192 COMPUTER AIDED ENGINEERING DESIGN
r(t,θ) = (t) + rt[– cos θ(– cos t, – sin t, 0) + sin(^22) +
θ
ab
(b sin t,–b cos t,a)]
Figure 6.21 (a) Helix tube (a = 2, b,r = 0.5) (b) Seashell (a = 2, b = 1, r = 0.2)
Figure 6.22 Corrugated sheet
(c) Corrugated sheet: A sinusoidal curve on the x-z plane along x-axis is given by
(t) = (t, 0, a sin t)
Sweeping this curve along y-axis by a distance v gives
r(, ) =
1000
010
0010
0001
0
sin
1
= sin
1
t
t
at
t
v at
v v
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
Note the use of homogeneous co-ordinate system. A plot of the corrugated surface is shown here with
a = 1 and v = 25 and t ranging from 0 to 8π.
6
4
2
0
–2
0
2 –2
0
2
15
10
5
0
–4
–2
0
2
4 –5–2.5
(^0) 2.5
20
10
0
20
10
0
0.0
- 0.1