Computer Aided Engineering Design

DESIGN OF SURFACES 223

or in matrix form

r 3 (u,v)

= [ ( ) ( ) ( ) ( )]

(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

0

1

2

3

φφφφ

φ

φ

φ

φ

vvvv

PPs s

PPs s

tt

tt





⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

u

u

u

u

(7.44)

The overall bi-cubic Coon’s patch is given by

r(u,v) = [φ 0 (v) φ 1 (v) φ 2 (v) φ 3 (v)] [b 0 (u) b 1 (u) t 0 (u) t 1 (u)]T

+ [ ( ) ( ) ( ) ( )]

()

()

()

()

0101

0

1

2

3

aassvvvv

φ

φ

φ

φ

u

u

u

u

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ –

[() () () ()]

(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

0

1

2

3

φφφφ

φ

φ

φ

φ

vvvv

PPs s

PPs s

tt

tt





⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

u

u

u

u

(7.45)

To verify from above that the other boundary conditions are met, we see that

r(0,v) = [φ 0 (v) φ 1 (v) φ 2 (v) φ 3 (v)] [b 0 (0) b 1 (0) t 0 (0) t 1 (0)]T

+ ( ) – [ ( ) ( ) ( ) ( )]

(0)

(0)

0 0123 = ( )

00

01

0

1

a 0

P

P

t

t

vvvvvvφφφφ a

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

r(1,v) = [φ 0 (v) φ 1 (v) φ 2 (v) φ 3 (v)] [b 0 (1) b 1 (1) t 0 (1) t 1 (1)]T

+ ( ) – [ ( ) ( ) ( ) ( )]

(1)

(1)

1 0123 = ( )

10

11

0

1

a 1

P

P

t

t

vvvvvvφφφφ a

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ∂

∂u

r(u,v) = [φ 0 (v) φ 1 (v) φ 2 (v) φ 3 (v)] ∂

∂

∂

∂

∂

∂

∂

∂

⎡

⎣⎢

⎤

u ⎦⎥

u

u

u

u

u

u

0101 () () () ()u

T

bbtt