Computer Aided Engineering Design

SPLINES 149

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–3

–1 –3

–3

–2 –3 –2

–1

–1 – 2 –1 –2

–2

–1 – 2

tt

tt

tt

tt

tt

tt

tt

tt

tt

tt

i

ii

i

iii

i

iii

i

i i

i

ii

⎡ δδ

⎣⎢

⎤

⎦⎥

δδδi i

i i i

tt

–1 tt–1

+

• 
  


• 
  

⎡

⎣⎢

⎤

⎦⎥

or

Nt

tt

tttt

tt t t

tttt

tttt

i

i

iiii

i

ii

iiii

i i

3,

–3^2

–1 –3 – 2 –3

–2

–3 –1

–1 –3 –1 – 2

() = (– ) –2

(– )(– )

+

( – )( – )

(– )(– )

+

(– )( – )

(

δ

ttt t ti– )(iii–2 –1– )–2 i–1

⎡

⎣⎢

⎤

⎦⎥

δ

+

(– )

(– )( – )

2

–1 – 2

tt

tt tt

i

i i i i

δi (5.33)

Thus

Nt

tt

tt t t

tt t t

tt tt

tttt

i t

i

i ii i

i

i i

i i i i

ii

3, +1

–2

2

–2 –1 –2

–1

–2

–2 –1

() = (– ) +1 –1

(– )( – )

+

( – )( – )

(– )( – )

+

(– )( – )

(

δ

iiii i

ttt i

+1– )( – )–1 –1

⎡

⎣⎢

⎤

⎦⎥

δ

+

(– )

( – )( – )

+1

2

+1 +1 –1

+1

tt

tttt

i

i i ii

δi

Nt

tt

tttt

tt t t

tttt

tttt

i

i

iii i

i

ii

iiii

i i

3, +2

–1

2

+1 –1 –1

–1 +1

+1 –1 +1

() = (– ) +2

( – )( – )

+

(– )( – )

(– )( – )

+

(– )( – )

(

δ

ttttti+2– )(i i+1 – )i i+1

⎡

⎣⎢

⎤

⎦⎥

δ

+

(– )

(– )( – )

+2

2

+2 +1 +2

+2

tt

tttt

i

iiii

δi

and

Nt

tt

tttt

ttt t

tttt

tttt

i t

i

i i i i i

i i

i i ii

ii

3, +3 i

2

+2 +1 +1

+2

+2 +2 +1

+3 +1

+3

() =

(– )

(– )( – ) +

• )( – )
  (– )( – ) +

–– )

δ (–

(()(

tt tii i+1)(+ 2 – +1) i+2

⎡

⎣⎢

⎤

⎦⎥

δ

+

(– )

(– )( – )

+3^2

+3 +2 +3 +1

+3

tt

tttt

i

iiii

δi

Since for t∈ [ti,ti+1),δi+1 = 1 and δi–1 = δi = δi+2 = δi+3 = 0, we have

N3,i+1(t) + N3,i+2(t) + N3,i+3(t) =

(– )

(– )( – )

+1^2

+1 +1 –1

tt

tttt

i

i i ii

+

(– )( – )

( – )( – )

–1 +1

+1 –1 +1

tt t t

tttt

ii

iiii

+

(– )( – )

(– )( – )

+

(– )

( – )( – )

+2

+2 +1

2

+2 +1

tttt

tttt

tt

tttt

ii

iiii

i

iiii

=

( – )( – )

(– )( – )

+

(– )( – )

( – )( – )

+1 +1 –1

+1 +1 –1

+2

+2 +1

tttt

tttt

ttt t

tttt

iii

i i ii

i i i

i i i i

=

(– )

(– ) +

(– )

(– ) =

(– )

(– ) = 1

+1

+1 +1

+1

+1

tt

tt

tt

tt

tt

tt

i

i i

i

i i

i i

i i

(F) For m + 1 number of knots, degree p–1 basis functions and n + 1 number of control
points, m = n + p
Forn+1 control points and hence basis functions of order p, this property puts a limit on the number