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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. Previous << 1 .. 70 71 72 73 74 75 < 76 > 77 78 79 80 81 82 .. 90 >> Next the curve becomes:
p(t)=m-t)2 2t(\—t) t2]
= [(l-t)2 2t(l — f) t2]
-*(-ë
-Ri + 1 _
1 1
î
2 2 0 1 0
1 1
0 Ó Ó,
ĂÉ;-2Č
Ri
Ri + 2
J-(l — 02 -t2 + t+\- J-12
2 2 2
R<-2H R<
Ri+7
= LN0t3(t) NiAt) N2,3ą
rRi-2n Ri
\—Ri + 2_1
(6.147)
(refer to Eqs. (6.17) and (6.18)). This means that the function expresses a uniform Á-spline curve of type (1) and order 3 with respect to 0-u Îë, 02,
Qn-č Qn+i(Q-i=2Qo-Qu Qn+i=2Qn-Qn-1).
6.16 Interpolation of a Sequence of Points with a 5-Spline Curve
Consider the generation of ŕ Á-spline curve (curve type (3)) by interpolation of a sequence of points P0, Pl9 ..., Pn. The following discussion is for the case of M = 4 (degree 3).
The relation among the points to be interpolated, interior knots and extended knots (with * symbol) is shown in Fig. 6.56. From property © of ŕ Á-spline function, there are 3 nonzero Á-spline functions at each knot. We call the
326
6. The Á-Spline Approximation
tn% = = ln% - an
pn\tn( = tn%= an)
P° t0( = tr=a0)
(*=«*=(*= a0
t0, tu......, tn : Interior knots
i f, t * ..., t*+ 6 : Extended knots
Fig. 6.56. Interpolation of a sequence of points with a Á-spline curve (curve type (3)) (case of M = 4)
Fig. 6.57. Nonzero Á-spline functions at ? = ?*
3 nonzero functions at knot ?*, Nt^3A(t), ^_2>4(?) and iV-_14(?) (refer to Fig. 6.57). Therefore, in order for the curve to interpolate point Pt_3 at knot tf, the following equation must hold:
This equation represents n +1 conditions, but there are n + 3 unknown vectors
• • •, 6n + 2> so we are 2 conditions short. So it is necessary to add 2 conditions. We will set the 2-nd derivative vectors of the generated curve P(t) equal to Î at both ends: at knot :
(6.148)
NoA(ao) Qo + N1A(a0) Qx + N2A(ao) Qi — O
(6.149)
at knot t *+ ç:
Nn,Aai) Qn + Nn+1A(an) Qn+l + Nn + 2A(an) Qn + 1 = O.
(6.150)
6.17 Matrix Expression of Â-Spline Curves
327
For values of extended knots, it is sufficient to take the sum of distances between points (Fig. 6.56):
Qo> Qu Qn + 2 can be found by solving Eqs. (6.148), (6.149) and (6.150), thus determining the Â-spline curve that passes through the specified points P0, P1}
,P„
6.17 Matrix Expression of Ë-Spline Curves
Let us espress the B-spline curve segment given by function (6.102) by a matrix, as we expressed a Bezier curve segment in Sect. 5.1.5 ®23). If we set M = n+ 1 in order to establish a correspondence with a Bezier curve, the curve segment (6.102) becomes:
i + 1
f*+4 = aI + 1 = Z ci (i = 0, 1, ...,n-2)
(6.151)
f*+* + 3 = e«= Z CJ O'= 0, 1,2,3)
m = lN0,n + i(t) Nl>n+1(t) ... iVn,n + 10)]
Qi-1 " = [<" t-' ... t lb a
Qi+n-i
(6.152)
where ó is an (n+ 1) x (n+ 1) matrix.
Setting M = n +1 in formula (6.101) gives:
Therefore, we have:
6. The Â-Spline Approximation
6.18 Expression of the Functions C0 0(t), C0>1(f), C10(t) and Ń1Ä({) (n+1^
(-1)C
0
0
(-dm";1) (-C1
n +1 0
n+1
; : - (-i)° —Ę':1)
= [tn t" 1 ... t 1] ó
y = bij} ib j = 0, 1, 2, ..., n)
n +1 0
(6.154)
6.18 Expression of the Functions Q0(^), C0i(f), Cuo(t) and Cu(r) by 5-Spline Functions28)
The blending functions that we used to make the higher-order derivative vectors of a Coons surface continuous (refer to Sects. 3.3.2 and 3.3.3) can be expressed using B-spline curves.
In the following discussion we consider functions which satisfy the blending function conditions that will make the 2nd derivative vectors continuous:
[Co>o(0) C0t0(l) Co>o(0) C0.0(1) Co.o(0) C0>0(1)] = [1 0 0 0 0 0]
[Ńîä(0) Ńîä(1) Ńîä(0) Ńîä(1) Co.J0) Ń0ä(1)] = [0 1 0 0 0 0]
[C1;o(0) Clf0(l) Clfo(0) Clf0(l) CliO(0) c1>0(l)] = [0 0 1 0 0 0]
[Ń1Ä(0) Ń1Ä(1) Ń1Ä(0) Ń1Ä(1) Ń1Ä(0) Ń1Ä(1)] = [0 0 0 1 0 0]
(6.155)
If we wish to derive ordinary polynomials which will satisfy these conditions, they will be 5-th degree polynomials such as those derived in Sect. 3.2.3.
Let us express C0 0(?), Ńîä(?), C1>0(t) and ŃlA(t) as uniform cubic B-spline curves of type (1) (refer to Sect. 6.1.1). Take the coordinates of Q0, Qu ..., Q5 in Fig. 6.58 to be:
330
6. The Á-Spline Approximation
Fig. 6.58. Expression of the function COtO(0 in terms of Á-spline functions
fio
Qi
Qi
Q3
Q*
Qs
1
1
T
0 1
1
1
Ó
2
0
3
1 0
4
0
3
In this case, 3 curve segments are produced by the sequence of vertices Q0, Qx, Qi* 6ç> 6s* From the properties of a Á-spline curve, it is clear that the
curve which is made up of these segments satisfies the condition on function C00(t) in Eq. (6.155). Let us write these 3 functions as XC0 0(t), 2Q),o(0 and 3^0,0 (0*
Now we introduce the local parameter w(O^w^l) shown in Fig. 6.58. The curve segment P^u) (i= 1, 2, 3) formed by the vertices Qt-U Qb Qi+1 and Qi + 2 is:
Pi(u) — [t iCo,o(0]
'-1 3 -3 ă
1 ˙ , 3 -6 3 0
— [č č č 1] 6 -3 0 3 0
1 4 1 0_
--(i-2) Qt_Uy
1
Qi+ l,y
1
— i 3
— (i + 1) Qi+2,ó
(6.156)
t — — (č — 1-J- i).
3
(6.157)
6.18 Expression of the Functions C0t0(t), C01(t), C10(t) and Cj j(t) 331
From Eq. (6.156) we have:
[lC0.o(f) 2^0,o(0 3^0,o(0] Previous << 1 .. 70 71 72 73 74 75 < 76 > 77 78 79 80 81 82 .. 90 >> Next 