Computer Aided Engineering Design

DESIGN OF SURFACES 211

rrrr

rrrr

rrrr

rrrr

rr r r

rr r r

rr r r

33 32 31 30

23 22 21 20

13 12 11 10

03 02 01 00

=

(1, 1) 1,^2

3

1,^1

3

(1, 0)

2

3

, 1^2

3

,^2

3

2

3

,^1

3

2

3

, 0

1

3

, 1^1

3

,^2

3

1

3

,^1

3

1

3

, 0

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

()()

()( )( )( )

()( )( )(()

()()

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

rr r r(0, 1) 0, ⎥

2

3

0,^1

3

(0, 0)

(7.23)

From Eqs. (7.22) and (7.23)

rrrr

rrrr

rrrr

rrrr

DDDD

DDDD

DDDD

DDD

33 32 31 30

23 22 21 20

13 12 11 10

03 02 01 00

33 32 31 30

23 22 21 20

13 12 11 10

03 02 01

=

1111

8

27

4

9

2

3 1

1

27

1

9

1

3

1

0001

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥ DD 00

1 8

27

1

27

0

1 4

9

1

9

0

1 23 13 0

11 1 1

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

(7.24)

using which

DDDD

DDDD

DDDD

DDDD

M

rrrr

rrrr

rrrr

rrrr

MM

33 32 31 30

23 22 21 20

13 12 11 10

03 02 01 00

16

33 32 31 30

23 22 21 20

13 12 11 10

03 02 01 00

= 16 , where 16 =

1111

8

27

4

9

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

T

22

3

1

1

27

1

9

1

3

1

0001

–1

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

The expression for a surface patch interpolating 16 uniformly spaced points is then

rM

rrrr

rrrr

rrrr

rrrr

( , ) = [ 1] M

1

32

16

33 32 31 30

23 22 21 20

13 12 11 10

03 02 01 00

16

3

2

uuuuv T

v

v

v

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

(7.25)

7.1.4 Bézier Surface Patches

Similar to Bézier curves employing Bernstein polynomials as weight functions with control points
(Chapter 4), a tensor product Bézier surface patch is given by

rr(, ) = () ()

=0 =0

uBuB

i

m

j

n

ij i

m

j

vvΣΣ n (7.26)

whererij,i = 0,... , m,j = 0,... , n are the control points and Buim() and Bjn()v are Bernstein
polynomials in parameters u and v. The control points form the control polyhedron orcontrol polynet
of the surface (Figure 7.6). For any u = u 0 ,r(u 0 ,v) is a Bézier curve of degree n. Likewise, for any
v = v 0 ,r(u,v 0 ) is a Bézicr curve of degree m. Eq. (7.26) may be written in the form