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