Computer Aided Engineering Design

210 COMPUTER AIDED ENGINEERING DESIGN

Figure 7.5 Bi-cubic surface patch in 16 point form

r(0, 1)

r(1/3, 1) r(2/3, 1)

r(1, 1)

r(0, 2/3)

r(0, 1/3)

r(0, 0) r(1, 0)

r(1/3, 0) r(2/3, 0)

r(1, 1/3)

r(1, 2/3)

r(1/3, 2/3) r(2/3, 2/3)

r(1/3, 1/3) r(2/3, 1/3)

( – 3.625) – 7.75 0

( – 3.625) 7.75 0

( – 3) 0 6

= 0

x

y

z

46.5 (x– 3.625) + 46.5(y– 3.625) = 0, that is x + y = 7.25

The surface area of the patch can be obtained by Eq. (7.19) as

ru(u,v)×rv(u,v) = 6 (–21u^2 + 16u + 5)i– 6(21u^2 – 26u)j

|ru(u,v)×rv(u,v) | = 6[(–21u^2 + 16u + 5)^2 + (21uu^22 – 26 ) ]

1

(^2) = f (u)
Sfudud
u
= ( ) = 54.64
=0
1
=0
1
∫∫v
v

7.1.3 Sixteen Point Form Surface Patch

For 16 uniformly spaced points on a surface patch, to fit a bi-cubic tensor product surface of the form
given in Eq. (7.5) with m = n = 3,

rD

DDDD

DDDD

DDDD

DDDD

( , ) = = [ 1]

1

=0

3

=0

3

32

33 32 31 30

23 22 21 20

13 12 11 10

03 02 01 00

3

2

uuuuu

ji ij

vvij

v

v

v

ΣΣ

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

(7.22)

has 16 unknowns Dij to be determined. Let u∈ [0, 1], v∈ [0, 1] and each interval be subdivided as
[0, 1/3, 2/3, 1]. The given sixteen points on the surface are (each rij is a triplet) such that