212 COMPUTER AIDED ENGINEERING DESIGN
r 00
r 01
r 02
r 03
r 13
r 23
r 33
r 30
r 10 r^20
r 11
r 21
r 31
r(ui,vj)
r 32
r 22
r 12
Figure 7.6 Schematic of a bi-cubic Bézier patch with its control polynet
[(1 – ) (1 – )... ]
..
..
......
......
..
..
–1
00 01 0( –1) 0
10 11 1( –1) 1
( –1)0 ( –1)1 ( –1)( –1) ( –1)
01 (–1)
umu ummmu
nn
nn
mm mnmn
mm mnmn
rr r r
rr r r
rr r r
rr r r
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎢⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
(1 – )
(1 – )–1
v
vv
v
n
n
n
n
M
(7.27)
For a bi-cubic Bézier surface patch, for instance, the equation above becomes
r
rrrr
rrrr
rrrr
rrrr
( , ) = [(1 – ) 3 (1 – ) 3 (1 – ) ]
(1 – )
3 (1 – )
3 (1 – )
3223
00 01 02 03
10 11 12 13
20 21 22 23
30 31 32 33
3
2
2
3
uuuuuuuv
v
vv
vv
v
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
= [ 1]
–1 3 –3 1
3–6 30
–3 3 0 0
1000
–1 3 –3 1
3–6 30
–3 3 0 0
1000
32
00 01 02 03
10 11 12 13
20 21 22 23
30 31 32 33
uuu
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
rrrr
rrrr
rrrr
rrrr⎢⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
T
v
v
v
3
2
1