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7/28/2019 Bezier Curve Continuity
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P i e c e w i s e C u r v e s
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P i e c e w i s e B e z i e r C u r v e
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C o n t i n u i t y
n e o f t h e f u n d a m e n t a l c o n c e p t s
o m m o n l y u s e d c a s e s : C
0
, C
1
, C
2
, e t c .
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P o s i t i o n a l C o n t i n u i t y
a(u)b(u)
a(1)=b(0)
u varies from 0 to 1
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D e r i v a t i v e C o n t i n u i t y
a(u) b(u)a(1)=b(0)
a(1)=b(0)
u varies from 0 to 1
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G e n e r a l C o n t i n u i t y
n
c o n t i n u i t y : d e r i v a t i v e s ( u p t o n - t h ) a r e t h e s a m
n d p o i n t s
a
(i
)
( 1 ) = b
(i
)
( 0 )
h e r e i
= 0 : : : n
h e p r i o r d e n i t i o n i s f o r p a r a m e t r i c c o n t i n u i t y
a r a m e t r i c c o n t i n u i t y d e p e n d s o n p a r a m e t e r i z a t i o
a r a m e t e r i z a t i o n i s n o t u n i q u e
i e r e n t p a r a m e t r i c r e p r e s e n t a t i o n s m a y e x p r e s s t
a m e g e o m e t r y
e - p a r a m e t e r i z a t i o n c a n b e e a s i l y i m p l e m e n t e d
n o t h e r t y p e o f c o n t i n u i t y : g e o m e t r i c c o n t i n u i t y ,
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G e o m e t r i c C o n t i n u i t y
e p e n d o n t h e c u r v e g e o m e t r y
O N O T d e p e n d o n t h e u n d e r l y i n g p a r a m e t e r i z a t
0
: t h e s a m e j o i n t
1
: T w o c u r v e t a n g e n t s a t t h e j o i n t a l i g n , b u t m a
a v e t h e s a m e m a g n i t u d e
1
: i t i s C
1
a f t e r t h e r e p a r a m e t e r i z a t i o n
h i c h c o n d i t i o n i s s t r o n g e r ? ? ?
x a m p l e s
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G e o m e t r i c C o n t i n u i t y
zeroorder Gcontinuity
firstorder Gcontinuity
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P i e c e w i s e H e r m i t e C u r v e s
o w t o b u i l d a n i n t e r a c t i v e s y s t e m t o s a t i s f y v a r i o
o n s t r a i n t s
0
c o n t i n u i t y
a( 1 ) =
b( 0 )
1
c o n t i n u i t y
a( 1 ) =
b( 0 )
a
0
( 1 ) = b
0
( 0 )
1
c o n t i n u i t y
a( 1 ) =
b( 0 )
a
0
( 1 ) = b
0
( 0 )
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P i e c e w i s e H e r m i t e C u r v e s
a(0)
a(0)
a(1)
a(1)
b(0)b(0)
b(1)
b(1)
continuity conditions
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P i e c e w i s e H e r m i t e C u r v e s
piecewise hermite curves
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P i e c e w i s e B e z i e r C u r v e s
p
pp
p
q
q
q
q
0
1
2
3
0
1
2
3
piecewise Bezier curves
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P i e c e w i s e B e z i e r C u r v e s
0
c o n t i n u i t y
p
3
= q
0
1
c o n t i n u i t y
p
3
= q
0
p
3
;
p
2
= q
1
;
q
0
1
c o n t i n u i t y
p
3
= q
0
p
3
;
p
2
= ( q
1
;
q
0
)
2
c o n t i n u i t y
p
3
= q
0
p
3
;
p
2
= q
1
;
q
0
p
3
;
2 p
2
+ p
1
= q
2
;
2 q
1
+ q
0
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e o m e t r i c i n t e r p r e t a t i o n
2
c o n t i n u i t y
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P i e c e w i s e C
2
B e z i e r C u r v e s
piecewiseC2 continuousBezier curves
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C o n t i n u i t y S u m m a r y
0
: s t r a i g h t f o r w a r d , b u t n o t e n o u g h
3
: t o o c o n s t r a i n e d
i e c e w i s e c u r v e s w i t h h e r m i t e a n d B e z i e r
e p r e s e n t a t i o n s s a t i s f y i n g v a r i o u s c o n t i n u i t y c o n d i t
n t e r a c t i v e s y s t e m f o r C
2
i n t e r p o l a t i n g s p l i n e s u s i n
i e c e w i s e B e z i e r c u r v e s
d v a n t a g e s a n d d i s a d v a n t a g e s
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C
2
I n t e r p o l a t i n g S p l i n e s
v
v
v
v
v
0
1
2
3
4
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N a t u r a l S p l i n e s
n t e r p o l a t e a l l c o n t r o l p o i n t s
q u i v a l e n t t o a t h i n s t r i p o f m e t a l i n a p h y s i c a l s e
o r c e d t o p a s s t h r o u g h a s e t o f d e s i r e d p o i n t s
o l o c a l c o n t r o l ( g l o b a l c o n t r o l )
+ 1 c o n t r o l p o i n t s
p i e c e s
ne x t r a p o i n t s
(n
; 1 ) c o n d i t i o n s
e n e e d t w o a d d i t i o n a l c o n d i t i o n s
{ s p e c i f y d e r i v a t i v e s a t t w o e n d p o i n t s
{ s p e c i f y t h e t w o i n t e r n a l c o n t r o l p o i n t s t h a t d e
r s t c u r v e s p a n
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{ i n t e r a c t i v e s y s t e m
{ n a t u r a l e n d c o n d i t i o n s : s e c o n d - o r d e r d e r i v a t i v e s
e n d p o i n t s a r e d e n e d t o b e z e r o
d v a n t a g e s : i n t e r p o l a t i o n , C
2
i s a d v a n t a g e s : n o l o c a l c o n t r o l ( i f o n e p o i n t i s
h a n g e d , t h e e n t i r e c u r v e w i l l m o v e )
o w t o o v e r c o m e t h i s d r a w b a c k : B - S p l i n e s
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B - S p l i n e s M o t i v a t i o n
h e g o a l i s l o c a l c o n t r o l ! ! !
- s p l i n e s p r o v i d e l o c a l c o n t r o l
o n o t i n t e r p o l a t e c o n t r o l p o i n t s
2
c o n t i n u i t y
l t e r n a t i v e l y
a t m u l l - R o m S p l i n e s
e e p i n t e r p o l a t i o n s
i v e u p C
2
c o n t i n u i t y ( o n l y C
1
i s a c h i e v e d )
i l l b e d i s c u s s e d l a t e r ! ! !