Computer Aided Engineering Design

DESIGN OF SURFACES 233

Once the 16 data points for a bi-cubic Bézier patch are chosen, in choosing the data points for
adjacent patches, restrictions are strict. The 16 data points for the first patch, say A (Figure 7.24) can

Patch I

Patch II

Figure 7.23 Arrangement of polyhedral edges for gradient continuity (Case I)

Figure 7.24 Four adjacent Bézier patches

C

D

A

B

be chosen freely. For an adjacent patch B, four
data points get constrained from the positional
continuity requirement. For λ = λ 1 in Eq. (7.55),
four out of the remaining 12 are further constrained
to maintain the tangent plane continuity at the
common boundary. Thus, only 8 points for patch
B can be freely chosen. For patch C also adjacent
to patch A, similar is the case in that 8 points can
be freely chosen. For patch D adjacent to both B
and C, only 4 of the 16 data points can be freely
chosen.
An alternative gradient continuity condition,
and a solution to Eq. (7.53) can be:

Case II

∂

∂

∂

∂

∂

uu∂

(0, ) = ( ) (1, ) + ( ) (1, )rrrII vvvvI I

v

λμv (7.56)

whereμ(v) is another scalar function of v. Note that Eq. (7.56) satisfies the requirement in Eq. (7.53)

and is a more general solution than that in Eq. (7.54). Eq. (7.56) suggests that ∂
∂u
rII(0,v) or ruII

(0,v) lies in the same plane as ruI (1, v) and rvI(1,v), i.e., the tangent plane of patch I at the boundary
point concerned.
In matrix form, Eq. (7.56) can be written as

[0 0 1 0] = ( ) [3 2 1 0] + ( ) [1 1 1 1]

3

2

1

0

B B

II

B

TT

B B

I

B

TT

B B

I

B

T

2

MGMV λμvvMGMV MGM

v

v

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

(7.57)