Computer Aided Engineering Design

SPLINES 141

Fork= 2, y[xj, xj+ 1 , xj+ 2 ]=
()

• ()


• 
  

  ()

  
  

+1

+1

+2

+2

y

wx

y

wx

y

wx

j

j

j

j

j

′′ ′j

Here,w(x) = (x–xj)(x–xj+ 1 )(x–xj+ 2 ) so that

w′(x) = (x–xj+ 1 )(x–xj+ 2 ) + (x–xj)(x–xj+ 2 ) + (x–xj)(x–xj+ 1 )

Thus,

w′(xj) = (xj–xj+ 1 )(xj–xj+ 2 ),w′(xj+ 1 ) = (xj+ 1 – xj)(xj+ 1 – xj+ 2 ) and w′(xj+ 2 ) = (xj+ 2 – xj)(xj+2–xj+ 1 )

This gives

y[xj, xj+ 1 , xj+ 2 ]

=

(– )( – )

+

(– )( – )

+

+1 + 2 (– )( – )

+1

+1 +1 + 2

+2

+2 +2 +1

y

xx xx

y

xxxx

y

xxxx

j

j j j j

j

j j jj

j

j j jj

=

( – ) – ( – ) + ( – )

( – )( – )( – )

+2 +1 +1 +2 +2 +1

+2 +1 +2 +1

yx x y x x y x x

xxxxxx

j jj jjj jjj

j j j j jj

=

( – ) – ( – + – ) + ( – )

( – )( – )( – )

+ 2 +1 +1 + 2 +1 +1 + 2 +1

+2 +1 +2 +1

yx x y x x x x y x x

xxxxxx

j jj jjjjj jj j

j j j j jj

=

( – )( – ) + ( – )( – )

( – )( – )( – )

+ 2 +1 +1 +1 + 2 +1

+2 +1 +2 +1

xxyy xxyy

xxxxxx

jjj jjj jj

j j j j jj

=^1

(– )

(– )

(– )

• 
  

(– )

(– )

=

[ , ] – [ , ]

+2 (– )

+2 +1

+2 +1

+1

+1

+1 + 2 +1

xx +2

yy

xx

yy

xx

yx x yx x

j j xx

jj

jj

j j

j j

jj j j

j j

⎡

⎣

⎢

⎤

⎦

⎥

5.5.1 Divided Difference Method to Compute B-Spline Basis Functions
To compute a B-spline basis function of order musing divided differences, consider a truncated
power function (Figure 5.8(a))

ft t

t t

t

m

m

( ) = =

0

, 0

, < 0

+

–1

⎧ –1 ≥

⎨

⎪

⎩⎪

(5.18)

t

(a)

t

(b)

tj

t+m–1

(– )ttj +m–1

Figure 5.8 Plots of truncated power functions