Graphics Reference
In-Depth Information
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°