Graphics Reference
In-Depth Information
Figure 2.3.
Defining a rotation with polar coordinates.
y
p¢ = (r,a + q)
p = (r,a)
q
a
x
(
)
=
xr
¢=
cos
aq
+
r
cos
a
cos
q
-
r
sin
a
sin
q
(
)
=
y
¢=
r
s
in
aq
+
r
cos
a
s
in
q
+
r
sin
a
c
os
q.
(2.6)
Substituting (2.5) into (2.6) leads to
2.2.2.1. Theorem.
The equations for a rotation R about the origin through an angle
q are
xx
¢=
cos
q
-
y
sin
q
y
¢=
x
s
in
q
+
y
c
os
q
(2.7)
In particular, such a rotation is a linear transformation with matrix
cos
q
sin
q
Ê
Ë
ˆ
¯
.
(2.8)
-
sin
q
cos
q
2.2.2.2. Theorem.
Rotations about the origin are motions.
Proof.
This is proved by direct computations using the definition of a motion and
Theorem 2.2.2.1.
2.2.2.3. Example.
The equations for the rotation R through an angle p/3 are
1
2
3
2
x
¢=
x
-
y
3
2
1
2
y
¢=
x
+
y
.
Furthermore, notice that the inverse of a rotation through an angle q is just the rota-
tion through the angle -q, so that given a rotation it is easy to write down the equa-
tions for the inverse. In our example the equations for the inverse are
1
2
3
2
xx
=¢ +
y
¢
