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Advanced Ship Design Automation Lab.

http://asdal.snu.ac.kr Seoul

National Univ.

2009 Fall, Computer Aided Ship Design – Part2. Hull Form Modeling

Naval Architecture & Ocean Enginee ring

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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

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Ch 3. 곡면(Surfaces)

3.1 Parametric Surfaces 3.2 Bezier Surfaces

3.3 B-spline surfaces

- Chap3. Surface

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3.1 Parametric Surfaces

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3.1 Parametric Surfaces

곡면 양함수

음함수

매개 변수

2 2

2 x y

d

z    

2 2

2 2 y z d

x   







 sin

cos sin

cos cos

d z

d y

d x



)) , ( ), , ( ), , (

( x   y   z  

 r r

구를 2개의 매개변수 로 표현하 면 간단히 수식화 할 수 있다.

) , (  

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3.2 Bezier surfaces

3.2.1 Generation of Bezier surfaces by de Casteljau algorithm

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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

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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

0 b 0

1 b 0

b 11

b 10

u

v

P

u

q

u

 u 1

v

 v 1

u v

0 1

1 u v

) u 1

(  u b ₀ ₀  ^b ₁ ₀

P

u



) u 1

(  u b ₀ ₁  b ₁ ₁ q

u



v )

1 (  v 

) , ( v u

r  P

_u

q

u

.

) (

1

0

,

1 1 0 0

있다 수

얻을 곡면을

하는 꼭지점으로

을 점

계산하면 를

증가하며 까지

에서 를

1 0 1

0

, b , b , b b

r u,v v u

2x2개의 Bezier 조정점을 이용하여 Bi-linear Interpolation 으로 곡면식을 구할 수 있다.

v ) 1 (  v )

, ( v u

r 

u )

1 (  u b ₀ ₁  _b ₁ ₁

) 1

(  u b ₀ ₀  ⁽ ¹ ^ ^v ⁾ u ^b ₁ ₀

 v

 

 





1 1

0 0

1 0

b b

b

b 



 



  u

u) 1

 ⁽ ¹ ^ ^v) ^v  ( )

, ( v u

r 

u

 u 1

방법: u, v 방향으로‘de Casteljau algorithm’ 사용

- Chap3. Surface ^7/52

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3.2.1.2 Bi-quadratic Bezier Surface Patch

- Given: 3x3 Bezier control point

- Find: Points on bi-quadratic Bezier Surface Patch

u v

b

02

b

12

b

22

u

1 u  u

1 u 

u 1 u 

2

p

u

q

₂^u

2

r

u

b

01

b

11

b

21

u

1 u  u

1 u 

u 1 u 

1

p

u

q

₁^u

1

r

u

b

00

b

10

b

20

u

1 u  u

1 u 

u 1 u 

0

p

u

q

₀^u

0

r

u

 

00 10

10 2

0 0 0

1 1

u

u u



 

p q

b b

 

01 11

11 2

1 1 1

1 1

u

u u



 

p q

b b

 

02 12

12 2

2 2 2

1 1

u

u u



 

p q

b b

  0

0 û  1  u p 0 û  u q û r

   

2 2

00 10 20

2 2

01 11 21

2 2

02 12 22

0

1

2 1 2 1

1 2 1

u

u u u u

    

r r

b b b

b b

b b b

r

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

  1

1 û  1  u p 1 û  u q û r

  2

2 û  1  u p 2 û  u q û r

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3.2.1.2 Bi-quadratic Bezier Surface Patch

- Given: 3x3 Bezier control point

- Find: Points on bi-quadratic Bezier Surface Patch

u v

b

02

b

12

b

22 2

r

u

b

01

b

11

b

21 1

r

u

b

00

b

10

b

20 0

r

u

 

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

s

v

0

s

v

v 1 v 

v

1 v 

  ^{u v} ^,

r

v 1 v 

 

0 1

1 2

0

1 1 1

u u

v

v u u

v v

 

  

 s

s

r r

   u v ,  1  v  s 0 ^v  v s 1 ^v

r

    ²   2 2

0 1

1 2 1

, v û v v û v û

u v   r    r

r r

    ²   ² 1 ⁰ 2

1 2

, 1

u u u

v v v

v v

u

   

 

      

    r r r r

       

 

2 00 10 20

2 2

01 11 21

2 02 12 22

1 1 2 1 2 1

,

u

v v v v u u

u v

u

  

 

 

 

                 

b b b

b b r

b

3x3개의 조정점을 이용하여

Bi-Quadratic Bezier Patch를 구할 수 있다.

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3.2.1.3 Bi-cubic Bezier Surface Patch

- Given: 4x4 Bezier control point

- Find: Points on bi-cubic Bezier Surface Patch

u

 u 1

u ₁ _ _u

u

 u 1

u ¹ ^ ^u u

 u 1

u v

4x4개의 조정점을 이용하여 Bi-Cubic Bezier Patch를 구할 수 있다.

3 1 0 2 3 0 0

3 1 1 2 1 3 1

3 3 3 3 1

0 1 2 3 3

0 2 1 2 2 3 2 2

3 0 3 1 3 2 3 3 3 3

( )

( , ) ( ) ( ) ( ) ( ) ( )

( ) ( ) B v u v B u B u B u B u B v

B v B v

 

 

 

      

 

   

0,0 , ,0 ,

0, ,1 , ,

, , ,2 ,

, , , ,

b b b b

b b b b b

b b b b

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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 수 있다.

3 1 0 2 3 0 0

3 1 1 2 1 3 1

3 3 3 3 1

0 1 2 3 3

0 2 1 2 2 3 2 2

3 0 3 1 3 2 3 3 3 3

( )

( , ) ( ) ( ) ( ) ( ) ( )

( ) ( ) B v u v B u B u B u B u B v

B v B v

 

 

 

      

 

   

0,0 , ,0 ,

0, ,1 , ,

, , ,2 ,

, , , ,

b b b b

b b b b b

b b b b

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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

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3.4.1 Tensor product approach (1)

매우 탄력적인 재질의 고무 밀대라고 가정

나무판

End moving curve Directional curve

Moving curve

 makes a surface

Start moving curve

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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

moving Curve

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)

1

( v b

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

b

10

b

00

b

20

u

 0 v

 1 v

 







 







 







 









 







 







) (

2 2 2 1

2 0

22 21

20 12 11

10 02 01

00 2 1 0

v B

v v v

b b

b

b b

b

b b

b

b b b

b

02

b

01

b

21

b

22

b

11

) ( )

( )

( u ₀₀ B ₀ ² u ₁₀ B ₁ ² u ₂₀ B ₂ ² u

S b b b

b   

 Given 3x3 Points b _ij ,

 Generate start/end moving curves and directional curves in quadratic Bezier form

b

12

 0 u

 1 u

) ( )

( )

( u ₀₂ B ₀ ² u ₁₂ B ₁ ² u ₂₂ B ₂ ² u

E b b b

b   

)

S

(u b

)

E

(u b

)

0

( v b

)

2

( v b

) ( )

( )

( ₀₀ ₀ ² ₀₁ ₁ ² ₀₂ ₂ ²

0 v b B v b B v b B v

b   

) ( )

( )

( ₁₀ ₀ ² ₁₁ ₁ ² ₁₂ ₂ ²

1 v b B v b B v b B v

b   

) ( )

( )

( ₂₀ ₀ ² ₂₁ ₁ ² ₂₂ ₂ ²

2 v b B v b B v b B v

b   

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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

b

10

b

00

b

20

u

 0 v

 1 v

b

02

b

01

b

21

b

22

b

11

 Given 3x3 Points b _ij ,

 Moving curve can be

represented in the following form:

b

12

 0 u

 1 u

)

S

(u b

)

E

(u b

)

0

( v b

)

1

( v b

)

2

( v

) b

, ( v u

b ^b ⁽ û ^, ^v ⁾ ^ ^b ⁰ ⁽ ^v ⁾ ^B ⁰ ² ⁽ û ⁾ ^ ^b ¹ ⁽ ^v ⁾ ^B ¹ ² ⁽ û ⁾ ^ ^b ² ⁽ ^v ⁾ ^B ² ² ⁽ û ⁾

 

 







 









) (

) ( )

( )

(

2 1 0 2

2 2

1 2

0 v v v u

B u B u B

b b b

 







 







 







 









 







 







) (

2 2 2 1

2 0

22 21

20 12 11

10 02 01

00 2 1 0

v B

v v v

b b

b

b b

b

b b

b

b b b

 

 







 







 







 









) (

) ( )

( )

, (

2 2 2 1

2 0

22 21

20 12 11

10 02 01

00 2

2 2

1 2

0 v B

v B

v B u

B u B u B v

u

b b

b

b b

b

b b

  

 ²

0 2

0 2 2 ( ) ( )

j i

ij B u B v

b Bezier surface

control points

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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

b

10

b

00

b

30

u

 0 v

 1 v

 







 







 





 







 





 





) (

3 3 3 2 3 1

3 0

33 32

31 30

23 22

21 20

13 12

11 10

03 02

01 00

3 2 1 0

v B

v v v v

b b

b b b b

b

03

b

01

b

31

b

33

 Given 4x4 Points b _ij ,

 Generate start/end moving curves and directional curves in cubic Bezier form

b

13

 0 u

 1 u

) ( )

( )

( u ₀₃ B ₀ ³ u ₁₃ B ₁ ³ u ₂₃ B ₂ ³ u ₃₃ B ₃ ³ u

E b b b b

b    

)

S

(u b

)

E

(u b

)

0

( v b

)

3

( v b

) ( )

( )

( ₀₀ ₀ ³ ₀₁ ₁ ³ ₀₂ ₂ ³ ₀₃ ₃ ³

0 v b B v b B v b B v b B v

b    

)

1

( v

b b

₂

( v )

b

02

b

23

b

20

b

32

) ( )

( )

( u ₀₀ B ₀ ³ u ₁₀ B ₁ ³ u ₂₀ B ₂ ³ u ₃₀ B ₃ ³ u

S b b b b

b    

) ( )

( )

( ₁₀ ₀ ³ ₁₁ ₁ ³ ₁₂ ₂ ³ ₁₃ ₃ ³

1 v b B v b B v b B v b B v

b    

) ( )

( )

( ₂₀ ₀ ³ ₂₁ ₁ ³ ₂₂ ₂ ³ ₂₃ ₃ ³

2 v b B v b B v b B v b B v

b    

) ( )

( )

( ₃₀ ₀ ³ ₃₁ ₁ ³ ₃₂ ₂ ³ ₃₃ ₃ ³

3 v b B v b B v b B v b B v

b    

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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

b

10

b

00

b

30

u

 0 v

 1 v

b

03

b

01

b

31

b

33

b

11

b

13

 0 u

 1 u

)

S

(u b

)

E

(u b

)

0

( v b

)

1

( v b

)

3

( v b )

2

( v b

b

02

b

23

b

20

b

32

) , ( v u b

 Given 4x4 Points b _ij ,

 Moving curve can be

represented in the following form:

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

,

( u v b

₀

v B

₀³

u b

₁

v B

₁³

u b

₂

v B

₂³

u b

₃

v B

₃³

u

b    

 

 





 







) (

) ( ) ( ) ( ) ( ) (

3 2 1 0

3 3 3

2 3

1 3

0

v v v v u

B u B u B u B

b b b b

 







 







 





 







 





 





) (

3 3 3 2 3 1

3 0

33 32 31 30

23 22 21 20

13 12 11 10

03 02 01 00

3 2 1 0

v B

v v v v

b b b b

b b b

b  

 







 







 





 







) (

) ( )

( ) ( ) ( ) ( )

, (

3 3 3 2 3 1

3 0

33 32 31 30

23 22 21 20

13 12 11 10

03 02 01 00

3 3 3

2 3

1 3

0

v B

v B u

B u B u B u B v u

b b b b

b b b b b

  

 ³

0 3

0 3 3 ( ) ( )

j i

ij B u B v

b

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3.3 B-spline surfaces

3.3.1 Generation of B-spline surfaces by tensor-product approach 3.3.2 B-spline surface Interpolation

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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

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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 d _ij , u-knots, v-knots,

u-degree(=3), v-degree(=3),

 Generate start/end moving

curves and directional curves in cubic B-spline form:

d

10

d

40

d

00 ²⁰

d

30

00

u u₄0.4 u₅1

u2¹

u

u3 ⁷

u⁶ u u8

u

d

14

d

44

d

04

d

24

d

34

00 v v2¹

v v3

51 v v7⁶

v v8

v4

d

03

d

02

d

01

)

0

( v d

d

11

d

12

)

1

( v d

d

23

d

22

d

21

)

2

( v d

d

33

d

32

d

31

)

3

( v d

d

43

d

42

d

41

d

₄

( v )

 

00 0³

 

01 1³

 

02 2³

 

03 3³

 

04 4³

 

0

v  N v  d N v  N v  d N v  d N v

d d d

 

10 0³

 

11 1³

 

12 2³

 

13 3³

 

14 4³

 

1

v  N v  d N v  N v  d N v  d N v

d d d

 

₂₀ ₀³

 

₂₁ ₁³

 

₂₂ ₂³

 

₂₃ ₃³

 

₂₄ ₄³

 

2

v  N v  d N v  N v  d N v  d N v

d d d

 

30 0³

 

31 1³

 

32 2³

 

33 3³

 

34 4³

 

3

v  N v  d N v  N v  d N v  d N v

d d d

 

40 0³

 

41 1³

 

42 2³

 

43 3³

 

44 4³

 

4

v  N v  d N v  N v  d N v  d N v

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

21/52

(22)

@ SDAL

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 d _ij , u-knots, v-knots,

u-degree(=3), v-degree(=3),

 Moving curve can be

represented in the following form:

d

10

d

40

d

00 ²⁰

d

30

00

u u₄0.4 u₅1

u2¹

u

u3 ⁷

u⁶ u u8

u

d

14

d

44

d

04

d

24

d

34

00 v v2¹

v v3

51 v v7⁶

v v8

v4

d

03

d

02

d

01

)

0

( v d

)

1

( v

d d

₂

( v )

)

3

( v d

d

43

d

42

d

41

)

4

( v d

) , ( v u r

 

 





 





 





 







) (

) ( )

( ) ( ) ( ) ( ) ( ) , (

3 4 3 3 3 2 3 1 3 0

44 43 42 41 40

34 33 32 31 30

24 23 22 21 20

14 13 12 11 10

04 03 02 01 00

3 4 3

3 3

2 3

1 3

0

v N

v N u

N u N u N u N u N v u

d d d d d

r



 



⁵

0 5

0

3 3

( ) ( )

j i

ij

N u N v

d

 

0

   

1

   

2

           

3 3 3 3 3

0 1 2 3 3 4 4

, v N u v N u v N u v N u

u v    d   d v N u

r d d d

         

   

 

0

3 3 3 3 3

0 4

1 2

3 4

N u N u N u N

3

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

- Chap3. Surface ^22/52

(23)

@ SDAL

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

d10

d40

d00 d²⁰

d30

00

u u40.4 u51

u2¹

u

u3 uu⁷⁶

u8

u

d14

d44

d04

d24

d34

00 v v2¹

v v3

51 v v7⁶

v v8

v4

d03

d02

d01

)

0(v d

)

1(v

d d₂(v)

)

3(v d

d43

d42

d41

)

4(v d

) , ( v u r

(3차)

(9개)

(u 방향 5개, v 방향 5개)

23/52

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@ SDAL

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

d10

d40

d00 d²⁰

d30

00

u u40.4 u51

u2¹

u

u3 uu⁷⁶

u8

u

d14

d44

d04

d24

d34

00 v v2¹

v v3

51 v v7⁶

v v8

v4

d03

d02

d01

)

0(v d

)

1(v

d d₂(v)

)

3(v d

d43

d42

d41

)

4(v d

) , ( v u r

u 방향 Knot

v 방향 Knot

24/52

(25)

@ SDAL

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

d10

d40

d00 d²⁰

d30

00

u u40.4 u51

u2¹

u

u3 uu⁷⁶

u8

u

d14

d44

d04

d24

d34

00 v v2¹

v v3

51 v v7⁶

v v8

v4

d03

d02

d01

)

0(v d

)

1(v

d d₂(v)

)

3(v d

d43

d42

d41

)

4(v d

) , ( v u r

B-spline Basis Function 계산

(Cox-de Boor Recurrence Formula)

) ( )

( )

( ¹ ₁ ¹

1 1

1 N u

u u

u u u

u N u

u u u

N _i ⁿ

i n i

n n i

i i n

i n i

i





 







 



 

 

  

 ^

else

0 if ) 1

( ¹

0 i i

i

u u u u

N

u, v 방향을 구분하는 flag

25/52