Computer Aided Engineering Design

214 COMPUTER AIDED ENGINEERING DESIGN

Figure 7.7 (a) Example of a quadratic-cubic Bézier surface with control polynet and (b) change in
patch’s shape with relocation of two control points

xx x x

yy y y

zz z z

x

y

z

z

pupxp

pup p

pup p

= 0

• 1.5 0 3
  
  
  • 1 2 0
  
  
  
  


• 0.75 0 0

v = 0 = 0.75

v

⇒⇒

Also, at u = 0.5, v = 0.5

rr

rrr

u

uu u

= {0, 2, 0}; = {3, 0, 0}

= {0, 0, 0} = {0, 0, 0} = {0, 0, – 6}

v

vvv

⇒ G 11 = 4, G 12 = 0, G 22 = 9; L = M = 0, N = 6

K LN M

GG G

= = –

• 12 = 0

2

11 22 122

κκ H GN G L GM

GG G

=^1

2

( + ) =

+ – 2

2( – )

=^1

(^123)
11 22 12
11 22 12
κκ 2
⇒ κ 1 =^2
3
,κ 2 = 0
Hence, the radius of curvature (at rp) = 1/κ 1 = 1.5 and since the Gaussian curvature K = 0, the surface
is developable.
For control points r 10 and r 13 changed as (0, 1, 0.5) and (3, 1, 0.5) respectively, lifting them up by
0.5 units each along the z-direction, the new equation of the surface is
r( , ) = [(1 – ) 2 (1 – ) ]
(0, 0, 0) (1, 0, 1) (2, 0, 1) (3, 0, 0)
(0, 1, 0.5) (1, 1, 1) (2, 1, 1) (3, 1, 0.5)
(0, 2, 0) (1, 2, 1) (2, 2, 1) (3, 2, 0)
(1 – )
3 (1 – )
3 (1 – )
22
3
2
2
3
uuuuuv
v
vv
vv
v
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
The new shape of the surface is shown in Figure 7.7(b).
1
0.5
0
3
2
1
0 0 1
2
3
4
z
y x
1
0.5
0
z
3
2
1
0 0 1
2 3
4
x
y
(a) (b)