Computer Aided Engineering Design

DESIGN OF SURFACES 203

rD

DD D

DD D

DD D

( , ) = = [... 1]

1

=0 =0

–1

( –1) 0

( –1) ( –1)( –1) ( –1)

0 0( –1) 00

–1

uuuu

j

n

i

m

ij

ij m m

mn mn m

mn m n m

nn

n

n

vv

v

v

ΣΣ

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎡⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

(7.5)

wherem and n are user-chosen degrees in parameters u and v. For a bi-cubic surface patch, we need
to specify 16 sets of data as control points and/or slopes. Though we can model patches with degrees
inu and vgreater than 3 and can as well choose the degrees unequal (m≠n), like in Example 7.1,
for most applications, use of bi-cubic surface patches seems adequate. Cubic curve models developed
in previous chapters can now be extended to fit in the schema given in Eq. (7.5).

7.1.1 Ferguson’s Bi-cubic Surface Patch

From Eq. 4.7, a point r(u) on the Hermite-Ferguson curve is given by

r(u) = φ 0 r(0) + φ 1 r(1) + φ 2 ru(0) + φ 3 ru(1)

Here,r(0) and r(1) are two end points of the curve and ru(0),ru(1) are the end tangents. The Hermite
blending functions Φi(u), (i = 0, 1, 2, 3) are given below.

φ 0 = (2u^3 – 3u^2 + 1), φ 1 = (–2u^3 + 3u^2 ), φ 2 = (u^3 –2u^2 + u), φ 3 = (u^3 – u^2 )

In matrix form, the equation for the above curve is written as

r

r

r

r

r

r

r

r

r

() = [ ]

(0)

(1)

(0)

(1)

= [ 1]

2–2 1 1

–3 3 –2 –1

00 10

10 00

(0)

(1)

(0)

(1)

0123

uuuu^32

u

u

u

u

φφφφ

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

(7.6)

Figure 7.1 Example of a tensor product surface

3.5

3

2.5

2

1.5

1

0.5

0

2

1

0

–1 0

0.5

1

1.5

x

y

z