Computer Aided Engineering Design

220 COMPUTER AIDED ENGINEERING DESIGN

From Eqs. (7.35a) and (7.35b), we realize that the two boundary curves for r 3 (u,v),r 3 (u, 0) and r 3
(u, 1), are available that can be linearly blended along parameter v. Or

r 3 (u,v) = (1 –v)r 3 (u, 0) + vr 3 (u, 1)

= (1 –v)[(1 –u)P 00 + uP 10 ] + v[(1 –u)P 01 + uP 11 ]

= (1 –v)(1 –u)P 00 + u(1 –v)P 10 + (1 –u)vP 01 + uvP 11 (7.36)

We may as well attempt to meet boundary conditions using r(0,v) = a 0 (v) and r(1,v) = a 1 (v) to get

r(0,v) = (1 –v)b 0 (0) + vb 1 (0) + a 0 (v)–r 3 (0,v)

= (1 –v)P 00 + vP 01 + a 0 (v)–r 3 (0,v) = a 0 (v)

⇒ r 3 (0,v) = (1 –v)P 00 + vP 01 (7.37a)

and r(1,v) = (1 –v)b 0 (1) + vb 1 (1) + a 1 (v)–r 3 (1,v)

= (1 –v)P 10 + vP 11 + a 1 (v)–r 3 (1,v) = a 1 (v)

⇒ r 3 (1,v) = (1 –v)P 10 + vP 11 (7.37b)

and thereafter linearly blend r 3 (0,v) and r 3 (1,v) with respect to u. However, observe from Eq. (7.36)

Figure 7.14 (a) Bi-linear Coon’s patch and (b) constituents of the Coon’s patch

v

a 0 (v)

P 01 b 1 (u)

P 11

a 1 (v)

P 10

P (^00) b
0 (u)
u
(a)
r(u,v)
P 01
b 1 (u)
P 11
P 00
b 0 (u)
P 10
r 1 (u,v)
P 01 P^11
P (^00) r 3 (u,v) P 10
(b)
P 01 P 11
P 00
a 0 (v)
P 10
a 1 (v)
r 2 (u,v)