Ribs and FansRibs and Fans of Bézier Curves and Surfaces

Reporter: Dongmei Zhang

2007.11.21

Papers (Joo-Haeng Lee and Hyungjun Park)

• Ribs and Fans of Bézier Curves and Surfaces Computer-Aided Design & Applications 2005

• Geometric Properties of Ribs and Fans of a Bézier Curve

CKJC 2006, Hangzhou

• A Note on Morphological Development and Transformation of Bézier Curves based on Ribs and Fans

SPM 2007, Beijing

Definition and Decomposition

Ribs and Fans

of Bézier Curves and Surfaces

Computer-Aided Design & Applications 2005

Ribs and Fans

curve surface

Ribs of a Bézier Curve

• A Bézier curve:

• Rib control points:

• Rib (a Bézier curve of degree k):

0

( ) ( )n

ni i

i

b t b B t

, 0,1,...ni ib r i n

1 11 , 0 , 1 1k k k

i i i

k i ir r r i k k n

k k

0

( ) ( ), 1 1k

k k ki i

i

r t r B t k n

1k

ir

11kir

12kir

11kir

Examples (cubic Bézier curve)

1 11 , 0

k k ki i i

k i ir r r i k

k k

A cubic Bézier curve

Control points of ribs Rib curves

2 3 2 3 3 2 30 0 1 1 2 2 3

1, ( ),

2r r r r r r r

Especial property

0 0 , , 1k n k nk nr r r r k n

Fans of a Bézier Curve

• A Bézier curve:

• Fan control vectors:

• Fan (“a Bézier curve” of degree k):

0

( ) ( )n

ni i

i

b t b B t

, 0,1,...ni ib r i n

2 2 21 2

1( ), 0 , 0 22

k k k ki i i if r r r i k k n

0

( ) ( ), 0 2k

k k ki i

i

f t f B t k n

1k

ir

11kir

12kir

11kir

Examples (cubic Bézier curve)

Control vectors Fans

2 2 21 2

1( ), 02

k k k ki i i if r r r i k

1 3 3 30 1 0 2

1( )2

f r r r

1 3 3 31 2 1 3

1( )2

f r r r

0 2 2 20 1 0 2

1( )2

f r r r

Decomposition

• Theorem: A Bézier curve of degree n can be decomposed into a rib of degree n-1 and a fan of degree n-2.

1 2( ) ( ) ( ) 2 (1 ) ( )n n nb t r t r t t t f t

Proof (mathematical induction)

• Base step: n=21 2 1 20 0 1 2,r r r r

11 1 1 2 2

0 20

( ) ( ) (1 )i ii

r t r B t t r tr

0 2 2 20 1 0 2

1( )2

f r r r 0 0 2 2 20 1 0 2

1( ) ( )

2f t f r r r

1 0

2 2 2 2 20 2 1 0 2

2 2 2 2 20 1 2

2

( ) 2 (1 ) ( )

1(1 ) 2 (1 )( ( ))

2

(1 ) 2 (1 )

( )

r t t t f t

t r tr t t r r r

t r t t r t r

r t

Proof

• Induction hypothesis (n=k):

• n=k+1 :

1 2( ) ( ) 2 (1 ) ( )k k kr t r t t t f t

1 1( ) ( ) 2 (1 ) ( )k k kr t r t t t f t

11 1 1 11 2

0

1( ( )) ( )

2

kk k k ki i i i

i

r r r B t

1

1 1 1

0

( ) ( )k

k k ki i

i

f t f B t

Proof

0

( ) ( )k

k k ki i

i

r t r B t

1 11

0

1(( ) ) ( )

kk k ki i i

i

k i r ir B tk

1 11

0 0

1 1( ) ( ) ( )

k kk k k ki i i i

i i

k i r B t ir B tk k

1 1

1( ) (1 ) ( ), ( ) ( ), 0n n n ni i i i

n nB t t B t B t tB t i n

n i i

11 1 1 1

1 10 1

1 1( ) (1 ) ( ) ( )

k kk k k ki i i i

i i

kk i r t B t ir tB t

k k i k

1 11 1 1 1

20 0

(1 ) ( ) ( )k k

k k k ki i i i

i i

r t B t ir tB t

1

1 1 12

0

((1 ) ) ( )k

k k ki i i

i

t r tr B t

1( ) 2 (1 ) ( )k kr t t t f t 1( )kr t

Proof1( ) 2 (1 ) ( )k kr t t t f t

1 11 1 1 1 1 1 1

2 1 20 0

1((1 ) ) ( ) 2 (1 ) ( ( )) ( )

2

k kk k k k k k ki i i i i i i

i i

t r tr B t t t r r r B t

1

2 1 1 2 1 11 2

0

((1 ) 2 (1 ) ) ( )k

k k k ki i i i

i

t r t t r t r B t

1 2

1 2 1

0 0

( ( )) ( )k

k ki j j i

i j

r B t B t

2 2 21 2 1 1 2 1 1 2 1

0 0 1 10 0 0

( ( ) ( )) ... ( ( ) ( )) ... ( ( ) ( ))k k k k k kj j i j i j k j k j

j j j

r B t B t r B t B t r B t B t

1 2

1 2 1

0 0

( ( ) ( ))k

k ki j i j

i j

r B t B t

1

1( ) (1 ) ( ) ( )n n ni i iB t t B t tB t

2

1 2 1 1 2 1 2 11 2

0

( ) (1 ) ( ) 2 (1 ) ( ) ( ) ( ) ( )n n n n ni i i i j i j

j

B t t B t t t B t t B t B t B t

1( )kr t

Decomposition

• Theorem: A Bézier curve of degree n can be decomposed into a rib of degree n-1 and a fan of degree n-2.

1 2( ) ( ) ( ) 2 (1 ) ( )n n nb t r t r t t t f t

Decomposition

• Corollary: A Bézier curve of degree n can be decomposed into a single rib of degree l and a sequence of n-l fans of degrees from n-2 to l-1.

2

1

( ) ( ) ( ) 2 (1 ) ( ),

1 1

nn l k

k l

b t r t r t t t f t

l n

Surface case, ,

,0 0

( , ) ( ) ( )m n

m n m n m ni j i j

i j

r u v r B u B v

, , , ,

0, , *, ,0

{ ,..., } ( ) ( )m

m n m n m n m n mj m j j i j i

i

r r r u r B u

i 1i

2i 1i

1j

j

1j

2j

, , , ,,0 , ,* ,

0

{ ,..., } ( ) ( )n

m n m n m n m n ni i n i i j j

j

r r r v r B v

, 1, 2,*, *, *,( , ) ( , ) 2 (1 ) ( , )m n m n m nj j jr u v r u v u u p u v

1, , ,, , 1,

1(( 1 ) ), 0 11

m n m n m ni j i j i jr m i r ir i m

m

2, , , ,, 1, , 2,

1( ), 0 22

m n m n m n m ni j i j i j i jp r r r i m

, , 1 , 2,* ,* ,*( ) ( ) 2 (1 ) ( )m n m n m ni i ir v r v v v q v

, 1 , ,, , , 1

1(( 1 ) ), 0 11

m n m n m ni j i j i jr n j r jr j n

n

, 2 , , ,, , 1 , , 2

1( ), 0 22

m n m n m n m ni j i j i j i jq r r r j n

Composite transformation

Rib and its control points, ,

,0 0

( , ) ( ) ( )m n

m n m n m ni j i j

i j

r u v r B u B v

i 1i

2i 1i

1j

j

1j

2j

, 1, 1 1, 1, , 1,

1, 1 1, 1, 1 1, 1

1(( )(( ) )

(( ) ))

1 ,1

k l k l k li j i j i j

k l k li j i j

r l j k i r irkl

j k i r ir

k n l m

，

, , , ,0,0 0,0 ,0 ,0

, , , ,0, 0, , ,

,

,

k l m n k l m nk m

k l m n k l m nl n k l m n

r r r r

r r r r

Fan and its control vectors, ,

,0 0

( , ) ( ) ( )m n

m n m n m ni j i j

i j

r u v r B u B v

, 1, 2 1, 2 2, 1, , 1 , , 2

1, 2 1, 2 1, 21, 1 1, 1, 2

1 1(( )( ( ))

21

( ( ))),2

1 1,0 2

k l k l k l k li j i j i j i j

k l k l k li j i j i j

q k i r r rk

i r r r

k m l n

i 1i

2i 1i

1j

j

1j

2j , 2, 2 2, 2 2, 2, 1, 1 , 1 2, 1

2, 2 2, 2 2, 21, , 2,

2, 2 2, 2 2, 21, 2 , 2 2, 2

1( ( ))

21 1(( ( ))2 2

1( ( ))),

20 2,0 2

k l k l k l k li j i j i j i j

k l k l k li j i j i j

k l k l k li j i j i j

f r r r

r r r

r r r

k m l n

Decomposition

• Theorem: A Bézier surface of degree (m,n) can be decomposed into a rib of degree (m-1,n-1) and three fans.

, 1, 1 2, 1 1, 2

2, 2

1, 1 2, 2

( , ) ( , ) 2 (1 ) ( , ) 2 (1 )

4 (1 )(1 ) ( , )

( , ) ( , )

m n m n m n m n

m n

m n m n

r u v r u v u u p u v v v q

uv u v f u v

r u v u v

Proof

, ,,

0 0

( , ) ( ) ( )m n

m n m n m ni j i j

i j

r u v r B u B v

固定 v

在 u方向上

固定 u

在 v方向上, 1, 1 2, 2( , ) ( , ) ( , )m n m n m nr u v r u v u v

Decomposition

• Corollary: A Bézier surface of degree (m,n) can be decomposed into a single rib of degree (m-k,n-k) and a sequence of k composite fans.

, , 1 , 1

1

( , ) ( , ) ( , ),

1 ( , ) 1

km n m k n k m i n i

i

r u v r u v u v

k MIN m n

Example(bi-cubic Bézier surface)

Example(bi-cubic Bézier surface)

Examples (Bézier curve of degree 9)

• d

Examples (Bézier curve of degree 10)

Geometric Properties

Geometric Properties

of Ribs and Fans of a Bézier Curve

CKJC 2006, Hangzhou

• Composite fans

• Rib-invariant deformation

Composite fans

• Property 1: A Bézier curve of degree n can be composed into a straight line segment and a composite fan of degree n-2.

22 2 21 2

0

( ) ( ) 2 (1 ) ( ), ( ) ( )nn n nn n

iii

r t r t t t f t f t f B t

21

0

( ) ( ) 2 (1 ) ( )n

n r

r

r t r t t t f t

0

( ) ( )r

r r ri i

i

f t f B t

degree elevation

2 21 2

0

( ) ( ) 2 (1 ) ( )n nn n

iii

r t r t t t f B t

Composite fans

• Property 2: A straight line segment and a composite fan of degree n-2 can build a unique Bézier curve of degree n.

21

0

21 10 1 1

( ) 2 (1 ) ( ) ( ),

1 1 2 ( )( )

1 ( 1)

nn n ni i

i

nni i

r t t t f t r B t

n n n i n ir r r f

i i i n n

Proof

,1( )

nr t1( )r t

21

( 1)( 1)(1 ) ( ) ( )

( 1)( 2)n ni i

i n it t B t B t

n n

2 2 2

0

2 (1 ) ( )n n n

iii

t t f B t

0

* ( )n

ni

i

B t

21

0

21 10 1 1

( ) 2 (1 ) ( ) ( ),

1 1 2 ( )( )

1 ( 1)

nn n ni i

i

nni i

r t t t f t r B t

n n n i n ir r r f

i i i n n

degree elevation

Rib-invariant Deformation

• Property 3:

For a given Bézier curve of degree n, we can modify up to n-d control points while preserving a rib of degree d.

Moreover, if we specify n-d control points explicitly, we can determine the unknown d-1 control points uniquely.

Proof

• Initial Bézier curve:

• Rib of degree d:

• New Bézier curve:

0 0

, 0,1...,n n

j n ji i i i

i i

b l r l j d

0

( ) ( ), 0,1,...n

n n ni i

i

r t r B t i n

0

, 0,1...,n

d n jj i i

i

r r l j d

0

( ) ( )n

ni i

i

b t b B t

Example (quartic Bézier curve)

Example (curve of degree 9)

Applications

A Note on Morphological Development and Transformation of Bézier

Curves

based on Ribs and Fans

SPM 2007, Beijing

Morphological development

• To find a sequence of shapes that believed to represent a pattern of growth.

Morphological transformation

• To find a sequence curves that represents the pattern from one curve to another.

Morphological development

• Current shape:• Initial shape (simple, minimum features):• Developmental pattern:

( ) ( )nb s r s1( )r s

1( , ) ( ) ( , )d s t r s p s t

1( ,0) ( ), ( ,1) ( )nd s r s d s r s

DCF (development by composite fan)

• Linear development:1

,

21

1

( , ) (1 ) ( ) ( )

( ) (2 (1 ) ( ))

( ) ( , )

nc f

n

cf

d s t t r s tr s

r s t s s f s

r s p s t

, ( , ; , )

(1 ) ( ) ( )

( ) ( , )

n qc f

q n

qcf

d s t r r

t r s tr s

r s p s t

DCF(development by composite fan)

DFL(development by fan lines)

• Utilize each rib:1( , ; , ) ( ) ( )

nn q i

fl i qi q

d s t r r r s N t

DFL(development by fan lines)

DFL(development by fan lines)

DSC(development by spline curves)

• path: a smooth curve. ( , ; , ) ( ) ( )n

n q i n qsc i q

i q

d s t r r r s B t

DSC (development by spline curves)

Comparision

Morphological transformation

• Three methods (TLI,TCE,TDE).

• TLI (Transformation by linear interpolation). correspondence: index of control points.

TCE(by cubic blending and extrapolation)

• Two Bézier curves:• Lower ribs:

3 2 20 0 1( , ) (1 ) ( , ) ( , )

( , )c

t s t t t s t tt s t

t s t

( ), ( )m na br s r s

( ), ( )p qa br s r s

3 3 3 30 1 2 3( , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )m p q n

c a a b bt s t B t r s B t r s B t r s B t r s

10

11

12

( , ) (1 ) ( ) ( )

( , ) (1 ) ( ) ( )

( , ) (1 ) ( ) ( )

m pa a

p qa b

q nb b

t s t t r s tr s

t s t t r s tr s

t s t t r s tr s

2 1 10 0 1

2 1 11 1 2

( , ) (1 ) ( , ) ( , )

( , ) (1 ) ( , ) ( , )

t s t t t s t tt s t

t s t t t s t tt s t

1 31 0( , ) (1 ( )) ( , ) ( ) ( , )

( ) 1 (1 3 (1 ))cet s t e t t s t e t t s t

e t t t

TDE(by development and extrapolation)

10

11

12

( , ) (1 ) ( ) ( )

( , ) (1 ) ( ) ( )

( , ) (1 ) ( ) ( )

m pa a

p qa b

q nb b

t s t t r s tr s

t s t t r s tr s

t s t t r s tr s

2 1 2 2, 0 1 1 2( , ) ( , ; , ) ( ) ( , ) ( ) ( , ; , ) ( )m nd sc sc a sc bt s t d s t r p B t t s t B t d s t r q B t

3 1 2 1 2 1 20 0 0 1 1 2 2( , ) ( , ) ( ) ( , ) ( ) ( , ) ( )t s t t s t B t t s t B t t s t B t

2 1 2 2, 0 1 1 2( , ) ( , ; , ) ( ) ( , ) ( ) ( , ; , ) ( )m nd fl fl a fl bt s t d s t r p B t t s t B t d s t r q B t

1,* 1 ,*( , ) (1 ( )) ( , ) ( ) ( , )de dt s t e t t s t e t t s t

( ) 1 (1 (1 ))e t t t

( ) 1 (1 3 (1 ))e t t t

Comparision

Summary

• Definition: Ribs and Fans

• Decomposition. Bézier curve = rib + scaled fans.

• Property. Rib + composite fan= Bézier curve=rib.

Rib-invariant deformation.

• Morphological applications Development and transformation.

Thanks