28 COMPUTER AIDED ENGINEERING DESIGN
Example 2.3. Lamina ABCD with an inner point P with coordinates (4, 3), (3, 1), (8, 1), (7, 4) and
(5, 2) respectively is first rotated through 60° and then translated by (5, 4). In another sequence, the
trapezoid is first translated by (5, 4) and then rotated through 60°. The lamina acquires different
positions and orientations given and shown below for the two sequences of transformations.
For rotation and then translation using Eq. (2.5), we have
′
′
′
′
′
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
°°
°°
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎡
⎣
⎢
⎢
⎢
⎢
A
B
C
D
P
T T
=
cos 60 –sin 60 5
sin 60 cos 60 4
001
431
311
811
741
521
=
4.43 8.96 1
5.63 7.09 1
8.13 11.42 1
5.03 12.06 1
5.76 9.33 1
⎢⎢
⎢
⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
T
For translation and then rotation, Eq. (2.6) gives
A
B
C
D
P
*T T
*
*
*
*
=
cos 60 –sin 60 5 cos 60 – sin 60
sin 60 cos 60 5 cos 60 + 4 sin 60
00 1
431
311
811
741
521
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
°° ° °
°° ° °
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
4
=
- 1.56 11.30 1
- .33 9.43 1
2.17 13.76 1 - .93 14.40 1
- .33 9.43 1
- .19 0 11.66 1
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
T
The two different laminar positions and orientations are shown in Figure 2.5.
–5 0 5 10 15
(b) Translation before rotation
I,
translation
II, rotation
15
10
5
0
–2 0 2 4 6 8 10
(a) Rotation before translation
I, rotation
II, translation
12
10
8
6
4
2
Figure 2.5 An example depicting the significance of order in transformations