198 COMPUTER AIDED ENGINEERING DESIGN
–2
–1(^01)
2
4
3
2
1
0
–2
–1
0
1
2
8
4
2
0
6
4
2
–5
0
5
(a) (b)
Figure P6.1
(c) (d) (e)
(d): r( , ) = 2 cosh
1
2 cos , 2 cosh
1
uu uuuvvvv 2 sin , ; [–3, 3], [0, 2 ]
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟
⎧
⎨
⎩
⎫
⎬
⎭
∈∈π
(e): r( , ) = 2 sin cos , 2 sin sin , 2 cos + 2 ln tan 2 ; 2 ,
3
uuuu 2 , [0, 2 ]
u
vvvv⎛ u
⎝⎜
⎞
⎠⎟
⎧
⎨
⎩
⎫
⎬
⎭
∈⎡
⎣⎢
⎤
⎦⎥
π ∈
π
π
- Figure P6.2 shows a Mobius strip. Find the tangents and normal for the surface. Show that the normal at
(u, 0) has two different values at the same point, that is, ulim ( , 0) →πnu = (0, 0, – 1) and ulim→–πn( , 0)u =
(0, 0, 1) depending upon whether we move along v = 0 in the CCW direction or clockwise direction.
The equation of the surface is
r( , ) = cos + sin uu 2 cos , sin + sin 2 sin , cos 2 , [– , ], [– 0.5, 0.5]
u
uu
u
u
u
vvvvv⎜⎛⎝ ⎞⎠⎟ ⎝⎜⎛ ⎞⎠⎟ ⎛⎝⎜ ⎟⎞⎠ u
⎧
⎨
⎩⎫
⎬
⎭∈∈ππ