Computer Aided Engineering Design

DIFFERENTIAL GEOMETRY OF SURFACES 191

T() =

1000

0100

001

0001

v

v

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

Thus, the equation of the cylinder (in homogeneous co-ordinates) produced by this sweep is given by

r(u,v) = T(v)C(u) = [a cos u,b sin u,v, 1]T

Following examples illustrate some other sweep surfaces:

Example 6.7

(a) Helical tube: The surface of a helical tube is produced if we sweep a circular cross section of
radiusr along a helix γ(t) = (a cos t,a sin t,bt). The unit tangent t(t), binormal b(t), and normal n(t)
to the helical curve are given by

t

b

=

/

| / |

=

(– sin , cos , )

+

,

=

(– sin , cos , ) (– cos , – sin , 0)

(– sin , cos , ) (– cos , – sin , 0)

=

( sin , – cos , )

+

22

22

ddt

ddt

ata tb

ab

ata tb a tat

ata tb a tat

btb ta

ab





×

×

n = b×t = (– cos t, – sin t, 0)

The equation of the tube thus formed is given by

r(t,θ) = (t) + r[–n cos θ + b sin θ]

r(t,θ) = (t) + r[– cos θ(– cos t, – sin t, 0) + sin

(^22) +
θ
ab
(b sin t,–b cos t,a)]
(b) Seashell: A seashell is a helical tube but the radius r of the tube increases as the circular cross
section sweeps along the helical backbone curve. One may use the following model for the surface:
Figure 6.20 Cylinders (a) 0 ≤u≤ 2 πππππ and (b) (0 ≤u≤ 3 πππππ/2)
–2
–1
0
1
2
2
1.5
1
0.5
0
1
0.5
0

• 0.5
  –1

–2

–1

0

1

2

2

1.5

1

0.5

0.5

0

• 0.5
  –1

0

1