Graphics Reference
In-Depth Information
which is the inverse rotation tra
ns
form
R
−
β
or
R
−
β
. For example, the point
(
1
,
1
)
in
XY
, will have coordinates
(
√
2
,
0
)
in the frame of reference rotated 45°:
⎡
√
2
/
2
√
2
/
20
√
2
0
1
⎤
⎡
⎤
⎡
⎤
1
1
1
√
2
/
2
√
2
/
20
⎣
⎦
=
⎣
⎦
⎣
⎦
−
0
0
1
which is confirmed.
We have previously shown that two separate rotations of a point is equivalent
to a single composite rotation of a point. Similarly, it is a trivial exercise to prove
that two separate rotations of a frame is equivalent to single composite rotation of a
frame.
8.3.3 Rotated and Translated Frame of Reference
Having looked at translated and rotated frames of reference, let's combine the two
operations and develop a single transform. This is not too difficult to follow, so long
as we are careful with our definitions and diagrams.
When a point is rotated and translated we use the operation
P
=
T
t
x
,t
y
R
β
P.
We know that the transform for moving a frame of reference - whilst keeping a point
fixed - is the inverse of that used for moving points - whilst keeping the frame fixed.
Which suggests that the transform for a rotated and translated frame of reference is
the inverse of
T
t
x
,t
y
R
β
which is
(
T
t
x
,t
y
R
β
)
−
1
R
−
1
β
T
−
1
t
x
,t
y
=
and makes
P
=
R
−
1
β
T
−
1
t
x
,t
y
P
or
P
=
R
−
β
T
−
t
x
,
−
t
y
P.
Substituting the matrices for
R
−
β
and
T
−
t
x
,
−
t
y
we have
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
x
y
1
cos
β
sin
β
0
10
−
t
x
x
y
1
⎣
⎦
=
⎣
⎦
⎣
⎦
⎣
⎦
−
sin
β
cos
β
0
01
−
t
y
0
0
1
00
1
which simplifies to
⎡
⎤
⎡
⎤
⎡
⎤
x
y
1
cos
β
sin
β
−
t
x
cos
β
−
t
y
sin
β
x
y
1
⎣
⎦
=
⎣
⎦
⎣
⎦
.
−
sin
β
cos
βt
x
sin
β
−
t
y
cos
β
(8.1)
0
0
1
Equation (
8.1
) is the homogeneous matrix for converting points in the
XY
coordi-
nate system to the translated and rotated
X
Y
coordinate system.
