Computer Aided Engineering Design

190 COMPUTER AIDED ENGINEERING DESIGN

rvv = (–u cos v,–u sin v, 0), ruv = (–sinv, cos v, 0), N = ru×rv = (– cos v, –sinv,u)

nrrrrrr =

(– cos , – sin , )

1 +

, = = 1 + , = = , = = 0

2 11

2

2 22

2

12

vv

vvv

u

u

G u

u

uu⋅⋅⋅GuGu

L

uu

MNru

u

= uu = – u

1

1 +

, = = 0, = =

(^22) 1 +
rn⋅⋅⋅rnvvvn
From the above, Gaussion and mean curvatures can be calculated:
K GGLN
u
H
GN G L
GG
uu
= = –^1
(1 + )
, =

• =^1
  (^1122) (1 + )
  22
  11 22
  (^1122232)
  The surface of revolution is shown as a funnel in Figure 6.19. The parallels are the circles with
  u = u 0 , a constant, while meridians are the curves for v = v 0 , a constant.
  Figure 6.19 Funnel as a surface of revolution
  Meridians
  Parallels

6.9 Sweep Surfaces

A large number of objects created by engineers are designed with sweep surfaces. Common examples
arewash-basin, volute of a hydraulic pump,aircondition ducting,helical pipe, corrugated sheets and
many more. A sweep surface consists of “cross section curves” swept along a directrix curve or cross
section curves with Hermite or B-spline blending.
A cylinder may be regarded as a sweep surface. If one considers the elliptical cross section curve
lying on the x-y plane swept linearly along the z-axis, it will form a cylinder. The equation of the cross
section curve is given in homogeneous coordinates by

C(u) = [a cos u,b sin u, 0, 1]T

Sweeping the curve along the z-axis through a distance v will mean applying a transformation matrix