Computer Aided Engineering Design

228 COMPUTER AIDED ENGINEERING DESIGN

For slopes tij

t 01 = [min (| P 01 – P 00 |, | P 02 – P 01 |)]

PP

PP

02 00

02 00

• | – |

= [min (| (0, 1, 0) – (0, 0, 0) |, | (0, 1, 2) – (0, 1, 0)]

(0, 1, 2) – (0, 0, 0)

| (0, 1, 2) – (0, 0, 0) |

= 0,^1

5

,^2

5

⎛

⎝⎜

⎞

⎠⎟

t 11 = [min (| P 11 – P 10 |, | P 12 – P 11 |)]
PP
PP

12 10

12 10

• | – |

= [min (| (1, 1, 0) – (1, 0, 0) |, | (1, 1, 2) – (1, 1, 0)] (1, 1, 2) – (1, 0, 0)

| (1, 1, 2) – (1, 0, 0) |

= 0,^1

5

,^2

5

⎛

⎝

⎜

⎞

⎠

⎟

t 21 = [min (| P 21 – P 20 |, | P 22 – P 21 |)]
PP
PP

22 20

22 20

• | – |

= [min (| (2, 1, 0) – (2, 0, 0) |, | (2, 1, 4) – (2, 1, 0)]

(2, 1, 4) – (2, 0, 0)

| (2, 1, 4) – (2, 0, 0) |

= 0,^1

17

,^4

17

⎛

⎝

⎜

⎞

⎠

⎟

Repeated application of Eq. (7.12) with the geometric matrix in Eq. (7.47) results in the following
composite surface with four patches shown in Figure 7.18.
To avoid local flatness or bulging, we can
compute the twist vectors from the data given
instead of specifying them as zero. Computations
are done by imposing the C^2 continuity condition
at patch boundaries. For patch I in Figure 7.17,
from Eq. (7.12), we have

rI(u,v) = UMGIMTVT (7.48a)

withGI defined as

G

PP t t

PP t t

ss

ss

I

+1 +1

+1 +1 +1 +1 +1 +1

+1 +1

+1 +1 +1 +1 +1 +1

=

|

|

––|– –

|

|

ij ij ij ij

ij ij ij ij

ij ij ij ij

ij ij ij ij





⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

(7.48b)

The unknown slopes and twist vectors can be computed as follows:
ForC^2 continuity along the common boundary between patches I and II

∂

∂

∂

∂

2

2

I^2

2

(1, ) = (0, )II

uu

rrvv

⇒ [6 2 0 0 ] MGIMTVT = [0 2 0 0] MGIIMTVT

or [6 –6 2 4] GIMTVT = [– 6 6 – 4 –2] GIIMTVT

or [6 –6 2 4] GI = [– 6 6 – 4 –2] GII

Figure 7.18 A composite Ferguson patch using the

FMILL method

4

2

0

–2

1.5

1

0.5

0 0

0.5

1

1.5

2

y x

z