Computer Aided Engineering Design

224 COMPUTER AIDED ENGINEERING DESIGN

+ [ ( ) ( ) ( ) ( )]

()

()

()

()

0101

0

1

2

3

aassvvvv

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

u

u

u

u

u

u

u

u

φ

φ

φ

φ

–[ () () () ()]

(0) (0)

(1) (1)

(0) (1)

(0) (1)

0123

00 10 0 1

01 11 0 1

0 0 00 10

1 1 01 11

φφφφvvvv

PPs s

PPs s

tt

tt





⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

u

u

u

u

u

u

u

u

φ

φ

φ

φ

0

1

2

3

()

()

()

()

so that ∂
∂u

r(0,v) = [φ 0 (v) φ 1 (v) φ 2 (v) φ 3 (v)] [s 0 (0) s 0 (1) χχχχχ 00 χχχχχ 01 ]T + s 0 (v)

–[ () () () ()]

(0)

(1)

0123 = ( )

0

0

00

01

φφφφvvvvv 0

s

s

s





⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

and ∂
∂u

r(1,v) = [φ 0 (v) φ 1 (v) φ 2 (v) φ 3 (v)] [s 1 (0) s 1 (1) χχχχχ 10 χχχχχ 11 ]T

+ ( ) – [ ( ) ( ) ( ) ( )]

(0)

(1)

1 0123 = ( )

0

0

10

11

s 1

s

s

vvvvvvφφφφ s





⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

Example 7.8. The boundary curves of a Coon’s patch consist of four cubic Bézier curves with the
following control points

a 0 (v)≡r 00 = (0, 0, 0) r 01 =^1
32

,^1

32

,^1

3

⎛

⎝⎜

⎞

⎠⎟

r 02 = 2 –^1

32

,^1

32

,^1

3

⎛

⎝⎜

⎞

⎠⎟

r 03 = ( 2,0,0)

b 1 (u)≡r 10 = r 03 r 11 = r 02 r 12 = 2 –^1
32

,2–^1

32

,^1

3

⎛

⎝⎜

⎞

⎠⎟

r 13 = (2, 2,0)

a 1 (v)≡r 20 = r 13 r 21 = r 12 r 22 =^1
32

,2–^1

32

,^13

⎛

⎝⎜

⎞

⎠⎟

r 23 = (0, 2 , 0)

b 0 (u)≡r 30 = r 23 r 31 = r 22 r 32 = r 01 r 33 = r 00