Computer Aided Engineering Design

208 COMPUTER AIDED ENGINEERING DESIGN

Suudud

u

= |u( , ) ( , ) |

,v

∫∫ rrvvv× v (7.19)

For Ferguson’s bi-cubic patch, from Eq. (7.12)

r UM GM V r UMG M V

r UM M V r UM G M V

r UMG M V

u

uTT TT

uu

uu T T

u

uTT

TT

uu

uu

u

(, ) = (, ) = ( )

(, ) = (, ) = ( )

(, ) = ( )

vv

vv

v

v

v

v

v

vv

vv

(7.20)

whereMu and Muu are M 1 and M 2 , respectively in Eq. (4.9). Muu and Mvv are identical with the
difference that they are used with their respective parameter matrices U and V.

MMuuu = = M M

0000

6–6 3 3

–6 6 –4 –2

00 10

, = =

0000

0000

12 – 12 6 6

–6 6 –4 –2

vvv

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

(7.21)

Example 7.2. A Ferguson surface patch has the following geometric coefficients:

G =

(6, 0, 0) (6, 0, 6) (0, 0, 6) (0, 0, 6)

(0, 6, 0) (0, 6, 6) (0, 0, 6) (0, 0, 6)

(0, 5, 0) (0, 5, 0) 0 0

(– 5, 0, 0) (–5, 0, 0) 0 0

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

Determine the tangents, normal, Gaussian curvature, mean curvature, principal curvatures, and equation
of the tangent plane at r(u = 0.5, v = 0.5). Also determine whether the surface is developable and find
the surface area of the patch.
The patch is given by r(u,v) = UMGMTVT = [xyz] = [(7u^3 – 13u^2 + 6) (–7u^3 + 8u^2 + 5u) (6v)]
whose plot is shown in Figure 7.3. At (u = 0.5, v = 0.5), the co-ordinates are (3.625, 3.625, 3). Using
Eq. (7.20), the slopes at any point on the surface are given by

ru(u,v) = [(21u^2 – 26u) (–21u^2 + 16u + 5) 0]

rv(u,v) = [0 0 6]

In particular, at (u = 0.5, v = 0.5),

ru(0.5, 0.5) = [–7.75 7.75 0]

rv(0.5, 0.5) = [0 0 6]

The unit normal can be determined using

ijk

• 7.75 7.75 0 ij
  006

= 46.5 + 46.5