Computer Aided Engineering Design

40 COMPUTER AIDED ENGINEERING DESIGN

2.5.2 Scaling in Three-Dimensions

The scaling matrix can be extended from that in a two-dimensional case (Eq. 2.18) as

S =

000

000

00 0

0001

μ

μ

μ

x

y

z

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

(2.29)

whereμx,μy and μz are the scale factors along x,y and z directions, respectively. For uniform overall
scaling,μx = μy = μz = μ.

Alternatively,

S 1 =

1000

0100

0010

000 s

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

has the same uniform scaling effect as that of Eq. (2.29). To observe this, we may write

x

y

z

s

x

y

z

x

y

z

s

x s y s z s

* s

*

*

1

=

1000

0100

0010

000 1

=

1

=

(^10)
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
≡
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥
000
0 1 00
001 0
0001
1
s
s
x
y
z
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
(2.30)
comparing which with Eq. (2.29) for μx = μy = μz = μ yields μ =^1
s

. Figure 2.17 shows uniform
scaling of a cylindrical primitive.

x

z

y

Figure 2.16 Rotation of a disc about its axis