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Fig. 8.4 The X Y axial
system translated (t x ,t y ) and
rotated β
A quick test confirms that T t x ,t y R β R 1
T 1
t x ,t y =
I ,i.e.
β
T t x ,t y R β R 1
T 1
T t x ,t y T 1
t x ,t y =
t x ,t y =
I
β
or
cos β
sin β x
cos β
sin β
t x cos β t y sin β
100
010
001
=
.
sin β
cos β y
sin β
cos βt x sin β t y cos β
0
0
1
0
0
1
Figure 8.4 shows how the frame of reference X T Y T is created for the intermedi-
ate translated frame, followed by a rotation of β about the new origin, ending with
the final frame of reference X Y .
Let's test ( 8.1 ) with the example shown in Fig. 8.5 where
β
=
90°
(t x ,t y )
=
( 10 , 5 )
(x, y)
=
( 9 , 6 )
1
1
1
01
5
9
6
1
=
10 10
00
(8.2)
1
and we see that ( 9 , 6 ) in XY becomes ( 1 , 1 ) in X Y .
Fig. 8.5 The X Y axial
system is translated ( 10 , 5 )
and rotated 90°
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