212 COMPUTER AIDED ENGINEERING DESIGN
r 00r 01r 02r 03r 13r 23r 33r 30r 10 r^20r 11r 21r 31r(ui,vj)
r 32r 22r 12Figure 7.6 Schematic of a bi-cubic Bézier patch with its control polynet[(1 – ) (1 – )... ]..
..
......
......
..
..–100 01 0( –1) 0
10 11 1( –1) 1( –1)0 ( –1)1 ( –1)( –1) ( –1)
01 (–1)umu ummmunn
nnmm mnmn
mm mnmnrr r r
rr r rrr r r
rr r r⎡⎣⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎢⎢⎤⎦⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎡⎣⎢
⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥
⎥(1 – )
(1 – )–1v
vvvn
nnn
M
(7.27)For a bi-cubic Bézier surface patch, for instance, the equation above becomes
rrrrr
rrrr
rrrr
rrrr( , ) = [(1 – ) 3 (1 – ) 3 (1 – ) ](1 – )
3 (1 – )
3 (1 – )322300 01 02 03
10 11 12 13
20 21 22 23
30 31 32 333
2
2
3uuuuuuuvv
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
10003200 01 02 03
10 11 12 13
20 21 22 23
30 31 32 33uuu⎡⎣⎢
⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥
⎥⎡⎣⎢
⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥
⎥⎡⎣⎢
⎢
⎢
⎢rrrr
rrrr
rrrr
rrrr⎢⎢⎤⎦⎥
⎥
⎥
⎥
⎥⎡⎣⎢
⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥
⎥T
v
v
v3
21