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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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MRBRMl = McBcMTc.
Solving this equation for Bc gives:
Bc = (MclMR)BR{M^MR)T.
6.2 Uniform Bi-Cubic Â-Spline Surfaces
253
We also have:
Therefore:
Bc =
1 2 Ó Ó 1 Ó 0
1 0 6 2 1
Ó Ó
Ur = — 0 2 1 Ó 0
- 1 0 -ó 0 l Ó_
6(0,0) 6(0,1) 6. (0,0) 6w(0,l)“
6(1,0) 6(1Ä) 6w(i,o) 6w(U)
6,(0,0) 6,(0,1) ft» Ä0.0) 6uw(0,l)
6,(1,0) 6»(U) Qu, „(1,0) ftw(U)_
' 1 ó 2 1 Ó Ó 0
0 1 2 l
Ó Ó Ó
1 ~y 1 0 — 2 0
0 - — 0 2 l Ó_
"6i-i,j-: i à-».. ft -1,7+1 Qi-\,j + 2
6,., i-l å,., Qi,j+1 Qi,J + 2
6^1,- 1 6i+l,j 6l+i,J+i 6i + 1,j + 2
_6t + 2,j- 1 Qi + 2,j 6t + 2, j + 1 6[ + 2, j + 2 _
Ó 0 2 0
2 l 0 1 1
Ó 1 Ó î ~Ó
1 Ó Z Ó 1 Ó 0
0 1 Ó 0 1 Ó
This implies:
ltJ-l+^,rl+gft + l,r
(6.44)
254 6. The Â-Spline Approximation
2/1 2 1 \
+I(ga-1,J+Ia.J+ia+1,Jj
1/1 2 1 \
+ 6(6ft-1J+1 + 3ft'J+1 + 6ft + 1J + 1j
1/1 2 1 \
å(0’1)=á1áé--+çå-+áà^)
2/1 2 1 \
+ çËá^ Qi-i,j+i+jQi,j+i+-^Qi+i.j+i J
1/1 2 1 \
+ 6(6a-1'J+2 + 3a'J+2 + 6ft+1-J+2j
e(1,0)= á (ia-J_1 +7a+1-J_1 + á a+2'J_1)
2 Ë ë 2 ë 1 ë A
+ ç1áà'+çà+,-'+áà+2^
1/1 2 1 \
+ 6«,j + l+y 6« + l,j + l+ g-6« + 2,J+l J
å(1,1)=á(á^+çà+1’'+áà+^) (6-44)
2/1 2 1 \
+ Qx,j + \+^Qx + \.j+\+-^Qx+2,j + \ J
+K^a-j+2+ia+i'j+2+ia+2-j+2)
a(0’0)=Kla+1'j-,“ia-1'j-1)
2/1 1 \ 1/1 1 \
+ 3 \2^i + Uj ~2 ^“1’7 + 6 (^y 6, + i.j + i-2-6,-i,; + iJ
a¹‘)=Kyft^-ia-J
2/1 1 \ 1 /1 1 \
+yl óQi+i,j+i~~^Qi-i,j+i Qi+i’j+2~~2 0-1~^ç+2J
1/1 1 \ 2/1 1 \ ^(1»0)=6Uft+2-J"1_2e,*J-V + 3V2ft+2-J_2a-V
^/1 1
^ 1 2 Qi+2,j+i 2 Q,j + i
6.2 Uniform Bi-Cubic 5-Spline Surfaces 255
1/1 1 \ 2/1 1 \
Qu(U i) = Qi+2,j~~^Qi,} j+^y^-Qi+2 j+i-~^-Qi,j+ij
+Kia+2'J+2_2e"J+2)
1/1 2 1 \
Qw(0,0)-~ Ql-i,j+i+—Qi,j+i+—Ql+i,j+1j
1/1 2 1 \
Qw(0, 1) = — Qi-1,j + 2 +y Qi,j + 2+ ~^Qi + 1,j + 2J
1/1 2 1 \
¦2l6ft-1'J+3ftj+6a + ,-JJ
1/1 2 1 \
0w(l,O)-y(^- Ql'j+i+—Qi+i,j+1 +—Qi+2,j+iJ
1/1 2 l \
“21áé^-1 + ç&+1^1 + áé+2^7
1/1 2 ë 1 \
Qwi 1, l)==2\6^,’J+2+y , + l’J + 2+i^, + 2fJ + 2) (6ËÀ)
1/1 2 1 \
~2\~6Q'-j+JQ, + u’ + ~6®‘*2'j)
2 (^2 fii + ij-1 j®'-1-'-1)
e.»(o,D=i({a+i,J+2-ya-i.J+2) a„(i,o)=i(ie(+2,J+1-{e,J+1)
256
6. The Â-Spline Approximation
(6.44)
Next, let us find the relation between the B-spline surface given by formula (6.41) and a Bezier bi-cubic surface patch (5.118). Equating formulas (6.42) and (5.118) gives:
Solving this equation for BB gives:
Bb = (Mb1Mr)Br(Mb'Mr)t.
We also have:
1 2 1
Ó Ó 6
0
0
M-'MR= 0
1 2
Ó Ó 1 2
Î
Î
6 3 6
Therefore:
6.2 Uniform Bi-Cubic Â-Spline Surfaces
257
a-i., Qi-i,j+i
X eM-i a., ft.,+1
O + i,j-i O + l,; a+iIJ+i
_0i + 2,j — 1 Qi + 2,j Qi + 2,j + 1
Qi-1,j + 2
Qi,J + 2 Qi + 1,j + 2
1 6~ 0 0 0
2 2 1 1
Ó Ó Ó J
l i 2 2
Ó Ó Ó Ó
0 0 0 1 ò
This implies:
1/1 2 1
å°0_á(áé-1^1 + çå«-1 + áà+1^
2/1 2 1
+ ç(áå,-1''+çé''+áéí''
1/1 2 1
Qi-i,j+i+jQi.j+i+-jrQi+i.j+i
2 /1
6°i-y^ 6rU + y6,j+gfl + i,j
1/1 2 1 C?02=y (jg É-1,;+3-áó+^É+1,.
2/1 2 ë 1
+ “ I ~tQi-\,j+1 + òáó + 1 +7“ 0 + l,7 + l
3 \6
1/1 2 1 e-=6Ua--+ie-+6a^.
2/1 2 1 + ft-ij + i+ jftj + i + gft + ij + i
1/1 2 1 + á1áà-è+2 + çÉ-+2+áÉ+1’'+2
1 /2
2 /2
å10“á1çà-'-1 + çà+1"-ó+ç1çÉ"+çà+1-
1 /2
+ 6\3^,4l + 3®,lj
(6.45)
6. The Â-Spline Approximation
2 Qn=j (fe-+ia+i'j) 4( 2 1 ^ 6t,7 + l 2 ^ + 1>7 + 1
l &2=3 4( 2 1 ^ ^^t,7 + l + 1,7 + 1
1 å.ç-g (f^+ja+u) *!( 2 1 ^ St^+i-^^ 61+1,j+i
+ K!a-4a + 1,7 + 2 )
a°=6 /1 2 (^y Q^j-i+jQi+i ¦ ^ f'y' ^ ^
+ + 1,7 + 1 )
2 621 — ó '+È 1 2 ó Si,j + 1 +y Qi + l,j + l
1 0.22 = ó (j- 6«,7 + y fi. + l^j '4( 1 2 ó 61,7+ 1 + ó 61 + 1,7 + 1
1 e23-g ( 3" 3" +1 l+f( 1 2 ó 61, J + 1 + ó Ql + 1,7 +1
+ 1 /1 2 6l>7 + 2+y6l + l,7 + 2 )
630 = 6 /1 2 (^r Q1,j-i+jQ1+i,j-i +
2/1 2 1 +iUa-,+Ta+i'j+7
i Ë
+ 6l6ftj*1 + 3e,,lj+l + 6a,2j
2/1 2 1
^31\á" &+i-/+ 6^^t+2’-i /1
+yUft-J+1+3a+1-J+1 + 6eH
1/1 2 1 ^32 = J\6 @l’J + J@l + 1’J + ~6@l+2’-
6.2 Uniform Bi-Cubic Á-Spline Surfaces
åçç"á(áé-'+çà+1''+!à+2''
(6.45)
2 / 1
+ 3^6 Q’J + l + ^ @1 + 1’J + 1 ^ Qi + 2,j + \
1 2 1 + \^)Q,J + 2+^Qi + 1,j + 2+ -^Qi + 2,j + 2
6.2.2 Determination of a Point on a Surface by Finite Difference Operations
When a 5-spline surface patch is given by formula (6.41), then, using the abbreviation NiA(t) = Nt{t), the finite difference matrices À, Â, Ñ and D given by Eqs. (1.113), (1.114), (1.115) and (1.116) are, respectively, as follows.
Finite difference matrix A is:
^(0,0)
r^4(0>0)
1
-ðÄ2Ðü(Ü0)
S3
À3Ðü](0,0)
N0 (0)
N i(0)
N2(0)
N3(0)
]rAN0( 0) ^AN,( 0) ^AN2( 0) \aN3( 0)
0 0 0 0
^2N0(0) — ^H2N2(0) -^-A2N3(0)
^3-zl3No(0) -jy A3N, (0) ^À3Ì2(0) 0)
re,-,.,-, a-i., 6t lj+1 ??i-1,j + 2 "No(0r
a.,-, a., a.7+i Qi,J + 2 N,(0)
6t + i,j-i Qi+i.j+1 + 1,7 + 2 iV2(0)
_6t + 2,j-l 6i + 2,j 6t + 2,j + l 6t+2,j + 2_ _iY3(0)_
(6.46)
Denoting differences in the w-direction by V, finite difference matrix Â is 1
1
-F/y0,0)
^¹(0,0))
-pV(A2PtJ{ 0,0))
<S4
P(zl3P[7(0,0))
^Vo(0)
N i(0)
N2(0)
N3( 0) r AN3 (0)
1
]rAN0( 0) -ANM ]rAN2( 0)
Î Î Î î
^zl2No(0) J2N,(0) j2n2(0) ^2N3(0)
^/l3No(0) Jj-J3N,(0) ^3JV2(0) ^3N3(0)
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