Graphics Reference
In-Depth Information
equals the identity matrix,
R
−
1
β
and because
R
β
R
β
R
β
:
=
⎡
⎤
cos
β
sin
β
0
R
−
1
β
⎣
⎦
=
−
sin
β
cos
β
0
0
0
1
confirms that
R
β
is orthogonal.
7.3.3 Rotate a Point About an Arbitrary Point
Now let's see how to rotate a point
(x, y)
about an arbitrary point
(t
x
,t
y
)
.The
strategy involves making the point of rotation a temporary origin, which is achieved
by subtracting
(t
x
,t
y
)
from the coordinates
(x, y)
respectively. Next, we perform a
rotation about the temporary origin, and finally, we add
(t
x
,t
y
)
back to the rotated
point to compensate for the original subtraction. Here are the steps:
1. Subtract
(t
x
,t
y
)
to create a new temporary origin:
x
1
=
x
−
t
x
y
1
=
y
−
t
y
.
2. Rotate
(x
1
,y
1
)
about the temporary origin by
β
:
x
2
=
(x
−
t
x
)
cos
β
−
(y
−
t
y
)
sin
β
y
2
=
−
+
−
t
y
)
cos
β.
3. Add
(t
x
,t
y
)
to the rotated point
(x
2
,y
2
)
to return to the original origin:
x
=
(x
t
x
)
sin
β
(y
x
2
+
t
x
y
=
y
2
+
t
y
x
=
(x
−
t
x
)
cos
β
−
(y
−
t
y
)
sin
β
+
t
x
y
=
(x
−
t
x
)
sin
β
+
(y
−
t
y
)
cos
β
+
t
y
.
Simplifying, we obtain
x
=
x
cos
β
−
y
sin
β
+
t
x
(
1
−
cos
β)
+
t
y
sin
β
y
=
x
sin
β
+
y
cos
β
+
t
y
(
1
−
cos
β)
−
t
x
sin
β
and in matrix form we have
⎡
⎤
⎡
⎤
⎡
⎤
x
y
1
cos
β
−
sin
β
x
(
1
−
cos
β)
+
t
y
sin
β
x
y
1
⎣
⎦
=
⎣
⎦
⎣
⎦
.
sin
β
cos
β
y
(
1
−
cos
β)
−
t
x
sin
β
(7.1)
0
0
1
For example, if we rotate the point
(
2
,
1
)
, 90° about the point
(
1
,
1
)
the transform
(
7.1
) becomes
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
.
−
1
2
1
0
12
100
001
2
1
1
