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G e o m e t r i c G e o m e t r i c G e o m e t r i c G e o m e t r i c M o d e l i n g M o d e l i n g M o d e l i n g M o d e l i n g a n d a n da n da n d P r o c e s s i n g P r o c e s s i n g P r o c e s s i n g P r o c e s s i n g 2 0 2 02 02 0 1 2 1 21 21 2
( G M P ( G M P ( G M P ( G M P 2 0 1 2 ) 2 0 1 2 ) 2 0 1 2 ) 2 0 1 2 ) J u n e J u n e J u n e J u n e 1 8 - 2 4 , 1 8 - 2 4 , 1 8 - 2 4 , 1 8 - 2 4 , 2 0 1 2 , 2 0 1 2 , 2 0 1 2 , 2 0 1 2 , H u a n g s h a n , H u a n g s h a n , H u a n g s h a n , H u a n g s h a n , C h i n a C h i n a C h i n a C h i n a
T u t o r i a l s T u t o r i a l s T u t o r i a l s T u t o r i a l s J u n e J u n e J u n e J u n e 1 8 - 1 9 , 1 8 - 1 9 , 1 8 - 1 9 , 1 8 - 1 9 , 2 0 1 2 2 0 1 2 2 0 1 2 2 0 1 2
U n i v e r s i t y U n i v e r s i t y U n i v e r s i t y U n i v e r s i t y o f o fo fo f S c i e n c e S c i e n c e S c i e n c e S c i e n c e a n d a n da n da n d T e c h n o l o g y T e c h n o l o g y T e c h n o l o g y T e c h n o l o g y o f o fo fo f C h i n a C h i n a C h i n a C h i n a
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T u t o r i a l s T u t o r i a l s T u t o r i a l s T u t o r i a l s o f o fo fo f G M P G M PG M PG M P 2 0 1 2 2 0 1 2 2 0 1 2 2 0 1 2
JJJJ uuuu n en en en e 1 8 1 81 81 8
9 : 3 0 a . m . - 1 1 : 3 0 a . m . K a i H o r m a n n G e n e r a l i z e d B a r y c e n t r i c C o o r d i n a t e s
1 1 : 3 0 a . m . - 1 : 3 0 p . m . L u n c h
1 : 3 0 p . m . - 3 : 3 0 p . m . D a v i d X i a n f e n g G u C o m p u t a t i o n a l C o n f o r m a l G e o m e t r y , T h e o r y ,
A l g o r i t h m a n d A p p l i c a t i o n
3 : 3 0 p . m . - 4 : 0 0 p . m . C o f f e e b r e a k
4 : 0 0 p . m . - 6 : 0 0 p . m . Y i n g H e
S h i - Q i n g X i n
D i s c r e t e G e o d e s i c s
JJJJ uuuu n en en en e 1 9 1 91 91 9
8 : 3 0 a . m . - 1 2 : 0 0 a . m . M y u n g S o o K i m
G e r s h o n E l b e r
N u m e r i c a n d S y m b o l i c P r o c e s s i n g i n G e o m e t r i c
M o d e l i n g
1 2 : 0 0 a . m . - 1 : 3 0 p . m . L u n c h
1 : 3 0 p . m . - 3 : 3 0 p . m . W e n p i n g W a n g C o m p u t a t i o n a n d A p p l i c a t i o n s o f C e n t r o i d a l V o r o n o i
T e s s e l l a t i o n s
3 : 3 0 p . m . - 4 : 0 0 p . m . C o f f e e b r e a k
4 : 0 0 p . m . - 5 : 0 0 p . m . B e r t J t t l e r I s o g e o m e t r i c A n a l y s i s
C o m m i t t e e C o m m i t t e e C o m m i t t e e C o m m i t t e e
C o n f e r e n c e C o n f e r e n c e C o n f e r e n c e C o n f e r e n c e C o - C h a i r s : C o - C h a i r s : C o - C h a i r s : C o - C h a i r s :
F a l a i C h e n U n i v e r s i t y o f S c i c e n c e a n d T e c h n o l o g y o f C h i n a , C h i n a
M y u n g - S o o K i m S e o u l N a i t o n a l U n i v e r s i t y , K o r e a
T u t o r i a l T u t o r i a l T u t o r i a l T u t o r i a l C o m m i t t e e : C o m m i t t e e : C o m m i t t e e : C o m m i t t e e :
L i g a n g L i u ( C h a i r , U S T C , C h i n a )
W e n p i n g W a n g ( T h e U n i v e r s i t y o f H o n g K o n g , C h i n a )
J i a n s o n g D e n g ( U S T C , C h i n a )
Z h o u w a n g Y a n g ( U S T C , C h i n a )
O r g a n i z a t i o n O r g a n i z a t i o n O r g a n i z a t i o n O r g a n i z a t i o n C o m m i t t e e : C o m m i t t e e : C o m m i t t e e : C o m m i t t e e :
Z h o u w a n g Y a n g ( C h a i r , U S T C , C h i n a )
W e i h u a T o n g ( U S T C , C h i n a ) X i n L i ( U S T C , C h i n a )
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1 .1 .1 .1 . T u t o r i a l T u t o r i a l T u t o r i a l T u t o r i a l 1 111 ( 9 : 3 0 a . m . - 1 1 : 3 0 a . m . , ( 9 : 3 0 a . m . - 1 1 : 3 0 a . m . , ( 9 : 3 0 a . m . - 1 1 : 3 0 a . m . , ( 9 : 3 0 a . m . - 1 1 : 3 0 a . m . , J u n e J u n e J u n e J u n e 1 8 ) 1 8 )1 8 )1 8 )
T i t l e : T i t l e : T i t l e : T i t l e : G e n e r a l i z e d B a r y c e n t r i c C o o r d i n a t e s
L e c t u r e : L e c t u r e : L e c t u r e : L e c t u r e : K a i H o r m a n n T i m e : T i m e : T i m e : T i m e : 9 : 3 0 a . m . - 1 1 : 3 0 a . m . , J u n e 1 8
A b s t r a c t : A b s t r a c t : A b s t r a c t : A b s t r a c t :
I n 1 8 2 7 , A u g u s t F e r d i n a n d M b i u s p u b l i s h e d h i s s e m i n a l w o r k o n t h e b a r y c e n t r i c c a l c u l w h i c h
p r o v i d e s a n o v e l a p p r o a c h t o a n a l y t i c g e o m e t r y . O n e k e y e l e m e n t i n h i s w o r k i s t h e i d e a o f
b a r y c e n t r i c c o o r d i n a t e s w h i c h a l l o w t o w r i t e a n y p o i n t i n s i d e a t r i a n g l e a s a u n i q u e c o n v e x
c o m b i n a t i o n o f t h e t r i a n g l e s v e r t i c e s . T h e s e t r i a n g u l a r b a r y c e n t r i c c o o r d i n a t e s a r e l i n e a r a n d p o s s e s s
t h e L a g r a n g e p r o p e r t y , a n d a r e t h e r e f o r e c o m m o n l y u s e d t o l i n e a r l y i n t e r p o l a t e v a l u e s g i v e n a t t h e
v e r t i c e s o f a t r i a n g l e . M b i u s a l s o n o t i c e d t h a t t h i s c o n s t r u c t i o n e x t e n d s n i c e l y t o l i n e a r i n t e r p o l a t i o n
o f d a t a g i v e n a t t h e v e r t i c e s o f a d - d i m e n s i o n a l s i m p l e x , a n d b y g i v i n g u p p o s i t i v i t y o f t h e c o o r d i n a t e s , w e c a n e v e n e x t r a p o l a t e t h e d a t a t o e v e r y p o i n t i n d d i m e n s i o n s .
W h i l e b a r y c e n t r i c c o o r d i n a t e s a r e u n i q u e f o r s i m p l i c e s , t h e y c a n b e g e n e r a l i z e d i n s e v e r a l w a y s t o
a r b i t r a r y p o l y g o n s a n d p o l y t o p e s i n h i g h e r d i m e n s i o n s , a n d o v e r t h e p a s t f e w y e a r s , a n u m b e r o f
r e c i p e s f o r s u c h g e n e r a l i z e d b a r y c e n t r i c c o o r d i n a t e s h a v e b e e n d e v e l o p e d . A s t h e y a r e u s u a l l y g i v e n
i n c l o s e d f o r m a n d c a n b e e v a l u a t e d e f f i c i e n t l y , t h e y h a v e m a n y u s e f u l a p p l i c a t i o n s , e . g . i n c o m p u t e r
g r a p h i c s , c o m p u t e r a i d e d g e o m e t r i c d e s i g n , a n d i m a g e p r o c e s s i n g .
I n t h i s t u t o r i a l , w e d i s c u s s t h e t h e o r e t i c a l b a c k g r o u n d o f g e n e r a l i z e d b a r y c e n t r i c c o o r d i n a t e s a n d
p r e s e n t h a n d s - o n e x a m p l e s , r a n g i n g f r o m c o l o u r i n t e r p o l a t i o n a n d i m p r o v e d P h o n g s h a d i n g t o i m a g e
w a r p i n g a n d m e s h d e f o r m a t i o n s .
SSSS y l l a b u s y l l a b u s y l l a b u s y l l a b u s : ::: ( 2 h o u r s w i t h a 2 0 - m i n u t e c o f f e e b r e a k )
1 . I n t r o d u c t i o n ( 5 m i n u t e s )
2 . T h e e s s e n c e o f b a r y c e n t r i c c o o r d i n a t e s ( 1 5 m i n u t e s )
3 . T h r e e - p o i n t c o o r d i n a t e s ( 1 5 m i n u t e s )
4 . B a r y c e n t r i c d a t a i n t e r p o l a t i o n ( 1 5 m i n u t e s )
C o f f e e b r e a k ( 2 0 m i n u t e s )
5 . T r a n s f i n i t e b a r y c e n t r i c i n t e r p o l a t i o n ( 1 5 m i n u t e s ) 6 . B a r y c e n t r i c m a p p i n g s ( 1 5 m i n u t e s )
7 . M e s h d e f o r m a t i o n s ( 1 5 m i n u t e s )
8 . W r a p - u p ( 5 m i n u t e s )
L e c t u r e r L e c t u r e r L e c t u r e r L e c t u r e r ssss s h o r t s h o r t s h o r t s h o r t b i o g r a p h y b i o g r a p h y b i o g r a p h y b i o g r a p h y : :::
K a iK a iK a iK a i H o r m a n n H o r m a n n H o r m a n n H o r m a n n i s a n a s s o c i a t e p r o f e s s o r i n t h e f a c u l t y o f i n f o r m a t i c s a t t h e U n i v e r s i t y o f L u g a n o . H e
r e c e i v e d a P h . D . i n c o m p u t e r s c i e n c e f r o m t h e U n i v e r s i t y o f E r l a n g e n i n 2 0 0 2 a n d s p e n t t w o y e a r s a s
a p o s t d o c t o r a l r e s e a r c h f e l l o w a t t h e M u l t i - R e s M o d e l i n g G r o u p a t C a l t e c h a n d t h e C N R I n s t i t u t e o f
I n f o r m a t i o n S c i e n c e a n d T e c h n o l o g i e s i n P i s a , b e f o r e j o i n i n g t h e f a c u l t y a t C l a u s t h a l U n i v e r s i t y o f
T e c h n o l o g y a s a n a s s i s t a n t p r o f e s s o r i n 2 0 0 4 . D u r i n g t h e w i n t e r t e r m 2 0 0 7 / 2 0 0 8 h e v i s i t e d t h e B e r l i n
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M a t h e m a t i c a l S c h o o l a t F r e i e U n i v e r s i t t B e r l i n a s a B M S s u b s t i t u t e p r o f e s s o r .
H i s r e s e a r c h i n t e r e s t s a r e f o c u s s e d o n t h e m a t h e m a t i c a l f o u n d a t i o n s o f g e o m e t r y p r o c e s s i n g
a l g o r i t h m s a s w e l l a s t h e i r a p p l i c a t i o n s i n c o m p u t e r g r a p h i c s a n d r e l a t e d f i e l d s . I n p a r t i c u l a r , h e i s
w o r k i n g o n p a r a m e t e r i z a t i o n o f m e s h e s , s u r f a c e r e c o n s t r u c t i o n f r o m p o i n t c l o u d s , b a r y c e n t r i c c o o r d i n a t e s f o r a r b i t r a r y p o l y g o n s , a n d s u b d i v i s i o n o f c u r v e s a n d s u r f a c e s .
P r o f . H o r m a n n h a s p u b l i s h e d o v e r 5 0 p a p e r s i n t h e p r o f e s s i o n a l l i t e r a t u r e a n d i s a n a c t i v e m e m b e r o f
A C M S i g g r a p h a n d S I A M . H e s e r v e d o n m o r e t h a n 2 0 o f t h e l e a d i n g g r a p h i c s a n d g e o m e t r y
c o n f e r e n c e p r o g r a m m e c o m m i t t e e s a n d i s a f r e q u e n t r e v i e w e r f o r i n t e r n a t i o n a l f u n d i n g a g e n c i e s a n d
t h e t o p j o u r n a l s i n h i s f i e l d . M o r e o v e r , h e i s a n a s s o c i a t e e d i t o r o f C o m p u t e r A i d e d G e o m e t r i c
D e s i g n , C o m p u t e r G r a p h i c s F o r u m , a n d t h e D o l o m i t e s R e s e a r c h N o t e s o n A p p r o x i m a t i o n .
L i n k s t o ( a l m o s t ) a l l p a p e r s o n t h e t o p i c c a n b e f o u n d o n t h e w e b p a g e : h t t p : / / w w w . i n f . u s i . c h / h o r m a n n / b a r y c e n t r i c / i n d e x . h t m l h t t p : / / w w w . i n f . u s i . c h / h o r m a n n / b a r y c e n t r i c / i n d e x . h t m l h t t p : / / w w w . i n f . u s i . c h / h o r m a n n / b a r y c e n t r i c / i n d e x . h t m l h t t p : / / w w w . i n f . u s i . c h / h o r m a n n / b a r y c e n t r i c / i n d e x . h t m l
P r o f . H o r m a n n p l a n s t o p r e p a r e s o m e h a n d s - o n e x e r c i s e s o r d e m o n s t r a t i o n s i n M a x i m a , w h i c h i s
s i m i l a r t o M a p l e b u t f r e e a n d a v a i l a b l e o n b a s i c a l l y a l l p l a t f o r m s . T h e a t t e n d e e s c a n d o w n l o a d a n d
i n s t a l l i t f r o m w e b p a g e :
h t t p : / / m a x i m a . s o u r c e f o r g e . n e t / h t t p : / / m a x i m a . s o u r c e f o r g e . n e t / h t t p : / / m a x i m a . s o u r c e f o r g e . n e t / h t t p : / / m a x i m a . s o u r c e f o r g e . n e t /
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2 .2 .2 .2 . T u t o r i a l T u t o r i a l T u t o r i a l T u t o r i a l 2 222 ( 1 : 3 0 p . m . - 3 : 3 0 p . m . , ( 1 : 3 0 p . m . - 3 : 3 0 p . m . , ( 1 : 3 0 p . m . - 3 : 3 0 p . m . , ( 1 : 3 0 p . m . - 3 : 3 0 p . m . , J u n e J u n e J u n e J u n e 1 8 ) 1 8 )1 8 )1 8 )
T i t l e : T i t l e : T i t l e : T i t l e : C o m p u t a t i o n a l C o n f o r m a l G e o m e t r y , T h e o r y , A l g o r i t h m a n d A p p l i c a t i o n
L e c t u r e : L e c t u r e : L e c t u r e : L e c t u r e : D a v i d X i a n f e n g G u T i m e : T i m e : T i m e : T i m e : 1 : 3 0 p . m . - 3 : 3 0 p . m . , J u n e 1 8
A b s t r a c t : A b s t r a c t : A b s t r a c t : A b s t r a c t :
A c c o r d i n g t o K l e i n s E r l a n g e n p r o g r a m : d i f f e r e n t g e o m e t r i e s s t u d y t h e i n v a r i a n t s u n d e r d i f f e r e n t
t r a n s f o r m a t i o n g r o u p s . I n G e o m e t r i c M o d e l i n g a n d P r o c e s s i n g f i e l d , t o p o l o g y , R i e m a n n i a n g e o m e t r y
a n d s u r f a c e d i f f e r e n t i a l g e o m e t r y h a v e b e e n e x t e n s i v e l y a p p l i e d . T h i s t u t o r i a l f o c u s e s o n c o n f o r m a l
g e o m e t r y , w h i c h s t u d i e s t h e i n v a r i a n t s u n d e r a n g l e p r e s e r v i n g t r a n s f o r m a t i o n g r o u p . C o n f o r m a l
g e o m e t r y i s m o r e r i g i d t h a n t o p o l o g y a n d m o r e f l e x i b l e t h a n R i e m a n n i a n g e o m e t r y , w h i c h o f f e r s
p o w e r f u l t o o l s f o r f u n d a m e n t a l p r o b l e m s i n g e o m e t r i c m o d e l l i n g a n d p r o c e s s i n g , i n c l u d i n g s h a p e
m a t c h i n g , r e g i s t r a t i o n , g e o m e t r i c c l a s s i f i c a t i o n a n d s o o n .
C o n f o r m a l g e o m e t r i c m e t h o d s h a v e t h e f o l l o w i n g m e r i t s :
1 . U n i f i c a t i o n : A l l t h e s u r f a c e s i n r e a l l i f e c a n b e e v e n t u a l l y u n i f i e d t o o n e o f t h r e e c a n o n i c a l
s h a p e s , t h e s p h e r e , t h e p l a n e o r t h e h y p e r b o l i c d i s k .
2 . D i m e n s i o n R e d u c t i o n : A l l 3 D g e o m e t r i c p r o c e s s i n g p r o b l e m s a r e c o n v e r t e d t o 2 D i m a g e
p r o c e s s i n g p r o b l e m s .
3 . I n f o r m a t i o n P r e s e r v a t i o n : A l l t h e d e f o r m a t i o n s p r e s e r v e s t h e o r i g i n a l g e o m e t r i c i n f o r m a t i o n ,
t h e r e i s n o i n f o r m a t i o n l o s s d u r i n g t h e t r a n s f o r m a t i o n .
4 . G e n e r a l T r a n s f o r m a t i o n s : C o n f o r m a l g e o m e t r i c m e t h o d i s c a p a b l e o f m o d e l l i n g a n d
p r o c e s s i n g a l l t h e h o m e o m o r p h i s m s a m o n g s u r f a c e s .
T h e r e f o r e , c o n f o r m a l g e o m e t r i c m e t h o d s a r e v a l u a b l e f o r p r a c t i c a l a p p l i c a t i o n s .
T h i s t a l k w i l l i n t r o d u c e t h e m a i n t h e o r e m s i n c o n f o r m a l g e o m e t r y , i n c l u d i n g H o d g e t h e o r y , s u r f a c e
u n i f o r m i z a t i o n t h e o r e m , R i c c i f l o w t h e o r y ; m a i n c o m p u t a t i o n a l a l g o r i t h m s , i n c l u d i n g h a r m o n i c
m a p p i n g , h o l o m o r p h i c f o r m s , d i s c r e t e R i c c i f l o w ; a n d m a i n a p p l i c a t i o n s , i n c l u d i n g s u r f a c e
p a r a m e t e r i z a t i o n , s p l i n e f i t t i n g , s u r f a c e r e g i s t r a t i o n , g e o m e t r i c c l a s s i f i c a t i o n .
SSSS y l l a b u s y l l a b u s y l l a b u s y l l a b u s : ::: ( 2 h o u r s )
1 . I n t r o d u c t i o n ( 8 m i n u t e s ) 2 . P h a s e - S h i f t i n g G e o m e t r i c A c c q u i s i t i o n ( 8 m i n u t e s )
3 . H o m o t o p y T h e o r y ( 8 m i n u t e s )
4 . H o m o l o g y / C o h o m o l o g y T h e o r y ( 8 m i n u t e s )
5 . H o d g e T h e o r y ( 8 m i n u t e s )
6 . H a r m o n i c M a p p i n g ( 8 m i n u t e s )
7 . H o l o m o r p h i c f o r m M e t h o d ( 8 m i n u t e s )
8 . R i c c i F l o w T h e o r y ( 8 m i n u t e s )
9 . S u r f a c e u n i f o r m i z a t i o n T h e o r e m ( 8 m i n u t e s )
1 0 . B e l t r a m i - E q u a t i o n a n d Q u a s i - C o n f o r m a l M a p p i n g ( 8 m i n u t e s )
1 1 . C o n f o r m a l M o d u l e / T e i c h m u l l e r S p a c e ( 8 m i n u t e s )
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1 2 . S u r f a c e P a r a m e t e r i z a t i o n ( 8 m i n u t e s )
1 3 . M a n i f o l d S p l i n e ( 8 m i n u t e s )
1 4 . C o n f o r m a l W e l d i n g f o r 2 D S h a p e S p a c e ( 8 m i n u t e s )
1 5 . S u r f a c e R e g i s t r a t i o n ( 8 m i n u t e s )
L e c t u r e r L e c t u r e r L e c t u r e r L e c t u r e r ssss s h o r t s h o r t s h o r t s h o r t b i o g r a p h y b i o g r a p h y b i o g r a p h y b i o g r a p h y : :::
D a v i d D a v i d D a v i d D a v i d X i a n f e n g X i a n f e n g X i a n f e n g X i a n f e n g G u G uG uG u g o t h i s P h D f r o m H a r v a r d i n 2 0 0 2 i n C o m p u t e r S c i e n c e a n d n o w i s a n a s s o c i a t e
p r o f e s s o r i n S t a t e U n i v e r s i t y o f N e w Y o r k a t S t o n y B r o o k . I n h i s P h D t h e s i s , u n d e r t h e s u p e r v i s i o n
o f h i s a d v i s o r , a F i e l d s m e d a l i s t P r o f . S h i n g - T u n g Y a u , D a v i d i n v e n t e d t h e a l g o r i t h m f o r c o m p u t i n g
t h e c o n f o r m a l s t r u c t u r e o f s u r f a c e s w i t h g e n e r a l t o p o l o g i e s b a s e d o n H o d g e t h e o r y . L a t e r , b y
c o l l a b o r a t i n g w i t h m a t h e m a t i c i a n s , D a v i d i n v e n t e d t h e a l g o r i t h m s f o r c o m p u t i n g s u r f a c e
u n i f o r m i z a t i o n b a s e d o n R i c c i f l o w t h e o r y . B y c o l l a b o r a t i n g w i t h c o m p u t e r s c i e n t i s t s a n d m e d i c a l
d o c t o r s , D a v i d h a s a p p l i e d c o m p u t a t i o n a l c o n f o r m a l m e t h o d s i n m a n y f i e l d s i n e n g i n e e r i n g a n d
m e d i c i n e , i n c l u d i n g c o m p u t e r g r a p h i c s , c o m p u t e r v i s i o n , g e o m e t r i c m o d e l l i n g a n d p r o c e s s i n g ,
w i r e l e s s s e n s o r n e t w o r k , c o m p u t a t i o n a l t o p o l o g y a n d m e d i c a l i m a g i n g .
T h e l e c t u r e n o t e s c a n b e d o w n l o a d e d f r o m m y w e b p a g e :
h t t p : / / w w w . c s . s u n y s b . e d u / ~ g u / l e c t u r e s / i n d e x . h t m l h t t p : / / w w w . c s . s u n y s b . e d u / ~ g u / l e c t u r e s / i n d e x . h t m l h t t p : / / w w w . c s . s u n y s b . e d u / ~ g u / l e c t u r e s / i n d e x . h t m l h t t p : / / w w w . c s . s u n y s b . e d u / ~ g u / l e c t u r e s / i n d e x . h t m l
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3 .3 .3 .3 . T u t o r i a l T u t o r i a l T u t o r i a l T u t o r i a l 3 333 ( 4 : 0 0 p . m . - 6 : 0 0 p . m . , ( 4 : 0 0 p . m . - 6 : 0 0 p . m . , ( 4 : 0 0 p . m . - 6 : 0 0 p . m . , ( 4 : 0 0 p . m . - 6 : 0 0 p . m . , J u n e J u n e J u n e J u n e 1 8 ) 1 8 )1 8 )1 8 )
T i t l e : T i t l e : T i t l e : T i t l e : D i s c r e t e G e o d e s i c s
L e c t u r e L e c t u r e L e c t u r e L e c t u r e r s
r sr sr s :
::: Y i n g H e a n d S h i - Q i n g X i n T i m e : T i m e : T i m e : T i m e : 4 : 0 0 p . m . - 6 : 0 0 p . m . , J u n e 1 8
A b s t r a c t : A b s t r a c t : A b s t r a c t : A b s t r a c t :
C o m p u t i n g d i s c r e t e g e o d e s i c s o n p o l y g o n a l m e s h e s p l a y s a n i m p o r t a n t r o l e i n m a n y a p p l i c a t i o n s i n
c o m p u t e r g r a p h i c s a n d c o m p u t a t i o n a l g e o m e t r y , s u c h a s s u r f a c e p a r a m e t e r i z a t i o n , r e m e s h i n g ,
n o n - r i g i d r e g i s t r a t i o n , s h a p e s e g m e n t a t i o n , e t c . T h i s t u t o r i a l c o v e r s b o t h t h e f u n d a m e n t a l a n d
s t a t e - o f - t h e - a r t t e c h n i q u e s i n d i s c r e t e g e o d e s i c s , w h i c h i n c l u d e g e o d e s i c d i s t a n c e f i e l d , g e o d e s i c
p a t h s / l o o p s , g e o d e s i c o f f s e t s , a n d a l l - p a i r s g e o d e s i c d i s t a n c e q u e r y . S o m e d e m o s a r e a l s o s h o w n i n
t h e t u t o r i a l t o i l l u s t r a t e h o w u s e r i n t e r a c t i o n e n h a n c e s t h e m o d e l i n g a n d p r o c e s s i n g o f g e o m e t r i c
s h a p e s .
SSSS y l l a b u s y l l a b u s y l l a b u s y l l a b u s : ::: ( 2 h o u r s w i t h a 2 0 - m i n u t e c o f f e e b r e a k )
1 . I n t r o d u c t i o n ( 5 m i n u t e s )
2 . C o n c e p t s & d a t a s t r u c t u r e s ( 1 5 m i n u t e s )
3 . C l a s s i c a l d i s c r e t e g e o d e s i c a l g o r i t h m s ( 3 0 m i n u t e s )
4 . A l l - p a i r s g e o d e s i c s ( 1 0 m i n u t e s )
C o f f e e b r e a k ( 2 0 m i n u t e s )
5 . G e o d e s i c p a t h s / l o o p s ( 2 0 m i n u t e s )
6 . G e o d e s i c o f f s e t s ( 1 5 m i n u t e s )
7 . W r a p - u p ( 5 m i n u t e s )
L e c t u r e r L e c t u r e r L e c t u r e r L e c t u r e r ssss s h o r t s h o r t s h o r t s h o r t b i o g r a p h y b i o g r a p h y b i o g r a p h y b i o g r a p h y : :::
Y i n g Y i n g Y i n g Y i n g H e H eH eH e i s a n A s s o c i a t e P r o f e s s o r a t S c h o o l o f C o m p u t e r E n g i n e e r i n g , N a n y a n g T e c h n o l o g i c a l
U n i v e r s i t y . H i s r e s e a r c h i n t e r e s t s f a l l i n t o t h e g e n e r a l a r e a s o f v i s u a l c o m p u t i n g a n d h e i s p a r t i c u l a r l y
i n t e r e s t e d i n t h e p r o b l e m s t h a t r e q u i r e g e o m e t r i c a n a l y s i s a n d c o m p u t a t i o n . H e h a s a p p l i e d g e o m e t r i c
a p p r o a c h e s t o a w i d e s p e c t r u m o f e n g i n e e r i n g f i e l d s , i n c l u d i n g c o m p u t e r g r a p h i c s , m u l t i m e d i a ,
w i r e l e s s s e n s o r n e t w o r k a n d m e d i c a l i m a g i n g . F o r d e t a i l s , p l e a s e v i s i t
h t t p : / / w w w . n t u . e d u . s g / h o m e / y h e h t t p : / / w w w . n t u . e d u . s g / h o m e / y h e h t t p : / / w w w . n t u . e d u . s g / h o m e / y h e h t t p : / / w w w . n t u . e d u . s g / h o m e / y h e
S h i - Q i n g S h i - Q i n g S h i - Q i n g S h i - Q i n g X i n X i nX i nX i n g o t h i s P h D d e g r e e i n Z h e j i a n g U n i v e r s i t y i n 2 0 0 9 . A f t e r t h a t , h e h a s b e e n w o r k i n g a s
a r e s e a r c h f e l l o w a t N a n y a n g T e c h n o l o g i c a l U n i v e r s i t y i n S i n g a p o r e , w o r k i n g w i t h D r . Y i n g H e . I n
t h e p a s t s e v e r a l y e a r s , h e f o c u s e d o n d i s c r e t e g e o d e s i c a n d i t s a p p l i c a t i o n s . H i s r e s e a r c h i n t e r e s t s
i n c l u d e c o m p u t e r g r a p h i c s , c o m p u t a t i o n a l g e o m e t r y a n d m o b i l e c o m p u t i n g .
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4 .4 .4 .4 . T u t o r i a l T u t o r i a l T u t o r i a l T u t o r i a l 4 444 ( 8 : 3 0 a . m . - 1 2 : 0 0 a . m . , ( 8 : 3 0 a . m . - 1 2 : 0 0 a . m . , ( 8 : 3 0 a . m . - 1 2 : 0 0 a . m . , ( 8 : 3 0 a . m . - 1 2 : 0 0 a . m . , J u n e J u n e J u n e J u n e 1 9 ) 1 9 )1 9 )1 9 )
T i t l e : T i t l e : T i t l e : T i t l e : N u m e r i c a n d S y m b o l i c P r o c e s s i n g i n G e o m e t r i c M o d e l i n g
L e c t u r e r s : L e c t u r e r s : L e c t u r e r s : L e c t u r e r s : M y u n g S o o K i m a n d G e r s h o n E l b e r T i m e : T i m e : T i m e : T i m e : 8 : 3 0 a . m . - 1 2 : 0 0 a . m . , J u n e 1 9
A b s t r a c t : A b s t r a c t : A b s t r a c t : A b s t r a c t :
I n r e c e n t y e a r s s e v e r a l r e s e a r c h g r o u p s w e r e i n v e s t i g a t i n g t o o l s t h a t c a n m a k e g e o m e t r i c m o d e l i n g
m o r e s t a b l e a n d e f f i c i e n t . T o o l s t o p r o c e s s s p l i n e a n d N U R B s c u r v e s a n d s u r f a c e s i n v a r i o u s
o p e r a t i o n s w e r e d e s i g n e d t h a t a r e f a r m o r e r o b u s t .
I n t h i s t u t o r i a l w e w i l l s t a r t b y p r e s e n t i n g a s u b d i v i s i o n b a s e d s o l v e r , w h i c h i s b a s e d o n a s p l i n e
r e p r e s e n t a t i o n , f o r s o l v i n g s e t s o f n o n - l i n e a r m u l t i v a r i a t e c o n s t r a i n t s . T h i s s o l v e r e x p l o i t s w e l l
k n o w n p r o p e r t i e s o f t h e B e z i e r a n d B - s p l i n e r e p r e s e n t a t i o n s a n d i s a b l e t o i s o l a t e a l l r o o t s , g l o b a l l y .
W e w i l l t h e n s h o w h o w s u c h a s o l v e r c a n b e u s e d t o g r e a t l y e n h a n c e o u r a n a l y s i s a n d s y n t h e s i s o f g e o m e t r i c m o d e l i n g c a p a b i l i t i e s , d e m o n s t r a t i n g i t s u s e i n a v a r i e t y o f m o d e l i n g a p p l i c a t i o n s .
T h e s e c o n d h a l f o f t h i s t u t o r i a l w i l l d i v e i n t o o p t i m i z i n g s e v e r a l g e o m e t r i c o p e r a t i o n s i n c l u d i n g
s y m b o l i c t o o l s i n o f f s e t o p e r a t i o n s a n d t r i m m i n g o f o f f s e t s , M i n k o w s k i s u m s , s e l f - i n t e r s e c t i o n s
d e t e c t i o n s a n d e l i m i n a t i o n s , b i s e c t o r s / t r i s e c t o r s s y m b o l i c / n u m e r i c c o m p u t a t i o n a s w e l l a s c o n s i d e r
e f f i c i e n c y i n n u m e r i c g e o m e t r i c c o m p u t a t i o n s , i n c l u d i n g t h e u s e o f f r e e f o r m C o o n s p a t c h e s a s
b o u n d i n g s u r f a c e s .
S y l l a b u s : S y l l a b u s : S y l l a b u s : S y l l a b u s : ( 3 . 5 h o u r s w i t h a 3 0 - m i n u t e c o f f e e b r e a k )
1 . I n t r o d u c t i o n ( 5 m i n u t e s )
2 . S u b d i v i s i o n b a s e d M V s o l v e r s ( 3 5 m i n u t e s )
3 . E x a m p l e s u s i n g t h e M V s o l v e r i n g e o m e t r i c m o d e l i n g ( 5 0 m i n u t e s )
C o f f e e b r e a k ( 3 0 m i n u t e s )
5 . O f f s e t s , M i n k o w s k i s u m s , a n d s w e e p s ( 3 0 m i n u t e s )
6 . B i s e c t o r s , t r i s e c t o r s , a n d V o r o n o i d i a g r a m s i n R2 a n d R3 ( 3 0 m i n u t e s )
7 . E f f i c i e n c y i n n u m e r i c g e o m e t r i c p r o c e s s i n g ( 3 0 m i n u t e s )
8 . W r a p - u p ( 5 m i n u t e s )
L e c t u r e r s L e c t u r e r s L e c t u r e r s L e c t u r e r s s h o r t s h o r t s h o r t s h o r t b i o g r a p h y : b i o g r a p h y : b i o g r a p h y : b i o g r a p h y :
G e r s h o n G e r s h o n G e r s h o n G e r s h o n E l b e r E l b e r E l b e r E l b e r i s a p r o f e s s o r i n t h e C o m p u t e r S c i e n c e D e p a r t m e n t , T e c h n i o n , I s r a e l . H i s r e s e a r c h i n t e r e s t s s p a n c o m p u t e r a i d e d g e o m e t r i c d e s i g n s a n d c o m p u t e r g r a p h i c s . P r o f . E l b e r r e c e i v e d a B S c
i n c o m p u t e r e n g i n e e r i n g a n d a n M S c i n c o m p u t e r s c i e n c e f r o m t h e T e c h n i o n , I s r a e l i n 1 9 8 6 a n d 1 9 8 7 ,
r e s p e c t i v e l y , a n d a P h D i n c o m p u t e r s c i e n c e f r o m t h e U n i v e r s i t y o f U t a h , U S A , i n 1 9 9 2 . H e i s a
m e m b e r o f t h e A C M .
P r o f . E l b e r h a s s e r v e d o n t h e e d i t o r i a l b o a r d o f t h e C o m p u t e r A i d e d D e s i g n , C o m p u t e r G r a p h i c s
F o r u m , T h e V i s u a l C o m p u t e r , G r a p h i c a l M o d e l s , a n d t h e I n t e r n a t i o n a l J o u r n a l o f C o m p u t a t i o n a l
G e o m e t r y & A p p l i c a t i o n s a n d h a s s e r v e d i n m a n y c o n f e r e n c e p r o g r a m c o m m i t t e e s i n c l u d i n g S o l i d
M o d e l i n g , S h a p e M o d e l i n g , G e o m e t r i c M o d e l i n g a n d P r o c e s s i n g , P a c i f i c G r a p h i c s , C o m p u t e r
G r a p h i c s I n t e r n a t i o n a l , a n d S i g g r a p h . P r o f . E l b e r w a s o n e o f t h e p a p e r c h a i r s o f S o l i d M o d e l i n g
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2 0 0 3 a n d S o l i d M o d e l i n g 2 0 0 4 , a n d o n e o f t h e c o n f e r e n c e c h a i r s o f S o l i d a n d P h y s i c a l M o d e l i n g
2 0 1 0 . H e h a s p u b l i s h e d o v e r 1 5 0 p a p e r s i n i n t e r n a t i o n a l c o n f e r e n c e s a n d j o u r n a l s a n d i s o n e o f t h e
a u t h o r s o f a b o o k t i t l e d " G e o m e t r i c M o d e l i n g w i t h S p l i n e s - A n I n t r o d u c t i o n " .
E l b e r c a n b e r e a c h e d a t t h e T e c h n i o n , I s r a e l I n s t i t u t e o f T e c h n o l o g y , D e p a r t m e n t o f C o m p u t e r S c i e n c e , H a i f a 3 2 0 0 0 , I S R A E L .
E m a i l : g e r s h o n @ c s . t e c h n i o n . a c . i l , W W W : h t t p : / / w w w . c s . t e c h n i o n . a c . i l / ~ g e r s h o n h t t p : / / w w w . c s . t e c h n i o n . a c . i l / ~ g e r s h o n h t t p : / / w w w . c s . t e c h n i o n . a c . i l / ~ g e r s h o n h t t p : / / w w w . c s . t e c h n i o n . a c . i l / ~ g e r s h o n ,
F a x : 9 7 2 - 4 - 8 2 9 - 5 5 3 8 .
M y u n g M y u n g M y u n g M y u n g S o o S o oS o oS o o K i K iK iK immmm i s a P r o f e s s o r o f t h e S c h o o l o f C o m p u t e r S c i e n c e a n d E n g i n e e r i n g . I n t h e p a s t , h e
s e r v e d a s t h e C I O o f S e o u l N a t i o n a l U n i v e r s i t y a n d t h e D i r e c t o r o f U n i v e r s i t y C o m p u t e r C e n t e r . H e
a l s o s e r v e d a s t h e D i r e c t o r o f I n s t i t u t e o f C o m p u t e r T e c h n o l o g y a n d a s t h e H e a d o f S c h o o l o f
C o m p u t e r S c i e n c e a n d E n g i n e e r i n g , S e o u l N a t i o n a l U n i v e r s i t y . H i s r e s e a r c h i n t e r e s t s a r e i n
c o m p u t e r g r a p h i c s a n d g e o m e t r i c m o d e l i n g . P r o f . K i m r e c e i v e d B S a n d M S d e g r e e s f r o m S e o u l N a t i o n a l U n i v e r s i t y i n 1 9 8 0 a n d 1 9 8 2 , r e s p e c t i v e l y . H e c o n t i n u e d h i s g r a d u a t e s t u d y a t P u r d u e
U n i v e r s i t y , w h e r e h e r e c e i v e d a n M S d e g r e e i n a p p l i e d m a t h e m a t i c s i n 1 9 8 5 a n d M S a n d P h D
d e g r e e s i n c o m p u t e r s c i e n c e i n 1 9 8 7 a n d 1 9 8 8 , r e s p e c t i v e l y . S i n c e t h e n u n t i l 1 9 9 8 , h e w a s w i t h t h e
D e p a r t m e n t o f C o m p u t e r S c i e n c e , P O S T E C H , K o r e a . P r o f . K i m s e r v e s / s e r v e d o n t h e e d i t o r i a l
b o a r d s o f C o m p u t e r - A i d e d D e s i g n , C o m p u t e r A i d e d G e o m e t r i c D e s i g n , C o m p u t e r G r a p h i c s F o r u m ,
a n d I n t ' l J o f S h a p e M o d e l i n g . H e a l s o e d i t e d m o r e t h a n t e n s p e c i a l i s s u e s o f j o u r n a l s s u c h a s
C o m p u t e r - A i d e d D e s i g n , C o m p u t e r A i d e d G e o m e t r i c D e s i g n , G r a p h i c a l M o d e l s , J o f V i s u a l i z a t i o n
a n d C o m p u t e r A n i m a t i o n , T h e V i s u a l C o m p u t e r , a n d I n t ' l J o f S h a p e M o d e l i n g . W i t h t w o o t h e r
e d i t o r s , G e r a l d F a r i n a n d J o s e f H o s c h e k , h e e d i t e d H a n d b o o k o f C o m p u t e r A i d e d G e o m e t r i c D e s i g n ,
N o r t h - H o l l a n d , 2 0 0 2 . I n 2 0 1 0 , h e s e r v e d a s a t e c h n i c a l p r o g r a m c o - c h a i r o f t h e A C M S y m p o s i u m o n
S o l i d a n d P h y s i c a l M o d e l i n g , H a i f a , I s r a e l . I n 2 0 1 1 , h e a l s o s e r v e d a s a p r o g r a m c o - c h a i r o f t h e
S I A M C o n f e r e n c e o n G e o m e t r i c a n d P h y s i c a l M o d e l i n g , O r l a n d o , U S A .
P r o f . K i m c a n b e r e a c h e d a t S e o u l N a t i o n a l U n i v e r s i t y , S c h o o l o f C o m p u t e r S c i e n c e a n d E n g i n e e r i n g ,
S e o u l 1 5 1 - 7 4 4 , K o r e a .
E m a i l : m s k i m @ s n u . a c . k r , W W W : h t t p : / / h t t p : / / h t t p : / / h t t p : / / c s e . s n u . a c . k r / m s k i m c s e . s n u . a c . k r / m s k i m c s e . s n u . a c . k r / m s k i m c s e . s n u . a c . k r / m s k i m ,
F a x : + 8 2 - 2 - 8 7 1 - 4 9 1 2 .
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5 .5 .5 .5 . T u t o r i a l T u t o r i a l T u t o r i a l T u t o r i a l 5 555 ( 1 : 3 0 p . m . - 3 : 3 0 p . m . , ( 1 : 3 0 p . m . - 3 : 3 0 p . m . , ( 1 : 3 0 p . m . - 3 : 3 0 p . m . , ( 1 : 3 0 p . m . - 3 : 3 0 p . m . , J u n e J u n e J u n e J u n e 1 9 ) 1 9 )1 9 )1 9 )
T i t l e : T i t l e : T i t l e : T i t l e : C o m p u t a t i o n a n d A p p l i c a t i o n s o f C e n t r o i d a l V o r o n o i T e s s e l l a t i o n s
L e c t u r e r : L e c t u r e r : L e c t u r e r : L e c t u r e r : W e n p i n g W a n g T i m e : T i m e : T i m e : T i m e : 1 : 3 0 p . m . - 3 : 3 0 p . m . , J u n e 1 9
A b s t r a c t : A b s t r a c t : A b s t r a c t : A b s t r a c t :
C e n t r o i d a l V o r o n o i T e s s e l l a t i o n ( C V T ) i s a s p e c i a l g e o m e t r i c s t r u c t u r e t h a t h a v e b e e n r e c e i v i n g
m u c h r e s e a r c h i n t e r e s t d u e t o i t s w i d e s p r e a d a p p l i c a t i o n s i n s c i e n c e a n d e n g i n e e r i n g . I n t h i s 2 - h o u r
t u t o r i a l , I ' l l s t a r t w i t h a b r i e f i n t r o d u c t i o n t o r e l a t e d c o n c e p t s a n d p r e v i o u s w o r k s . T h e n I s h a l l
p r e s e n t s o m e r e c e n t a d v a n c e s o n t h e c o m p u t a t i o n a n d a p p l i c a t i o n s o f C V T :
1 ) A f a s t q u a s i - N e w t o n m e t h o d f o r c o m p u t i n g C V T ;
2 ) H i g h q u a l i t y m e s h g e n e r a t i o n ;
3 ) M o d e l i n g m i n i m a l s u r f a c e s a n d C o n s t a n t M e a n C u r v a t u r e s u r f a c e s ; 4 ) C i r c l e p a c k i n g a n d c i r c l e c o v e r a g e ;
5 ) A c o m p a r a t i v e s t u d y o n C V T a n d O D T ( o p t i m a l D e l a u n a y t r i a n g u l a t i o n ) , w h i c h i s p r o m i s i n g
a l t e r n a t i v e t o C V T f o r t e t r a h e d r a l m e s h g e n e r a t i o n .
K e y r e f e r e n c e s :
1 ) Q i a n g D u , V a n c e F a b e r a n d M a x G u n z b u r g e r , C e n t r o i d a l V o r o n o i T e s s e l l a t i o n s : A p p l i c a t i o n s
a n d A l g o r i t h m s , S I A M R e v i e w , V o l . 4 1 , N o . 4 ( D e c . , 1 9 9 9 ) , p p . 6 3 7 - 6 7 6
2 ) Q i a n g D u , M a x G u n z b u r g e r , L i l i J u , A d v a n c e s i n S t u d i e s a n d A p p l i c a t i o n s o f C e n t r o i d a l
V o r o n o i T e s s e l l a t i o n s , N u m e r i c a l M a t h e m a t i c s - - A J o u r n a l o f C h i n e s e U n i v e r s i t i e s . 2 0 1 0 .
3 ) Y . L i u , W . W a n g , B . L e v y , F . S u n , D . M . Y a n , L . L u a n d C . L . Y a n g , O n c e n t r o i d a l V o r o n o i
t e s s e l l a t i o n - - e n e r g y s m o o t h n e s s a n d f a s t c o m p u t a t i o n , A C M T r a n s a c t i o n s o n G r a p h i c s , v o l . 2 8 .
n o . 4 , ( 2 0 0 9 ) , p p . 1 - 1 7 .
L e c t u r e r s L e c t u r e r s L e c t u r e r s L e c t u r e r s s h o r t s h o r t s h o r t s h o r t b i o g r a p h y : b i o g r a p h y : b i o g r a p h y : b i o g r a p h y :
W e n p i n g W e n p i n g W e n p i n g W e n p i n g W a n g W a n g W a n g W a n g i s P r o f e s s o r o f C o m p u t e r S c i e n c e a t T h e U n i v e r s i t y o f H o n g K o n g . H i s r e s e a r c h
c o v e r s c o m p u t e r g r a p h i c s , v i s u a l i z a t i o n , a n d g e o m e t r i c c o m p u t i n g . H e h a s r e c e n t l y f o c u s e d o n m e s h
g e n e r a t i o n a n d s u r f a c e m o d e l i n g f o r a r c h i t e c t u r a l d e s i g n . H e i s j o u r n a l a s s o c i a t e e d i t o r o f C o m p u t e r
A i d e d G e o m e t r i c D e s i g n ( C A G D ) , C o m p u t e r s & G r a p h i c s , a n d I E E E T r a n s a c t i o n s o n V i s u a l i z a t i o n
a n d C o m p u t e r G r a p h i c s ( T V C G ) . H e h a s s e r v e d a s p r o g r a m c h a i r o f s e v e r a l i n t e r n a t i o n a l c o n f e r e n c e s , i n c l u d i n g P a c i f i c G r a p h i c s 2 0 0 3 , A C M S y m p o s i u m o n P h y s i c a l a n d S o l i d M o d e l i n g
( S P M 2 0 0 6 ) , a n d I n t e r n a t i o n a l C o n f e r e n c e o n S h a p e M o d e l i n g ( S M I 2 0 0 9 ) .
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6 .6 .6 .6 . T u t o r i a l T u t o r i a l T u t o r i a l T u t o r i a l 6 666 ( 4 : 0 0 p . m . - 5 : 0 0 p . m . , ( 4 : 0 0 p . m . - 5 : 0 0 p . m . , ( 4 : 0 0 p . m . - 5 : 0 0 p . m . , ( 4 : 0 0 p . m . - 5 : 0 0 p . m . , J u n e J u n e J u n e J u n e 1 9 ) 1 9 )1 9 )1 9 )
T i t l e : T i t l e : T i t l e : T i t l e : I s o g e o m e t r i c A n a l y s i s
L e c t u r e : L e c t u r e : L e c t u r e : L e c t u r e : B e r t J t t l e r T i m e : T i m e : T i m e : T i m e : 4 : 0 0 p . m . - 5 : 0 0 p . m . , J u n e 1 9
A b s t r a c t : A b s t r a c t : A b s t r a c t : A b s t r a c t :
I s o g e o m e t r i c A n a l y s i s ( I G A ) i s a n e w a p p r o a c h t o n u m e r i c a l s i m u l a t i o n , w h i c h w a s i n t r o d u c e d b y
H u g h e s e t a l . i n 2 0 0 5 . I t h a s t h e p o t e n t i a l t o b r i d g e t h e g a p b e t w e e n g e o m e t r i c d e s i g n , w h i c h i s o f t e n
b a s e d o n N U R B S ( n o n - u n i f o r m r a t i o n a l B - s p l i n e s ) a n d f i n i t e - e l e m e n t - b a s e d n u m e r i c a l s i m u l a t i o n .
T h e m a i n i d e a o f I G A i s t o u s e t h e s a m e r e p r e s e n t a t i o n o f t h e g e o m e t r y b o t h f o r s i m u l a t i o n a n d f o r
d e s i g n , t h e r e b y e l i m i n a t i n g t h e c o n v e r s i o n p r o c e s s e s , s u c h a s m e s h g e n e r a t i o n . T h e t u t o r i a l w i l l
i n t r o d u c e t h e b a s i c c o n c e p t s o f N U R B S - b a s e d i s o g e o m e t r i c a n a l y s i s . I t i s b a s e d o n t h e p a p e r :
A . - V . V u o n g , C . H e i n r i c h , B . S i m e o n , I S O G A T : A 2 D t u t o r i a l M A T L A B C o d e f o r I s o g e o m e t r i c A n a l y s i s , C o m p . A i d e d G e o m . D e s i g n 2 7 ( 2 0 1 0 ) , 6 4 4 - 6 5 5
T h e M A T L A B s o u r c e s a r e a v a i l a b l e f r o m :
h t t p : / / w w w - m 2 . m a . t u m . d e / b i n / v i e w / A l l g e m e i n e s / E X C I T I N G h t t p : / / w w w - m 2 . m a . t u m . d e / b i n / v i e w / A l l g e m e i n e s / E X C I T I N G h t t p : / / w w w - m 2 . m a . t u m . d e / b i n / v i e w / A l l g e m e i n e s / E X C I T I N G h t t p : / / w w w - m 2 . m a . t u m . d e / b i n / v i e w / A l l g e m e i n e s / E X C I T I N G
S y l l a b u s : S y l l a b u s : S y l l a b u s : S y l l a b u s : ( 1 h o u r )
1 . I n t r o d u c t i o n ( 5 m i n u t e s )
2 . M o d e l p r o b l e m ( 1 0 m i n u t e s )
3 . G a l e r k i n p r o j e c t i o n ( 1 0 m i n u t e s )
4 . I G A b a s i c s ( 1 0 m i n u t e s )
5 . T r a n s f o r m a t i o n t o t h e p a r a m e t r i c d o m a i n ( 1 0 m i n u t e s )
6 . M a t l a b C o d e ( 5 m i n u t e s )
7 . E x a m p l e s ( 5 m i n u t e s )
8 . C l o s i n g r e m a r k s ( 5 m i n u t e s )
L e c t u r e r L e c t u r e r L e c t u r e r L e c t u r e r ssss s h o r t s h o r t s h o r t s h o r t b i o g r a p h y : b i o g r a p h y : b i o g r a p h y : b i o g r a p h y :
B e r t B e r t B e r t B e r t J JJJ t t l e r t t l e r t t l e r t t l e r i s p r o f e s s o r o f m a t h e m a t i c s a t t h e J o h a n n e s K e p l e r U n i v e r s i t y o f L i n z , A u s t r i a . H i s
r e s e r a r c h i n t e r e s t s i n c l u d e C o m p u t e r A i d e d G e o m e t r i c D e s i g n a n d I s o g e o m e t r i c A n a l y s i s . F r o m
2 0 0 8 t o 2 0 1 2 h e c o o r d i n a t e d t h e E u r o p e a n p r o j e c t E x a c t G e o m e t r y S i m u l a t i o n f o r O p t i m i z e d
D e s i g n o f V e h i c l e s a n d V e s s e l s w h i c h a p p l i e d I s o g e o m e t r i c A n a l y s i s t o s e v e r a l i n d u s t r i a l a p p l i c a t i o n s c e n a r i o s .
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Computer Aided Geometric Design 27 (2010) 644655
Contents lists available at ScienceDirect
Computer Aided Geometric Design
www.elsevier.com/locate/cagd
ISOGAT: A 2D tutorial MATLAB code for Isogeometric Analysis
A.-V. Vuong, Ch. Heinrich , B. Simeon
Technische Universitt Mnchen, Centre for Mathematical Sciences, Boltzmannstrae 3, 85748 Garching, Germany
a r t i c l e i n f o a b s t r a c t
Article history:
Available online 18 June 2010
Keywords:
Isogeometric Analysis
MATLAB
NURBS
Exact geometry
Tutorial code
A tutorial 2D MATLAB code for solving elliptic diffusion-type problems, including Poissons
equation on single patch geometries, is presented. The basic steps of Isogeometric Analysisare explained and two examples are given. The code has a very lean structure and has been
kept as simple as possible, such that the analogy but also the differences to traditional
finite element analysis become apparent. It is not intended for large-scale problems.
2010 Elsevier B.V. All rights reserved.
1. Introduction
Over the last years, the new field of Isogeometric Analysis (Hughes et al., 2005) has demonstrated its potential to bridge
the gap between Computer Aided Design (CAD) and the Finite Element Method (FEM). The main feature of IsogeometricAnalysis is the usage of one common geometry representation for creating CAD models, for meshing, and for numerical
simulation. In this way, a seamless integration of all computational tools within a single design loop comes into reach.
However, the numerical analysts are typically not familiar with the elegant and powerful algorithms of computational
geometry, and on the other hand in the CAD community there is still little awareness of the specific requirements in
numerical methods. In order to further broaden the common platform of Isogeometric Analysis, we present here a tutorial
2D MATLAB code for solving diffusion-type problems on single patch geometries. The basic steps are explained and two
examples given. The code has a very lean structure and has been kept as simple as possible. It is not intended for large-
scale problems but for serving as illustration and also for teaching.
As basic reference for Isogeometric Analysis and the usage of Non-Uniform Rational B-Splines (NURBS) as basis functions
in the FEM, we recommend the recent monograph (Cottrell et al., 2009). Standard references on splines and CAD are Farin
(2002) and Piegl and Tiller (1997). Additionally, more advanced techniques like T-spline refinement are discussed in Drfel
et al. (2010) and Bazilevs et al. (2010). An analysis of specific quadrature rules can be found in Hughes et al. (2010) while
swept volume techniques and a study of corresponding parameterizations are presented in Aigner et al. (2009). We finallyalso mention the work on isogeometric fluid-structure interaction (Bazilevs et al., 2008), stress calculation (Vuong and
Simeon, submitted for publication) and electromagnetics (Buffa et al., 2010).
The paper is organized as follows: Section 2 introduces the model problem of Poissons equation, which serves as basis
for the whole paper. How the infinite-dimensional problem is discretized by means of the Galerkin projection can be found
in Section 3. After having presented the necessary fundamentals of Isogeometric Analysis in Section 4, we show how the
occurring integrals are transformed to the parametric domain in Section 5. The subsequent sections demonstrate algorithmic
d l b d f d b h h f h d d h bl f h ff
http://dx.doi.org/10.1016/j.cagd.2010.06.006http://www.sciencedirect.com/http://www.elsevier.com/locate/cagdhttp://www.elsevier.com/locate/cagdhttp://www.sciencedirect.com/http://dx.doi.org/10.1016/j.cagd.2010.06.006