Computer Aided Ship design -Part II. Hull Form Modeling- · 2018. 1. 30. · 2009 Fall, Computer...
Transcript of Computer Aided Ship design -Part II. Hull Form Modeling- · 2018. 1. 30. · 2009 Fall, Computer...
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
Naval A
rch
itectu
re &
Ocean
En
gin
eerin
g
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
Computer Aided Ship design
-Part II. Hull Form Modeling-
September, 2009
Prof. Kyu-Yeul Lee
Department of Naval Architecture and Ocean Engineering,Seoul National University of College of Engineering
학부3학년 교과목“전산선박설계(Computer Aided ship design)”강의 교재
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
Naval A
rch
itectu
re &
Ocean
En
gin
eerin
g
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
Ch 3. 곡면(Surfaces)
3.1 Parametric Surfaces
3.2 Bezier Surfaces
3.3 B-spline surfaces
- Chap3. Surface
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
Naval A
rch
itectu
re &
Ocean
En
gin
eerin
g
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.1 Parametric Surfaces
- Chap3. Surface
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.1 Parametric Surfaces
곡면
양함수
음함수
매개
변수
222 yxdz
2222 dzyx
sin
cossin
coscos
dz
dy
dx
)),(),,(),,(( zyxrr
구를 2개의 매개변수 로 표현하면 간단히 수식화 할 수 있다.
),(
- Chap3. Surface
4/52
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
Naval A
rch
itectu
re &
Ocean
En
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eerin
g
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.2 Bezier surfaces
3.2.1 Generation of Bezier surfaces by de Casteljau algorithm
- Chap3. Surface
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
Naval A
rch
itectu
re &
Ocean
En
gin
eerin
g
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.2.1 Generation of Bezier surfaces
by de Casteljau algorithm
3.2.1.1 Bi-linear Bezier Surface Patch
3.2.1.2 Bi-quadratic Bezier Surface Patch
3.2.1.3 Bi-Cubic Bezier Surface Patch
- Chap3. Surface
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.2.1.1 Bi-linear Bezier Surface Patch- Given: 2x2 Bezier control point- Find: Points on bi-linear Bezier Surface Patch
00b
10b
11b
10b
u
v
uP
uq
u
u1
v
v1
u
v
0 1
1
u
v
u)1( u 00b 01bu
P
u)1( u 10b 11bu
q
v)1( v ),( vur u
P uq
.
)(
1 0 ,
1100
있다수얻을곡면을하는꼭지점으로
을점계산하면를
증가하며까지에서를
1010 b,b,b,br u,v
vu
2x2개의 Bezier 조정점을 이용하여 Bi-linear Interpolation 으로 곡면식을 구할 수 있다.
v
)1( v),( vur u)1( u 10b
11b
)1( u u00b 01b)1( v
v
11
00
10
10
bb
bb
u
u)1( vv)1( ),( vur
u
u1
방법: u, v 방향으로 ‘de Casteljau algorithm’ 사용
- Chap3. Surface7/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.2.1.2 Bi-quadratic Bezier Surface Patch- Given: 3x3 Bezier control point- Find: Points on bi-quadratic Bezier Surface Patch
u
v
방법: u, v 방향으로 ‘de Casteljau algorithm’ 사용
02b
12b
22bu
1 u u
1 uu 1 u
2
up 2
uq
2
ur
01b
11b
21bu
1 u u
1 uu 1 u
1
up 1
uq
1
ur
00b
10b
20bu
1 u u
1 uu 1 u
0
up 0
uq
0
ur
00 10
10 2
0
0 0
1
1
u
u
u u
u u
p
q
b b
b b
01 11
11 2
1
1 1
1
1
u
u
u u
u u
p
q
b b
b b
02 12
12 2
2
2 2
1
1
u
u
u u
u u
p
q
b b
b b
00 01 uu uu u p qr
2 2
00 10 20
2 2
01 11 21
2 2
02 12 22
0
1
2
1 2 1
1 2 1
1 2 1
u
u
u
u u u u
u u u u
u u u u
r
r
b b b
b b
b b br
b
2
00 10 20
01 11 21
2
02 12 22
0
1
2
1
2 1
u
u
u
u
u u
u
r
r
b b
b
r
b
b b
b b b
11 11 uu uu u p qr
22 21 uu uu u p qr
- Chap3. Surface
8/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.2.1.2 Bi-quadratic Bezier Surface Patch- Given: 3x3 Bezier control point- Find: Points on bi-quadratic Bezier Surface Patch
u
v
방법: u, v 방향으로 ‘de Casteljau algorithm’ 사용
02b
12b
22b2
ur
01b
11b
21b1
ur
00b
10b
20b0
ur
2
00 10 20
01 11 21
2
02 12 22
0
1
2
1
2 1
u
u
u
u
u u
u
r
r
b b
b
r
b
b b
b b b
1
vs
0
vs
v
1 v
v
1 v
,u vrv
1 v
0 1
1 2
0
1
1
1
u uv
v u u
v v
v v
s
s
r r
r r
0 1, 1 v vu v v v s sr
2
2
2
0 11 2 1, u u uv v v vu v r rr r
2 2
0
1
2
1 2, 1
u
u
u
v v vv vu
r
r
r
r
2
00 10 202 2
01 11 21
2
02 12 22
1
1 2 1 2 1,
u
v v v v u uu v
u
b b b
b b b
b b
r
b
3x3개의 조정점을 이용하여Bi-Quadratic Bezier Patch를 구할 수 있다.
- Chap3. Surface
9/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.2.1.3 Bi-cubic Bezier Surface Patch- Given: 4x4 Bezier control point- Find: Points on bi-cubic Bezier Surface Patch
u
u1
u u1
u
u1
uu1 u
u1
u
v
4x4개의 조정점을 이용하여 Bi-Cubic Bezier Patch를 구할 수있다.
31 0 2 3 0 0
31 1 2 1 3 13 3 3 3 1
0 1 2 3 30 2 1 2 2 3 2 2
30 3 1 3 2 3 3 3 3
( )
( )( , ) ( ) ( ) ( ) ( )
( )
( )
B v
B vu v B u B u B u B u
B v
B v
0,0 , ,0 ,
0, ,1 , ,
, , ,2 ,
, , , ,
b b b b
b b b bb
b b b b
b b b b
방법: u, v 방향으로 ‘de Casteljau algorithm’ 사용
10/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.2.1.3 Bi-cubic Bezier Surface Patch- Given: 4x4 Bezier control point- Find: Points on bi-cubic Bezier Surface Patch
u
v
4x4개의 조정점을 이용하여 Bi-Cubic Bezier Patch를 구할 수있다.
31 0 2 3 0 0
31 1 2 1 3 13 3 3 3 1
0 1 2 3 30 2 1 2 2 3 2 2
30 3 1 3 2 3 3 3 3
( )
( )( , ) ( ) ( ) ( ) ( )
( )
( )
B v
B vu v B u B u B u B u
B v
B v
0,0 , ,0 ,
0, ,1 , ,
, , ,2 ,
, , , ,
b b b b
b b b bb
b b b b
b b b b
방법: u, v 방향으로 ‘de Casteljau algorithm’ 사용
11/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
Naval A
rch
itectu
re &
Ocean
En
gin
eerin
g
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.2.2 Generation of Bezier surfaces
by tensor-product approach
3.2.2.1 Tensor-product approach
3.2.2.2 Tensor-product biquadratic Bezier surface
3.2.2.3 Tensor-product bicubic Bezier surface
- Chap3. Surface
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.4.1 Tensor product approach (1)
매우 탄력적인 재질의고무 밀대라고 가정
나무판
End moving curve
Directional curve
Moving curve
makes a surface
Start moving curve
13/52
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.4.1 Tensor product approach (2)
u
v
r(u) 곡선을 v 방향으로 sweeping
u
v
r(v) 곡선을 u 방향으로 sweeping
moving curve가 일정한 차수의 Bezier curve
이고, moving curve의Bezier control points의
궤적을 나타내는 directional curve도 Bezier
curve일 때, 이러한 방법으로 생성되는 곡면을
“Tensor product Bezier surface” 라고 한다.
* CAGD, Ch14.3, Tensor Product Approach
moving Curve
directional
Curve
directional
Curve
moving Curve
- Chap3. Surface
14/52
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
)(1 vb
3.4.2 Tensor-product bi-quadratic Bezier surface (1)- Given: Control Points of bi-quadratic Bezier Surface- Find: Points on bi-quadratic Bezier Surface
10b
00b
20b
u
0v
1v
)(
)(
)(
)(
)(
)(
2
2
2
1
2
0
222120
121110
020100
2
1
0
vB
vB
vB
v
v
v
bbb
bbb
bbb
b
b
b
02b
01b
21b
22b11b
)()()()( 2
220
2
110
2
000 uBuBuBuS bbbb
Given 3x3 Points bij,
Generate start/end moving curves and directional curves in quadratic Bezier form
12b
0u
1u
)()()()( 2
222
2
112
2
002 uBuBuBuE bbbb
)(uSb
)(uEb
)(0 vb
)(2 vb
)()()()( 2
202
2
101
2
0000 vBvBvBv bbbb
)()()()( 2
212
2
111
2
0101 vBvBvBv bbbb
)()()()( 2
222
2
121
2
0202 vBvBvBv bbbb
- Chap3. Surface
15/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling- Chap3. Surface
3.4.2 Tensor-product bi-quadratic Bezier surface (2)- Given: Control Points of bi-quadratic Bezier Surface- Find: Points on bi-quadratic Bezier Surface
10b
00b
20b
u
0v
1v
02b
01b
21b
22b11b
Given 3x3 Points bij,
Moving curve can be represented in the following form:
12b
0u
1u
)(uSb
)(uEb
)(0 vb
)(1 vb
)(2 vb),( vub )()()()()()(),( 2
22
2
11
2
00 uBvuBvuBvvu bbbb
)(
)(
)(
)()()(
2
1
0
2
2
2
1
2
0
v
v
v
uBuBuB
b
b
b
)(
)(
)(
)(
)(
)(
2
2
2
1
2
0
222120
121110
020100
2
1
0
vB
vB
vB
v
v
v
bbb
bbb
bbb
b
b
b
)(
)(
)(
)()()(),(2
2
2
1
2
0
222120
121110
020100
2
2
2
1
2
0
vB
vB
vB
uBuBuBvu
bbb
bbb
bbb
b
2
0
2
0
22 )()(j i
jiij vBuBb Bezier surface
control points
16/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.4.2 Tensor-product bi-cubic Bezier surface (1)- Given: Control Points of bi-cubic Bezier Surface- Find: Points on bi-cubic Bezier Surface
10b
00b
30b
u
0v
1v
)(
)(
)(
)(
)(
)(
)(
)(
3
3
3
2
3
1
3
0
33323130
23222120
13121110
03020100
3
2
1
0
vB
vB
vB
vB
v
v
v
v
bbbb
bbbb
bbbb
bbbb
b
b
b
b
03b
01b31b
33b
Given 4x4 Points bij,
Generate start/end moving curves and directional curves in cubic Bezier form
13b
0u
1u
)()()()()( 3
333
3
223
3
113
3
003 uBuBuBuBuE bbbbb
)(uSb
)(uEb
)(0 vb
)(3 vb
)()()()()( 3
303
3
202
3
101
3
0000 vBvBvBvBv bbbbb
)(1 vb)(2 vb02b
23b
20b
32b
)()()()()( 3
330
3
220
3
110
3
000 uBuBuBuBuS bbbbb
)()()()()( 3
313
3
212
3
111
3
0101 vBvBvBvBv bbbbb
)()()()()( 3
323
3
222
3
121
3
0202 vBvBvBvBv bbbbb
)()()()()( 3
333
3
232
3
131
3
0303 vBvBvBvBv bbbbb
- Chap3. Surface
17/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling- Chap3. Surface
3.4.2 Tensor-product bi-cubic Bezier surface (2)- Given: Control Points of bi-cubic Bezier Surface- Find: Points on bi-cubic Bezier Surface
10b
00b
30b
u
0v
1v
03b
01b31b
33b11b
13b
0u
1u
)(uSb
)(uEb
)(0 vb
)(1 vb
)(3 vb
)(2 vb02b
23b
20b
32b
),( vub
Given 4x4 Points bij,
Moving curve can be represented in the following form:
)()()()()()()()(),( 3
33
3
22
3
11
3
00 uBvuBvuBvuBvvu bbbbb
)(
)(
)(
)(
)()()()(
3
2
1
0
3
3
3
2
3
1
3
0
v
v
v
v
uBuBuBuB
b
b
b
b
)(
)(
)(
)(
)(
)(
)(
)(
3
3
3
2
3
1
3
0
33323130
23222120
13121110
03020100
3
2
1
0
vB
vB
vB
vB
v
v
v
v
bbbb
bbbb
bbbb
bbbb
b
b
b
b
)(
)(
)(
)(
)()()()(),(
3
3
3
2
3
1
3
0
33323130
23222120
13121110
03020100
3
3
3
2
3
1
3
0
vB
vB
vB
vB
uBuBuBuBvu
bbbb
bbbb
bbbb
bbbb
b
3
0
3
0
33 )()(j i
jiij vBuBb
18/52
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
Naval A
rch
itectu
re &
Ocean
En
gin
eerin
g
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3 B-spline surfaces
3.3.1 Generation of B-spline surfaces by tensor-product approach
3.3.2 B-spline surface Interpolation
- Chap3. Surface
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
Naval A
rch
itectu
re &
Ocean
En
gin
eerin
g
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.1 Generation of B-spline surfaces
by tensor-product approach
3.3.1.1 Tensor-product bicubic B-spline surface
3.3.1.2 Programming Guide of Tensor-product bicubic B-spline surface
- Chap3. Surface
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.1.1 Tensor-product bicubic B-spline surface (1)- Given: Control Points of bicubic B-spline surface- Find: Points on bicubic B-spline surface
Given 5x5 Control Points dij, u-knots, v-knots, u-degree(=3), v-degree(=3),
Generate start/end moving curves and directional curves in cubic B-spline form:
10d
40d
00d20d
30d
00 u 15 u4.04 u
2u1u
3u7u6u
8u
u
14d
44d
04d
24d
34d
00 v
2v1v
3v
15 v
7v6v
8v
4v
03d
02d
01d
)(0 vd
11d
12d
)(1 vd
23d
22d
21d
)(2 vd
33d
32d
31d
)(3 vd
43d
42d
41d )(4 vd
3 3 3 3 3
00 0 01 1 02 2 03 3 04 40 N v N v N v N v N vv d d dd d d
3 3 3 3 3
10 0 11 1 12 2 13 3 14 41 N v N v N v N v N vv d d dd d d
3 3 3 3 3
20 0 21 1 22 2 23 3 24 42 N v N v N v N v N vv d d dd d d
3 3 3 3 3
30 0 31 1 32 2 33 3 34 43 N v N v N v N v N vv d d dd d d
3 3 3 3 3
40 0 41 1 42 2 43 3 44 44 N v N v N v N v N vv d d dd d d
3
00 01 02 03 04 0
3
10 11 12 13 14 1
3
20 21 22 23 24 2
3
30 31 32 33 34 3
3
40 41 42 43 44
0
2
4 4
1
3
N v
N v
N v
N v
N v
v
v
v
v
v
d
d
d
d d d d d
d d d d d
d
d
d
d d d d
d d d d d
d d d d d
- Chap3. Surface
21/52
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.1.1 Tensor-product bicubic B-spline surface (1)- Given: Control Points of bicubic B-spline surface- Find: Points on bicubic B-spline surface
Given 5x5 Control Points dij, u-knots, v-knots, u-degree(=3), v-degree(=3),
Moving curve can be represented in the following form:
10d
40d
00d20d
30d
00 u 15 u4.04 u
2u1u
3u7u6u
8u
u
14d
44d
04d
24d
34d
00 v
2v1v
3v
15 v
7v6v
8v
4v
03d
02d
01d
)(0 vd
)(1 vd)(2 vd
)(3 vd
43d
42d
41d
)(4 vd
),( vur
)(
)(
)(
)(
)(
)()()()()(),(
3
4
3
3
3
2
3
1
3
0
4443424140
3433323130
2423222120
1413121110
0403020100
3
4
3
3
3
2
3
1
3
0
vN
vN
vN
vN
vN
uNuNuNuNuNvu
ddddd
ddddd
ddddd
ddddd
ddddd
r
5
0
5
0
33 )()(j i
jiij vNuNd
0 1 2
3 3 3 3 3
0 1 2 33 4 4, v v v vN u N u N u N uu N uv v d dr d d d
0
3 3 3 3 3
0 4
1
21 2
3
4
3N u N u N u N u N u
v
v
v
v
v
d
d
d
d
d
3
00 01 02 03 04 0
3
10 11 12 13 14 1
3
20 21 22 23 24 2
3
30 31 32 33 34 3
3
40 41 42 43 44
0
2
4 4
1
3
N v
N v
N v
N v
N v
v
v
v
v
v
d
d
d
d d d d d
d d d d d
d
d
d
d d d d
d d d d d
d d d d d- Chap3. Surface
22/52
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.1.2 Programming Guide of Tensor-product bicubic B-spline surface - Member Variables of Class
class CBsplineSurface
{
public:
// member variables
int m_nDegree;
double* m_pKnot_U;
int m_nNumOfKnot_U;
double* m_pKnot_V;
int m_nNumOfKnot_V;
Vector** m_pCP;
int m_nNumOfCP_U;
int m_nNumOfCP_V;
// member functions
…
};
차수
u 방향 Knot와 개수
v 방향 Knot와 개수
Control Point, u, v 방향 개수
10d
40d
00d20d
30d
00 u 15 u4.04 u
2u1u
3u7u6u
8u
u
14d
44d
04d
24d
34d
00 v
2v1v
3v
15 v
7v6v
8v
4v
03d
02d
01d
)(0 vd
)(1 vd)(2 vd
)(3 vd
43d
42d
41d
)(4 vd
),( vur
(3차)
(9개)
(9개)
(u 방향 5개, v 방향 5개)
- Chap3. Surface
23/52
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.1.2 Programming Guide of Tensor-product bicubic B-spline surface - Member Functions of Class
class CBsplineSurface
{
public:
// member variables
…
// member functions
…
void SetKnot(double* pKnot_U, int nNumOfKnot_U, double* pKnot_V, int
nNumOfKnot_V);
double N(int n, int i, double u, int uv);
Vector GetPoint(double u, double v);
};
Knot 설정
10d
40d
00d20d
30d
00 u 15 u4.04 u
2u1u
3u7u6u
8u
u
14d
44d
04d
24d
34d
00 v
2v1v
3v
15 v
7v6v
8v
4v
03d
02d
01d
)(0 vd
)(1 vd)(2 vd
)(3 vd
43d
42d
41d
)(4 vd
),( vur
u 방향 Knot
v 방향 Knot
- Chap3. Surface
24/52
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.1.2 Programming Guide of Tensor-product bicubic B-spline surface - Member Functions of Class
class CBsplineSurface
{
public:
// member variables
…
// member functions
…
void SetKnot(double* pKnot_U, int nNumOfKnot_U, double* pKnot_V, int
nNumOfKnot_V);
double N(int n, int i, double u, int uv);
Vector GetPoint(double u, double v);
};
B-spline Basis Function 계산
10d
40d
00d20d
30d
00 u 15 u4.04 u
2u1u
3u7u6u
8u
u
14d
44d
04d
24d
34d
00 v
2v1v
3v
15 v
7v6v
8v
4v
03d
02d
01d
)(0 vd
)(1 vd)(2 vd
)(3 vd
43d
42d
41d
)(4 vd
),( vur
B-spline Basis Function 계산(Cox-de Boor Recurrence Formula)
)()()( 1
1
1
11
1 uNuu
uuuN
uu
uuuN n
i
ini
nin
i
ini
in
i
else 0
if 1)(
10 ii
i
uuuuN
u, v 방향을 구분하는 flag
- Chap3. Surface
25/52
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.1.2 Programming Guide of Tensor-product bicubic B-spline surface - Member Functions of Class
class CBsplineSurface
{
public:
// member variables
…
// member functions
…
void SetKnot(double* pKnot_U, int nNumOfKnot_U, double* pKnot_V, int
nNumOfKnot_V);
double N(int n, int i, double u, int uv);
Vector GetPoint(double u, double v);
};
Parameter u, v에 대한 곡면 상의 점 계산
10d
40d
00d20d
30d
00 u 15 u4.04 u
2u1u
3u7u6u
8u
u
14d
44d
04d
24d
34d
00 v
2v1v
3v
15 v
7v6v
8v
4v
03d
02d
01d
)(0 vd
)(1 vd)(2 vd
)(3 vd
43d
42d
41d
)(4 vd
),( vur
)(
)(
)(
)(
)(
)()()()()(),(
3
4
3
3
3
2
3
1
3
0
4443424140
3433323130
2423222120
1413121110
0403020100
3
4
3
3
3
2
3
1
3
0
vN
vN
vN
vN
vN
uNuNuNuNuNvu
ddddd
ddddd
ddddd
ddddd
ddddd
r
5
0
5
0
33 )()(j i
jiij vNuNd26/52
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.1.2 Programming Guide of Tensor-product bicubic B-spline surface - Member Function Example ‘GetPoint’
Vector CBsplineSurface::GetPoint(double u, double v){// return valueVector r_u_v(0.0, 0.0, 0.0);
// get curvefor (int i=0; i<m_nNumOfCP_U; i++){Vector r_v(0.0, 0.0, 0.0);
for (int j=0; j<m_nNumOfCP_V; j++){r_v = r_v + m_pCP[i][j] * N(m_nDegree, j, v, ID_V);
}
r_u_v = r_u_v + N(m_nDegree, i, u, ID_U) * r_v;}
return r_u_v;}
)(
)(
)(
)(
)(
)()()()()(),(
3
4
3
3
3
2
3
1
3
0
4443424140
3433323130
2423222120
1413121110
0403020100
3
4
3
3
3
2
3
1
3
0
vN
vN
vN
vN
vN
uNuNuNuNuNvu
ddddd
ddddd
ddddd
ddddd
ddddd
r
5
0
5
0
33 )()(j i
jiij vNuNd
Parameter u, v에 대한 곡면 상의 점 계산
)(d)(d)(d)(d)(d)(d 3
404
3
303
3
202
3
101
3
0000 vNvNvNvNvNv
27/52
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
Naval A
rch
itectu
re &
Ocean
En
gin
eerin
g
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.2 B-spline surfaces
Interpolation
3.3.2.1 bicubic B-spline surface interpolation 개요
3.3.2.2 bicubic B-spline surface interpolation 상세 과정
3.3.2.3 Sequences of Finding knots
3.3.2.4 Knot 간격차이가 주는 영향
3.3.2.5 Example of bicubic B-spline Surface Interpolation
- Chap3. Surface
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
p21
p00
p01
p02
p20
p22p10
p12
u
v
3.3.2.1 bicubic B-spline Surface Interpolation (1)- Given: 곡면상의 9개 점과 4 꼭지점에서의 u, v방향의 접선벡터- Find: Control Points of bicubic B-spline Surface
00 01 02 03 04
10 11 12 13 14
20 21 22 23 24
30 31 32 33 34
40 41 42 43 44
d d d d d
d d d d d
d d d d d
d d d d d
d d d d d
)(
)(
)(
)(
)(
3
4
3
3
3
2
3
1
3
0
vN
vN
vN
vN
vN
)()()()()( 3
4
3
3
3
2
3
1
3
0uNuNuNuNuN),( vur
5x5 개의 조정점을 구하면 곡면을 생성할 수 있다.- Chap3. Surface
29/52
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
곡선상의 점(Pi,j)과 접선벡터 (ti,j) 으로부터
중간 조정점(Ci,j)을 구한다.p21
p00
p01
p02
p20
p22p10
p12C01
C02
C03
u
v
00 00
01 00
02 01
03 02
04 02
1 0 0 0 0
3 30 0 0
0 0
3 30 0 0
0 0 0 0 1
s s
E E
C P
C t
C P
C t
C P
p00
p01
p02 t02
t00
3.3.2.1 bicubic B-spline Surface Interpolation (2)- Given: 곡면상의 9개 점과 4 꼭지점에서의 u, v방향의 접선벡터- Find: Control Points of bicubic B-spline Surface
p10
p11
p12 t12
t01
Bessel end condition으로 접선벡터(ti,j)를 구한다
Bessel end condition:
곡선을 지난는 3점을 2차식으로 보간(interpolation)
한 후, 곡선의 끝점에서의 1차미분값을 구하는 방법
- Chap3. Surface
p20
p21
p22 t22
t20
30/52
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
중간 조정점(Ci,j)과 접선 벡터 (ti,j)
으로부터 B-spline조정점(di,j)을 구한다.
3.3.2.1 bicubic B-spline Surface Interpolation (3)- Given: 곡면상의 9개 점과 4 꼭지점에서의 u, v방향의 접선벡터- Find: Control Points of bicubic B-spline Surface
02 02
12 02
22 12
32 22
42 22
1 0 0 0 0
d C3 30 0 0
d t
d C0 0
d t3 30 0 0
d C
0 0 0 0 1
s s
E E
p21
p00
p01
p02
p20
p22p10
p12C01
C02
C03
u
v
C22
C02
C12
t22
t02
- Chap3. Surface
p00
p10
p20
t00
t20
p02
p12
p22
t24
t04
C11
C01
C21
C23
C13
C03
t01
t03
t21
t23
Bessel end condition으로 접선벡터(ti,j)를 구한다 31/52
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.2.2 bicubic B-spline Surface Interpolation (4)- Given: 곡면상의 9개 점과 4 꼭지점에서의 u, v방향의 접선벡터- Find: Control Points of bicubic B-spline Surface
p00
p10
p20
p02p12
p22
p01
p21
p11
u
v
4 6
66
3.65.4
Given 곡면이 지나야할 점들의 좌표 점들은 사각형 grid 형태여야 함 (예, 2x2)
1. u 방향 knot를 결정 주어진 점들의 u방향 거리를 계산한다
- Chap3. Surface
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.2.2 bicubic B-spline Surface Interpolation (5)- Given: 곡면상의 9개 점과 4 꼭지점에서의 u, v방향의 접선벡터- Find: Control Points of bicubic B-spline Surface
p00
p10
p20
p02p12
p22
p01
p21
p11
u
v
04
10
0 6
12
5.40
9
Given 곡면이 지나야할 점들의 좌표 점들은 사각형 grid 형태여야 함 (예, 2x2)
1. u 방향 knot를 결정 주어진 점들의 u방향 거리를 계산한다 계산한 거리를 각 점별로 누적한다. 이 거리를 곡면의
u방향 knot라고 부른다
4 6
66
3.65.4
- Chap3. Surface
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2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.2.2 bicubic B-spline Surface Interpolation (6)- Given: 곡면상의 9개 점과 4 꼭지점에서의 u, v방향의 접선벡터- Find: Control Points of bicubic B-spline Surface
p00
p10
p20
p02p12
p22
p01
p21
p11
u
v
1
1
1
0.4
0.5
0.6
Given 곡면이 지나야할 점들의 좌표 점들은 사각형 grid 형태여야 함 (예, 2x2)
1. u 방향 knot를 결정 주어진 점들의 u방향 거리를 계산한다 계산한 거리를 각 점별로 누적한다. 이 거리를 곡면의
u방향 knot라고 부른다 마지막 점의 knot값으로 각 점의 knot값을 나누어 정
규화된 knot값을 계산한다
04
10
0 6
12
5.40
9
- Chap3. Surface
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2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.2.2 bicubic B-spline Surface Interpolation (7)- Given: 곡면상의 9개 점과 4 꼭지점에서의 u, v방향의 접선벡터- Find: Control Points of bicubic B-spline Surface
p00
p10
p20
p02p12
p22
p01
p21
p11
u
v
00.5
1
Given 곡면이 지나야할 점들의 좌표 점들은 사각형 grid 형태여야 함 (예, 2x2)
1. u 방향 knot를 결정 주어진 점들의 u방향 거리를 계산한다 계산한 거리를 각 점별로 누적한다. 이 거리를 곡면의
u방향 knot라고 부른다 마지막 점의 knot값으로 각 점의 knot값을 나누어 정
규화된 knot값을 계산한다 정규화된 knot값들을 v방향으로 평균 u방향 knot값
을 계산한다
0
0
0
1
1
10.4
0.5
0.6
- Chap3. Surface
35/52
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.2.2 bicubic B-spline Surface Interpolation (8)- Given: 곡면상의 9개 점과 4 꼭지점에서의 u, v방향의 접선벡터- Find: Control Points of bicubic B-spline Surface
p00
p10
p20
p02p12
p22
p01
p21
p11
u
v
Given 곡면이 지나야할 점들의 좌표 점들은 사각형 grid 형태여야 함 (예, 2x2)
1. u 방향 knot를 결정 주어진 점들의 u방향 거리를 계산한다 계산한 거리를 각 점별로 누적한다. 이 거리를 곡면의
u방향 knot라고 부른다 마지막 점의 knot값으로 각 점의 knot값을 나누어 정
규화된 knot값을 계산한다 정규화된 knot값들으로 v방향으로 평균 u방향 knot
값을 계산한다
2. u 방향 점들을 보간하는 B-spline 곡선을 계산 최종적인 u방향 knot와 점들을 지나는 B-spline 곡선
과 그 조정점을 계산한다
3. v 방향 knot 간격을 u뱡향과 동일한 방법으로 구함
4. u 방향 B-spline 곡선의 조정점을 보간하는 v방향B-spline 곡선을 계산 u방향 B-spline 곡선의 조점정에 대해서 1번과 같은
방법으로 v방향 knot를 결정한 후, 이를 이용하여 v방향 B-spline 곡선 (보라색점선) 을 생성한다. 이 곡선의 조정점 (빨간점) 이 최종 B-spline 곡면의 조정점이다
00.5
1
- Chap3. Surface
36/52
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
p12
00.5
1
3.3.2.2 bicubic B-spline Surface Interpolation (9)- Given: 곡면상의 9개 점과 4 꼭지점에서의 u, v방향의 접선벡터- Find: Control Points of bicubic B-spline Surface
p00
p10
p20
p02
p22
p01
p21
p11
u
v
Given 곡면이 지나야할 점들의 좌표 점들은 사각형 grid 형태여야 함 (예, 2x2)
1. u 방향 knot를 결정 주어진 점들의 u방향 거리를 계산한다 계산한 거리를 각 점별로 누적한다. 이 거리를 곡면의
u방향 knot라고 부른다 마지막 점의 knot값으로 각 점의 knot값을 나누어 정
규화된 knot값을 계산한다 정규화된 knot값들을 v방향으로 평균하여 최종적인 u
방향 knot값을 계산한다
2. u 방향 점들을 보간하는 B-spline 곡선을 계산 최종적인 u방향 knot와 점들을 지나는 B-spline 곡선
과 그 조정점을 계산한다
3. v 방향 knot 간격을 u뱡향과 동일한 방법으로 구함
4. u 방향 B-spline 곡선의 조정점을 보간하는 v방향B-spline 곡선을 계산 u방향 B-spline 곡선의 조점정에 대해서 1번과 같은
방법으로 v방향 knot를 결정한 후, 이를 이용하여 v방향 B-spline 곡선 (보라색점선) 을 생성한다. 이 곡선의 조정점 (빨간점) 이 최종 B-spline 곡면의 조정점이다
- Chap3. Surface
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling- Chap3. Surface
곡선상의 점,접선벡터와 위의 Knot
간격을 이용하여 B-spline 조정점을구할 수 있음
주어진 곡선상의 점들로부터 Chord
Length를 구함
3.3.2.3 Sequences of Finding Knot
B-spline 곡선에서의Knot 간격 예측
)2,....,3,2(,1
21
1
nipp
pp
ii
ii
i
i
즉, 주어진 곡선들의 Knot 간격이 모두 일정해야 함.
주어지는 곡면상의 점이 GRID와 같이 균일하게 주어져야 이상적인 곡면재현 가능
Tensor Product방식으로 정의된spline 곡면에서의 Knot 간격예측
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Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
점과 점 사이의 knot 간격은 그 점 사이를 지나가는 데 걸리는 시간과 같은 개념이다.
3.3.2.4 Knot 간격 차이가 주는 영향
0 0.5 1.5 6.5
곡선상의 점
Knot 간격
1
2
3
그러므로 knot간격비가 비슷할 수록 곡면의 왜곡이 적다.39/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
점과 점 사이의 knot 간격은 그 점 사이를 지나가는 데 걸리는 시간과 같은 개념이다.
3.3.2.4 Knot 간격 차이가 주는 영향
0 0.5 1.5 6.5
곡선상의 점
Knot 간격
1
2
3
그러므로 knot간격비가 비슷할 수록 곡면의 왜곡이 적다.40/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.2.5 Example of bicubic B-spline Surface Interpolation (1)- Given: Points on Surface- Find: bicubic B-spline Surface (Control Points of bicubic B-spline Surface)
일반 3차원 곡면 형상 예시
41/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.2.5 Example of bicubic B-spline Surface Interpolation (2)- Given: Points on Surface- Find: bicubic B-spline Surface (Control Points of bicubic B-spline Surface)
선박 선형 곡면 예시
42/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.2.5 Sample code of bicubic B-spline Surface Interpolation (1)
void BicubicBsplineSurface::Interpolate(Vector **pFittingPoint, int nU, int nV) {
// Generate u-Knot
if(m_UKnot) delete[] m_UKnot;
m_nUKnot = (m_nU - 2) + 2*(3+1);
m_UKnot = new double [m_nUKnot];
// Initial u-Knot
double** tmpUKnots;
tmpUKnots = new double*[nV];
for(int j = 0; j < nV; j++){
tmpUKnots[j] = new double[nU];
for(int i = 0; i < nU; i++){
tmpUKnot[j][i] = …; // chord length or centripetal
}
}
// generate average u-Knot
for(int i = 0; i < nU; i++){
m_UKnot[i] = 0;
for(int j=0; j<nV; j++) { m_UKnot[i] += tmpUKnot[j][i]; }
m_UKnot[i] = m_UKnot[i] / nV;
}
Chord length를 이용하여u, v 방향 Knot 생성
Knot간 간격의 평균값으로Knot 재 설정
43/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.2.5 Sample code of bicubic B-spline Surface Interpolation (2)
// Interpolate u-directional B-spline curve
CubicBsplineCurve* u_curve = new CubicBsplineCurve[nV];
for(int j = 0; j < nV; j++){
u_curve[j].SetKnot( m_UKnot );
u_curve[j].Interpolate( pFittingPoint[j], nU );
}
// Generate v-directional Fitting Point
int nvFittingPoint = u_curve[0].m_nControlPoint;
Vector** vFittingPoint = new Vector [ nvFittingPoint ];
for(int j=0; j < nvFittingPoint; j++){
vFittingPoint[j] = new Vector[ nV ];
for( int i = 0; i < nV; i++){
vFittingPoint[j][i] = u_curve[i].m_ControlPoint[j];
}
}
……………………..
p00
p10
p20
p02p12
p22
p01
p21
p11
u
v
00.5
1
u 방향 B-spline curve 생성
44/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
3.3.2.5 Sample code of bicubic B-spline Surface Interpolation (2)
// Interpolate u-directional B-spline curve
CubicBsplineCurve* u_curve = new CubicBsplineCurve[nV];
for(int j = 0; j < nV; j++){
u_curve[j].SetKnot( m_UKnot );
u_curve[j].Interpolate( pFittingPoint[j], nU );
}
// Generate v-directional Fitting Point
int nvFittingPoint = u_curve[0].m_nControlPoint;
Vector** vFittingPoint = new Vector [ nvFittingPoint ];
for(int j=0; j < nvFittingPoint; j++){
vFittingPoint[j] = new Vector[ nV ];
for( int i = 0; i < nV; i++){
vFittingPoint[j][i] = u_curve[i].m_ControlPoint[j];
}
}
……………………..
p12
00.5
1
p00
p10
p20
p02
p22
p01
p21
p11
u
v
u 방향 B-spline curve 의 조정점을Fitting Point로 하여v 방향 B-spline Curve 생성
45/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
Naval A
rch
itectu
re &
Ocean
En
gin
eerin
g
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
Ch 4. Term Project
Generation Ship hull surfaces by interpolating given points
- Given: Pi,j- Find : di,j
- Chap4. Term Project
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
4.1 단일 B-spline 곡면 patch를 이용한 선형곡면 생성 프로그램 구현
- Chap4. Term Project
47/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
4.2 간단한 요트 형상의 선수부 곡선그물망 형상
* 출처 : 서울대 조선해양공학과 2005년 2학년 교과목『 조선해양공학계획』강좌 중에 학생들이 설계한 선형
사각형 패치의 꼭지점 결정
- Chap4. Term Project
48/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
4.3 요트 형상의 곡선그물망으로부터 선형곡면 생성결과 (1)
Offset table 형식으로 점을 추출한 후,
이 점들로 부터 bicubic B-spline 선형곡면을 생성한 결과
정보 부족으로 인한 형상왜곡문제가 있음
6x14 points
- Chap4. Term Project
49/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
4.3 요트 형상의 곡선그물망으로부터 선형곡면 생성결과 (2)
점들의 x좌표 사이의 거리가 일정하도록 점을 추출한 후,
이 점들로부터 bicubic B-spline 선형곡면을 생성한 결과
11x11 points
50/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
4.4 구상선수를 갖는 단축선의 선수부 곡선그물망 형상
선수부
- Chap4. Term Project
51/52
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling
4.5 구상선수부 곡선그물망으로부터 선수부 선형곡면 생성결과
주어진 곡선그물망 이외의 보조선을 생성하여 곡선 보간에 적합한
점 data를 생성한 후, 점 data로부터 선수부 선형곡면을 생성한 결과
왜곡 감소
- Chap4. Term Project
52/52