Computer Aided Engineering Design

222 COMPUTER AIDED ENGINEERING DESIGN

⇒ r 3 (u, 0) = φ 0 (u)P 00 + φ 1 (u)P 10 + φ 2 (u)s 0 (0) + φ 3 (u)s 1 (0) (7.40a)

r(u, 1) = φ 0 (1)b 0 (u) + φ 1 (1)b 1 (u) + φ 2 (1)t 0 (u) + φ 3 (1)t 1 (u)

+φ 0 (u)a 0 (1) + φ 1 (u)a 1 (1) + φ 2 (u)s 0 (1) + φ 3 (u)s 1 (1) –r 3 (u, 1) = b 1 (u)

⇒ r 3 (u, 1) = φ 0 (u)P 01 + φ 1 (u)P 11 + φ 2 (u)s 0 (1) + φ 3 (u)s 1 (1) (7.40b)
Now,

∂

∂v

r(u,v) = ∂

∂v

r 1 (u,v) + ∂

∂v

r 2 (u,v)–∂

∂v

r 3 (u,v)

= ∂

∂v

φ 0 (v)b 0 (u) + ∂

∂v

φ 1 (v)b 1 (u) + ∂

∂v

φ 2 (v)t 0 (u) + ∂

∂v

φ 3 (v)t 1 (u)

+φ 0 (u) ∂

∂v

a 0 (v) + φ 1 (u) ∂

∂v

a 1 (v) + φ 2 (u) ∂

∂v

s 0 (v)

+φ 3 (u) ∂

∂v

s 1 (v)–∂

∂v

r 3 (u,v) (7.41)

The twist vectors χχχχχij, initially introduced in section 7.1.1, are the mixed derivatives defined as

ij ui j i j j ui

u

u

u

= ( , ) = ( ) = ( ) ui j, = 0, 1; = 0, 1

2

=, = = =

∂

∂∂

∂

∂

∂

v ∂

v

v

rstvvv (7.42)

Thus, for v = 0, realizing from Figure 7.15 that ∂
∂v
a 0 (0) = t (0) and ∂
∂v
a 1 (0) = t 0 (1), Eq. (7.41)

becomes

∂

∂v

r(u, 0) = t 0 (u) + φ 0 (u)t 0 (0) + φ 1 (u)t 0 (1) + φ 2 (u)χχχχχ 00 + φ 3 (u)χχχχχ 10 – ∂

∂v

r 3 (u, 0)

=t 0 (u)

⇒ ∂
∂v

r 3 (u, 0) = φ 0 (u)t 0 (0) + φ 1 (u)t 0 (1) + φ 2 (u)χχχχχ 00 + φ 3 (u)χχχχχ 10 (7.43a)

Similarly, for v = 1, noting that ∂
∂v
a 0 (1) = t 1 (0) and ∂
∂v
a 1 (1) = t 1 (1),

∂

∂v

r(u, 1) = t 1 (u) + φ 0 (u)t 1 (0) + φ 1 (u)t 1 (1) + φ 2 (u)χχχχχ 01 + φ 3 (u)χχχχχ 11 – ∂

∂v

r 3 (u, 1)

=t 1 (u)

⇒ ∂
∂v
r 3 (u, 1) = φ 0 (u)t 1 (0) + φ 1 (u)t 1 (1) + φ 2 (u)χχχχχ 01 + φ 3 (u)χχχχχ 11 (7.43b)

From Eqs. (7.40) and (7.43), we can use bi-cubic lofting with respect to v to get the corrected surface,
that is

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

∂v

r 3 (u, 0) + φ 3 (v) ∂

∂v

r 3 (u, 1)

or r 3 (u,v) = φ 0 (v) [φ 0 (u)P 00 + φ 1 (u)P 10 + φ 2 (u)s 0 (0) + φ 3 (u)s 1 (0)]

+φ 1 (v) [φ 0 (u)P 01 + φ 1 (u)P 11 + φ 2 (u)s 0 (1) + φ 3 (u)s 1 (1)]

+φ 2 (v) [φ 0 (u)t 0 (0) + φ 1 (u)t 0 (1) + φ 2 (u)χχχχχ 00 + φ 3 (u)χχχχχ 10 ]

+φ 3 (v) [φ 0 (u)t 1 (0) + φ 1 (u)t 1 (1) + φ 2 (u)χχχχχ 01 + φ 3 (u)χχχχχ 11 ]