Computer Aided Engineering Design

DESIGN OF SURFACES 225

To construct a bi-linear Coon’s patch, the four boundary curves are given by

a

r

r

r

r

a

r

r

r

r

0

32

00

01

02

03

1

32

20

21

22

23

() = [ 1]

–1 3 –3 1

3–6 30

–3 3 0 0

1000

, ( ) = [ 1]

–1 3 –3 1

3–6 30

–3 3 0 0

1000

vvvvvvvv

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

b

r

r

r

r

b

r

r

r

r

0 32

30

31

32

33

1 32

10

11

12

13

() = [ 1]

–1 3 –3 1

3–6 30

–3 3 0 0

1000

, ( ) = [ 1]

–1 3 –3 1

3–6 30

–3 3 0 0

1000

uuuu uuuu

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

which can be used directly with Eq. (7.38). Stepwise results are shown in Figure 7.16.

0.4

0.3

0.2

0.1

0

1.5

1

0.5

0 0 0.5

1

1.5

0.4

0.3

0.2

0.1

0

1.5 1

0.5

0 0 0.5

1 1.5

0.4

0.3

0.2

0.1

0

1.5

1

0.5

0 0 0.5

1

1.5

(a) Bézier boundary curves

(b)r 1 (u,v) (c)r 2 (u,v)

Figure 7.16 Bilinear Coon’s patch in Example 7.8

0.4

0.3

0.2

0.1

0

1.5 1

0.5

0 0 0.5

1

1.5

0.4

0.3

0.2

0.1

0

1.5 1

0.5

0 0 0.5

1 1.5

(d)r 3 (u,v) (e) Final patch r 1 (u,v) + r 2 (u,v)–r 3 (u,v)