Computer Aided Engineering Design

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
  
  
  
  


• .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