rC Polynomials By G. FARIN

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TR/91 January 1980

Bézier Polynomials over Triangles and the Construction of Piecewise

Cr Polynomials By

G. FARIN

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1.

Introduction

Bézier polynomials and their generalization to tensor-product

surfaces provide a useful tool in surface design (Bézier 1970, 1977;

Forrest 1972). They were developed as early as 1959 by de Casteljau at Citroën but owe their name to P. Bézier from Renault who was first to employ them in car body design in the late sixties.

De Casteljau 1959 also describes triangular patches, but these

scarcely received any attention until Sabin 1977. Farin 1979 generalizes and extends results obtained by de Casteljau and Sabin, sharing their restrictions to domains that consist of congruent triangles only.

The present paper restates some of the results of Farin 1979, including a short outline of the univariate case, and then generalizes them to Bezier polynomials defined over arbitrary triangles; formulas describing

continuity of adjacent triangular patches are provided.

Cr

The last two sections give applications of the theory: the ^C¹

Clough-Tocher scheme is generalized to the ^C² case and a formula for the dimension of the linear space of piecewise ^C^r polynomials (of degree n) is derived.

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2.

I U n i v a r i a t e B é z i e r P o l y n o m i a l s 1 . D e f i n i t i o n

A B é z i e r p o l y n o m i a l Bnφ i s d e f i n e d b y

∑=

=

φ ⁿ

0 i

n i i

n ] (t) b B (t)

[B (1)

w h e r e t h e Bn_i a r e B e r n s t e i n p o l y n o m i a l s n i o

; i n ) t 1 i ( i t ) n t n ( Bi ) 2

( ⎟⎟⎠ − − ≤ ≤

⎜⎜ ⎞

⎝

= ⎛

a n d i s t h e p i e c e w i s e l i n e a r f u n c t i o n j o i n i n g t h e p o i n t s φ

φ

≤

≤i n. 0

; )

(ni ,b_i is called the Bézier polygon associated with Bnφ*i⁾ ; t h e b_i a r e c a l l e d B é z i e r o r d i n a t e s o f Bnφ.

S i n c e t h e n s a t i s f y Bi

, 1 t 0

; n i 0

; 0 ) t n ( Bi ) 3

( ≤ ≤ ≤ ≤ ≤

, 1 ) t n (

0 i

ni B )

4

( ∑ =

=

*) In classical approximation theory, Bnφ is called the "Bernste in approximant" to φ (Davis 1975) the graph of Bnφ , 0≤t≤1 , l i e s i n t h e convex hull of the graph of φ. (Bézier 1970, Bézier 1976, Forrest 1970).

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3 . 2 . D e g r e e E l e v a t i o n

E v e r y p o l y n o m i a l o f d e g r e e n c a n b e w r i t t e n a s a p o l y n o m i a l o f

d e g r e e n + 1 ; l e t Eφ b e a p o l y g o n j o i n i n g p o i n t s

(

ni+1 ,b^*i

)

^, ⁰≤ⁱ≤ⁿ+¹^. I f

) )

(

^b

(

¹ ^b ^, ⁰ ⁱ ⁿ ¹ ^,

b ) 5

( _i ₁ _nⁱ ₁ _i

1 ni 1

n

* i

i = φ ₊ = ₊ ₋ + − ₊ ≤ ≤ +

i t i s e a s y t o s h o w t h a t φ

=

φ B ₊ E B

) 6

( _n _n ₁

3 . D e r i v a t i v e s

F o r t h e r - t h d e r i v a t i v e o f Bnφ w e f i n d

∑⁻

=

− Δ −

=

φ ⁿ ^r

0 i

r in r i

! ) r n ( n! dt n

dr r[B ] (t) b B (t)

) 7 (

T h i s y i e l d s i m m e d i a t e l y

r o

! ) r n ( n! dt n

dr r[B ] (0) b

) a 8

( φ = ₋ Δ

, r r n

! ) r n ( n! dt n

dr r[B ] (1) b

) b 8

( φ = ₋ Δ ₋

i . e . t h e r - t h d e r i v a t i v e a t a n e n d p o i n t d e p e n d s o n l y o n t h e ( r + 1 ) a d j a c e n t B é z i e r o r d i n a t e s .

W e a l s o n o t e t h a t f o r ζε[a,b] i n s t e a d o f tε[0,1], ( 7 ) b e c o m e s

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4 .

) r ( n Bi r

n 0

i rbi

) r a b ( 1

! ) r n ( n! )

( n ] B [ d r dr ) 9

( − ζ

∑−

= Δ

⋅ −

= − ζ ζ φ

4 . R e c u r s i v e A l g o r i t h m ( d e C a s t e l j a u 1 9 5 9 )

T h e Bⁿ_i s a t i s f y a r e c u r r e n c e r e l a t i o n

) t 1( n

1 Bi t ) t 1( n Bi ) t 1 ( ) t n ( Bi ) 10

( −

+ −

− −

=

).

n i or 0 i for 0 ) t n ( Bi with

( = < >

( 1 0 ) a l l o w s t o e x p a n d Bnφ i n t e r m s o f B e r n s t e i n p o l y n o m i a l s o f l o w e r d e g r e e :

∑⁻

=

− ≤ ≤

=

φ ⁿ ^r

0 i

r n i ri

n ] (t) b (t)B (t) , 0 r n,

B [ ) 11 (

w h e r e t h e b_i^r (t) a r e d e f i n e d b y

( 1 2 ) ^r ¹

1 i 1

r i i

r (t) (1 t) b tb

b ⁻

+

− +

−

=

b^o_i (t) = b_i . S i n c e

, ) t n( b0 ) t ( n ] B [ ) 13

( φ =

( 1 2 ) p r o v i d e s a n e a s y a n d s t a b l e a l g o r i t h m f o r t h e n u m e r i c a l e v a l u a t i o n o f [Bnφ](t). T h i s i s i l l u s t a r t e d i n f i g . 1

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5.

F i g . 1 c o n s t r u c t i o n o f [B3φ](₂¹) O n e c a n s h o w t h a t

r n i 0

n r

; 0 ) t r ( Bj r

0 j bi j )

t ir ( b ) 14

( ≤ ≤ −

≤

∑ ≤

= +

=

T h e r(t) c a n a l s o b e u s e d t o d e t e r mi n e t h e r - t h d e r i v a t i v e o f

bi Bnφ;

) t r ( Bi r

n 0 i

) t r( in

! b ) r n ( n! )

t ( n ] B [ dt r dr ) 15

( ∑−

=

− −

= φ

F o r r = 1 , ( 1 5 ) s t a t e s t h a t n 1(t) bo−

a n d n 1(t) b1−

d e t e r m i n e t h e t a n g e n t t o )

t ( n ] B

[ φ a n d , f o r r = 2 , t h a t n 2(t) b0−

, n 2(t)

b1− , n 2(t)

b2− d e t e r mi n e t h e o s c u l a t i n g p a r a b o l a .

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6.

y t i u n i t n o C C .

5 ^r−

S u p p o s e w e a r e g i v e n a B é z i e r p o l y g o n φ w i t h B é z i e r o r d i n a t e s b_i o v e r W e s e e k a B é z i e r p o l y g o n tε[0,1]. ψ w i t h B é z i e r o r d i n a t e s C_i o v e r ]ζε[1,2 s u c h t h a t t h e t w o p o l y n o m i a l s d e f i n e d b y a n d φ ψ f o r m a f u n c t i o n i n Cr[0,2]. F r o m ( 8 a ) a n d ( 8 b ) w e g e t t h e c o n d i t i o n s

. r 0

o ; n c

b )

16

( Δ ρ −ρ = Δρ ≤ρ≤

( 1 6 ) i m p l i e s t h a t , f o r f i x e d ρ, t h e B é z i e r p o l y n o m i a l s d e f i n e d b y + ρ

ρ

− ρ

− , b n 1 , ... , bn and co ,c1 ,...,c

bn c o i n c i d e s i n c e a l l t h e i r

d e r i v a t i v e s c o i n c i d e a t t = l r e s p . ς=0.

H e n c e

∑

∑ ^ρ

= ρ ρ

=

+ ρ ρ

− =

o

i i i

o

i bn iBi (t) c B (1).

s i n c e ς =t − 1. T h i s i s t r u e f o r a l l t , i . e . a l s o f o r t = 2 :

∑

∑ ^ρ

= ρ ρ

=

+ ρ ρ

− =

o

i i i

o

i bn iBi (2) c B (1) .

T h e r i g h t - h a n d s i d e e q u a l s cρ , a n d w e g e t .

r 0

; ) 2 ( B b

c ) 17

( _i

o i

i

n ≤ρ ≤

= ^ρ ^ρ

= −ρ+

ρ ∑

N o t e t h a t t h i s i s e q u i v a l e n t t o : . r ρ 0

; ρ (2)

ρ bn cρ

(18) = − ≤ ≤

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7.

W e c a n d e f i n e t h e s e c o n d B é z i e r p o l y n o m i a l o v e r [1,β ] i n s t e a d o f [ 1 , 2 ] ; i n t h i s c a s e , ( 1 8 ) b e c o m e s

. ) β ρ (

ρ bn cρ

(19) = −

T h u s w e h a v e a c o n d i t i o n f o r Cr−c o n t i n u i t y g i v e n b y ( 1 6 ) a n d a c o n s t r u c t i o n g i v e n b y ( 1 7 ) , N o t e t h a t f i g . 1 c a n b e i n t e r p r e t e d a s t h e c o n s t r u c t i o n o f t h e b_oⁿ , b₁ⁿ⁻¹ , ..., b^o_n f r o m t h e n

bo ,.

..

1 , bo o , bo

t o o b t a i n Cn− c o n t i n u i t y . W e a l s o n o t e t h a t t h e t w o c o r r e s p o n d i n g B é z i e r p o l y n o m i a l s c o i n c i d e w i t h t h e o r i g i n a l p o l y n o m i a l g i v e n b y

n . bo , ..

. o , b1 o , bo

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8.

I I B e z i e r P o l y n o m i a l s o v e r a T r i a n g l e 1 . D e f i n i t i o n

W e c o n s i d e r a t r i a n g l e T i n t h e p l a n e w i t h v e r t i c e s P1,P2,P3 a n d e d g e s e1,e2,e3 i n w h i c h w e a s s u m e b a r y c e n t r i c c o o r d i n a t e s d e f i n e d s u c h t h a t f o r e a c h p o i n t P i n t h e p l a n e

wP3 vP2

uP1

P = + +

w h e r e

, T P all for 1 w , v , u

0 ≤ ≤ ε

, 1 w v

u + + =

a n d

] . P3 P2 P1 [

2] P 1P P w [ ] , P3 P2 P1 [

3] P 1 P P v [ ] , P3 P2 P1 [

2] P 3 P P

u = [ = =

H e r e , [P3 P P2] d e n o t e s t h e a r e a o f t h e t r i a n g l e P₁, P, P₂ e t c .

W e d e f i n e B e r n s t e i n p o l y n o m i a l s

~ n

Bi (~)u o v e r T :

) w , v , u

~u (

) k ,j ,i

~i ( 1

w v

u j k n

k i j i

! k

! j

! i

! n n

i ~ u v w ;

~ (u) B )

20

( = ⁺₊ ⁺₊ ⁼₌ ₌⁼

S i n c e t h e (u_~)

~i

B n a r e t e r m s o f

, w v u )

w v u ( ) 21

( ⁱ ^j ^k

0 k , j ,

i j k n

i i!jn!!k!

n ∑

=≥ + +

= +

+

w e h a v e i m m e d i a t e l y

, 1 w , v , u 0

for

~)u 0 ( B )

22

( ⁿ

~i ≤ ≤ ≤

(18)

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9.

.

~) 1 u ( B~ )

23

( ⁿ

~i

ni ≡

∑

T h e s u m m a t i o n ∑ⁿ

i~

i n ( 2 3 ) i s s h o r t f o r t h e o n e u s e d i n ( 2 1 ) . T h e (n 1) (n 2)

12 + + p o l y n o m i a l s

~ ni

B f o r m a b a s i s f o r t h e l i n e a r s p a c e o f a l l b i v a r i a t e p o l y n o m i a l s o f d e g r e e n .

A B é z i e r p o l y n o m i a l o v e r T i s d e f i n e d b y

~) , u (

~ B

~)u b ( ] B [ ) 24 (

n

~i

n i ~

n^φ ⁼ ∑ _i

w h e r e i s t h e p i e c e w i s e l i n e a r f u n c t i o n d e t e r m i n e d b y t h e p o i n t s φ .

i) b n, ,k n , j ni

( T h e

b~i a r e c a l l e d B é z i e r o r d i n a t e s o f Bnφ; φ i s c a l l e d t h e B é z i e r n e t o f B_nφ.*) ( 2 2 ) a n d ( 2 3 ) i m p l y t h a t t h e g r a p h o f Bnφ; l i e s i n t h e c o n v e x h u l l o f t h e g r a p h o f φ. T h e s t r u c t u r e o f φ i s i l l u s t r a t e d i n f i g . 3 .

W e a l s o n o t e t h a t t h e b o u n d a r y c u r v e s o f Bnφ a r e t h e ( u n i v a r i a t e ) B é z i e r p o l y n o m i a l s d e t e r m i n e d b y t h e b o u n d a r y p o i n t s o f φ.

* ) T h i s n o t a t i o n i s c h o s e n t o b e l i k e t h e o n e i n t h e u n i v a r i a t e c a s e t o p o i n t o u t t h e s i m i l a r i t y o f b o t h m e t h o d s . N o c o n f u s i o n

s h o u l d a r i s e , h o w e v e r , s i n c e t h e m e a n i n g o f Bnφ , φ, e t c . w i l l b e c l e a r f r o m t h e c o n t e x t .

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10.

2 . D e g r e e e l e v a t i o n

E v e r y b i v a r i a t e p o l y n o m i a l o f d e g r e e n c a n b e w r i t t e n a s a p o l y n o m i a l o f d e g r e e n + 1 ; l e t Eφ b e a n e t d e t e r m i n e d b y p o i n t s

~*), bi 1, nk 1, n , j 1 ni

( + + +

i + j + k = n + 1 . I f

k , 1 j , bi 1 n k j , j , 1 bi 1 ni )

1 i n1

* ( b~i ) 25

( ⋅ = + + + +

= −

+

, 1 n k j i 1 ; k ,j , bi 1

nk+ + + + = +

+

i t i s e a s y t o s h o w t h a t . 1E Bn Bn

) 26

( φ = + φ

T h i s i s i l l u s t r a t e d i n f i g . 4 a n d t h e f o l l o w i n g e x a m p l e .

E x a m p l e 1 : T h e f o l l o w i n g t w o B é z i e r n e t s d e t e r m i n e t h e s a m e p o l y n o m i a l :

3 . D e r i v a t i v e s

L e t ~u = ~u(s) b e t h e e q u a t i o n o f a s t r a i g h t l i n e i n t e r m s o f t h e

b a r y c e n t r i c c o o r d i n a t e s o f T , e . g . w i t h t w o p o i n t s

~1 u 0 s

~u ) s 1 ( ) s

~(

u = − +

1.

~u 0,

~u

(22)

(23)

1 1 .

H e n c e

1 r for 0 )

s ( u )

27

( _~ _~

ds d

r

r = >

( h e r e , ~0 = ( 0 , 0 , 0 ) ) .

F o r r = 1 , w e s e t dsd ~u(s) u~⋅ =

S i n c e u + v + w = 1 w e h a v e u⋅ + v⋅ + w⋅ = 0.

T h e t e r m d e f i n e s a d i r e c t i o n w i t h r e s p e c t t o w h i c h w e c a n t a k e ⋅

~u

d i r e c t i o n a l d e r i v a t i v e s . W e s e t . dsd .:

D ^r_r

u~r =

F o r t h e B e r n s t e i n p o l y n o m i a l s n

~i

B w e g e t T h e o r e m 1 : S e t ~λ = (λ,μ,ν). T h e n

∑λ

−λ

⋅ −

− λ

⋅ =

r

~

~) u r ( n

~ B~i

~) u r ( B~

! ) r n (

! ) n

(~u n B~i r

~u D ) 28 (

R e m a r k s : ( a ) T h e t e r m r (~u) i s w e l l - d e f i n e d e v e n i f t h e s u m B~ ⋅

λ

o f t h e a r g u me n t s d o e s n o t e q u a l 1 . ( b ) F o r ~i,λ~ t h a t d o n o t s a t i s f y ( c o m p o n e n t w i s e ) , w e s e t n r (~u) 0.

~i ~

B − =

λ

~0 −

~

~i − λ≥ P r o o f

i ) F o r t r i p l e p r o d u c t s o f f u n c t i o n s o f o n e v a r i a b l e s t h e L e i b n i z f o r m u l a

r f( )(s).g( ) (s).h( ) (s)

~ ! ! !

! ) r

s ( h . ) s ( g . ) s ( r f ds

dr ∑ λ μ ν

λ λ μ ν

=

i s t r u e .

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1 2

i i ) B e c a u s e o f t h e l i n e a r i t y o f u ( s ) , v ( s ) , w ( s ) r e p e a t e d a p p l i c a t i o n s o f t h e c h a i n a n d p r o d u c t r u l e s y i e l d :

. etc u

) u )( f( ] ) s ( u [ f ds

d = λ ⋅ ⋅λ

λ λ

i i i ) S e t t i n g f ( u ( s ) ) = [u(s)]i etc., w e o b t a i n ]k ) s ( w j [ ] ) s ( v i [ ] ) s ( u

! [ k

! j

! i

! u dsr

dr

~u) n ( B~i r

~u

D⋅ =

ν

− μ

− λ ν −

− μ

− λ

−

⋅ λ

⋅λ

⋅λ λ λ μ ν

∑

= ^r _!^r^!_! _! u .v .w . ₍_i ₎_!₍_jⁿ^!₎_!₍_k ₎_! uⁱ v^j w^k

~

T h i s i m p l i e s

T h e o r e m 2 : T h e r - t h d i r e c t i o n a l d e r i v a t i v e w r t ⋅ o f a B é z i e r

~u p o l y n o m i a l Bnφ o v e r T i s g i v e n b y

∑ ∑⁻ _λ ^⋅ ⁻

λ +λ

= − φ

⋅

r n

i n r ~

~ i r r

! i ) r n (

! n n ~

ur

~

~ [B ] (u) b~ B (u) B (u)

D ) 29 (

W e n o t e t h a t t h i s c a n b e r e a r r a n g e d t o

. ) u ( B

b )

u ( B )

u ( ] [B D

(30) ^r

λ n r ~

i r

- n

i i λ

r ~

! λ r) (n

! n n ~

ur

~ ~

~

~ ~

~^⋅ ^φ ⁼ ₋ ∑ ^⋅ ∑ ⁺ ⁻

4 . R e c u r s i v e A l g o r i t h m ( d e C a s t e l j a u 1 9 5 9 )

L e t u s d e f i n e = ( 1 , 0 , 0 ) , = ( 0 , 1 , 0 ) , = ( 0 , 0 , 1 ) . a~1

a~2

a~3 L e m m a 3 : T h e B_iⁿ s a t i s f y a r e c u r r e n c e r e l a t i o n

n k j i ,

~) u 1 ( n

~a3 B~i . w

~) u 1 ( n

2

~a B~i v.

~) u 1 ( n

~a1 B~i u

~u) n ( B~i

(31) − − + + =

+ −

−−

⋅

= + −

P r o o f : U s e t h e i d e n t i t y

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ −

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛

j i n i n

! k

! i

! n

a n d t h e r e c u r s i o n f o r m u l a f o r b i n o m i a l c o e f f i c i e n t s .

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(27)

1 3

T h i s l e m m a a l l o w s t o e x p a n d Bnφ i n t e r m s o f B e r n s t e i n p o l y n o m i a l s o f l o w e r d e g r e e , w i t h p o l y n o m i a l c o e f f i c i e n t s r (~u):

b~i T h e o r e m 4 :

n r 0 , ) u ( B ) u ( b )

u ( ] [B

(32) ⁿ^-^r

i n r ~

~ i ri n ~

~

~ ≤ ≤

=

φ ∑ ⁻

w h e r e t h e r (~u) a r e d e f i n e d b y b~i

⎪⎪

⎩

⎪⎪

⎨

⎧

=

= +

− +

⋅ +

− + + +

−+

⋅

=

b~i

~u) 0 ( b~i

r - n k j i

;

~) u 1 ( r

~a3 b~i w

~) u 1 ( r

~a2 b~i v.

~) u 1 ( r

~a1 b~i u

~u) r ( b~i (33)

P r o o f i s b y i n d u c t i o n o n r . ( 3 2 ) i s t r u e f o r r = 0 . I n d u c t i o n :

∑ ⁻

=

φ ⁿ^-^r

i n r ~

~ i ir n ~

~

~ (u) B (u)

b )

u ( ] [B

∑⁻ ⁻⁻ ⁻ ⁺ ⁻⁻ ⁻ ⁺ ⁻⁻ ⁻

=

r n

i~

1 r n

~3

~ a 1 i

r n

~2

~i a 1

r na~1

~i

~ir (u~) [u.B (u~) v.B (u~ ) w B (u~)]

b )

31 (

) u ( B

] ) u ( b w ) u ( b v ) u ( b u

[ _i^r _a _~ ^r_i _a _~ _iⁿ ^r ¹ _~

1 - r - n

i r ~

a

i ^~ ^~² ^~ ^~³ ^~

~

1

~

−

− −

−

− + ⋅ + ⋅

⋅

= ∑

∑ ⁺ ⁻

= ⁿ^-^r^-¹

i n r-1 ~

~ i 1 ri )

33 (

~

~ (u) B (u)

b S i n c e

, ) u ( b ) u ( ]

[B ~

n 0

~

n ~

=

φ

( 3 3 ) p r o v i d e s a n a l g o r i t h m f o r t h e e v a l u a t i o n o f [B_nφ] (u_~) . T h i s i s i l l u s t r a t e d i n f i g u r e 5 .

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(29)

1 4 .

F i g u r e 5 : C o n s t r u c t i o n f o r [B ] (₃φ ¹₂^, ₄¹^,₄¹) T h e br (~u) h a v e a n e x p l i c i t f o r m s i m i l a r t o ( 1 4 ) :

~i

r.

- n k j i

;

~) u r (

~ r B

~ b~i ~

~) u r (

~i b

(34) + + =

∑ λ

λ −λ

=

T o p r o v e t h i s o n e c h e c k s t h a t ( 3 4 ) i s c o n s i s t e n t w i t h t h e r e c u r s i v e d e f i n i t i o n ( 3 3 ) o f t h e (~u).

~i br

W i t h ( 3 4 ) , w e c a n s i m p l i f y ( 3 0 ) t o

.

~) u r (

~ B

~) u r (

~

r n

~

! b ) r n (

! ) n

~u ( n ] B r [ D~u

(35) ⋅

∑ λ λ

−

− λ

=

⋅ φ

H e n c e t o t a k e t h e r t h ( d i r e c t i o n a l ) d e r i v a t i v e o f B_n , w e f i r s t p e r f o r m n - r s t e p s o f t h e e v a l u a t i o n a l g o r i t h m ( 3 3 ) t o o b t a i n t h e

~ r bnλ−

u~)

( a n d t h e n e v a l u a t e t h e B e z i e r p o l y n o m i a l ( 3 5 ) u s i n g t h e s a m e a l g o r i t h m , b u t n o w w i t h w e i g h t s u. , .v , w^. i n s t e a d o f u , v , w . F o r

) 35 . ( , 1

r = m e a n s t h a t t h e n 1(~u)

b~i− d e t e r m i n e t h e t a n g e n t p l a n e t o

~u) ( n ] B

[ φ - f o r r = 2 w e s e e t h a t t h e o s c u l a t i n g p a r a b o l o i d i s d e t e r m i n e d b y t h e n 2(~u) .

b~i−

A n o t h e r p o s s i b i l i t y t o c o m p u t e D_~.ur B_n i s g i v e n b y

,

~u) r (

n

~i

r n B~i

~) u r ( b~i

! ) r n (

! ) n

(~u n ] B r [

~u D ) 36

( ∑− ⋅ −

= −

⋅ φ

w h i c h i s p r o v e d f r o m ( 2 9 ) ( F a r i n 1 9 7 9 ) . W e h a v e t h u s a s e c o n d m e t h o d t o c o m p u t e t h e r t h d e r i v a t i v e o f Bnφ: f i r s t , p e r f o r m r s t e p s o f

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15.

a l g o r i t h m ( 3 3 ) w i t h w e i g h t s u^⋅, v^⋅, w^⋅ t o o b t a i n t h e r (u) , t h e n bi

e v a l u a t e t h e B é z i e r p o l y n o m i a l ( 3 6 ) u s i n g ( 3 3 ) w i t h w e i g h t s u , v , w . A c t u a l l y , o n e c a n s w i t c h f r o m o n e m e t h o d t o t h e o t h e r a t e a c h s t e p . *

L e t u s n o w e v a l u a t e d e r i v a t i v e s a c r o s s a b o u n d a r y o f T , s a y e3, t h i s i m p l i e s w = 0 . F r o m ( 3 6 ) w e s e e t h a t t h e r t h d e r i v a t i v e o f

i s a B é z i e r p o l y n o m i a l o f d e g r e e n - r w i t h B é z i e r o r d i n a t e s nφ

B ). u~

b_~i^r ( O n t h e b o u n d a r y e3, t h i s B é z i e r p o l y n o m i a l w i l l o n l y d e p e n d o n t h o s e b^r_i_~(u~) f o r w h i c h k = 0 .

T h e r e f o r e [B ] _e₃

~.

D_u^r _nφ | d e p e n d s o n l y o n t h e r + 1 p a r a l l e l s ( o f B é z i e r o r d i n a t e s ) t o e₃.

N o t e a l s o t h a t [B ] e3

D~u_&r _nφ | i s a n ( n - r ) t h d e g r e e u n i v a r i a t e p o l y n o m i a l

i n v :

) v r ( nj B

~u) r (

n 0 j

r b~i

! ) r n (

!

| n

3 3 e ] B

~[ D ) 37

( _u^r _n − −

− =

= ∑

φ &

&

w h e r e ~i3 i s s h o r t f o r ( n - r - j , j , 0 ) .

* T h e r e l a t i o n s h i p o f t h i s s t a t e m e n t w i t h t h e u n i v a r i a t e c a s e b e c o m e s c l e a r i f w e v i e w t e r m s i n ~.u a s g e n e r a l i z a t i o n s o f t h e d i f f e r e n c e o p e r a t o r Δ.

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