.4 W = JV6,3 (0 + JV7i3 (()
tg  l6 ri0  f7
5 999  η 6_5 9QQ
N6a(5.999)= - - - 0+ - - -  0.002 = 0
ξ  J 0  4
NiA(t)=~r NsMt)
110 r7 tu r8
5.999-4 6-5.999
... jv7>4(5.999) = - 6_- ¦ 0.002 + - - - -  0.998 = 0.003
N8A(t) = -^-N8t3(t) hi h
5.999-5
N8 4(5.999) = --  0.998 = 0.997 6  5
6.11 Differentiation of 5-Spline Curves
Denote a B-spline curve by:
Ο)= t NlM(t)Qj (6.118)
J = 1-M+1
where tt^t<tl+l.
NhM(t) can be expressed as (refer to Eq. (6.77)):
Nj,M(t) = M [t', tj+i> > tj+M]M[t', tj, ..., tJ+M_i]
so that:
^/,m(0 = (d/dt) (M[?; t7 +15 ..., tJ+M]  M [t; t}, ..., ?7 + m-i]}
=  (Ml) [Mj+, M , (0 - mjM _ t (0].
Therefore:
pm{t)={M-1) ? [Mj.M-iW-Mj+i.M-jWjej
j = i-M+ 1
= ("->) Z -^.«-.(06,- -  N,+ ,.M-,(0gjl =
j = i - M + 1 L <7 + M - 1 Lj lj + M lj+1 J
6. The 5-Spline Approximation *)
=(M-d t Γ-1 NjM_i(t)e._1 N)M_i(t)Qj 1
j = i  M + 2 + lj lj + M~ 1 lj J
= (M1) ? NhM^(t)
j = i-M + 2 ^ + M - 1 tj
=(m-l) j Wj.M-.wej11 (t,st<tl+l) (6.Θ9)
j = i-M + 2
where:
gji>= ^ ^ 1 . (6.120)
tj + μ  i tj
More generally, the r-th derivative of P(t) is:
/*> (t) = (M -1)... (M - γ) I Nj'M-r(t) Qf (6.121)
j
where:
Qr = Qj
n^-1)  0(r~v)
(r>0). (6.122)
If the knots are uniformly spaced, that is, if tj = t0+jh for all j, then, with V as the backward finite difference operator:
17Qj = Q?-Qf\
=(μ-1)λε}1)
V2Qj=VQ-VQj^
= (M  1) (M  2)h2Qf]
V3QJ=V2Q-V2Qj-l
= (M1) {M  l)h2(Q(2)  Q(2\)
= (M  1) (M 2) (M  3)h3 Qf\
Repeating a similar procedure, we find that, in general:
VrQj = (Ml) (M 2)... (M γ)ιγε}γ) (6.123)
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