Graphics Reference
In-Depth Information
Fig. 12.5
Rotating a point by
180°
m
ˆ
=
e
2
n
ˆ
=−
e
1
e
2
.
As the mirrors are separated by 90° the point
P
is rotated 180°:
p
= ˆ
p
=
e
1
−
n
mp
ˆ
m
ˆ
n
ˆ
=−
e
1
e
2
(
e
1
−
e
2
)
e
2
(
−
e
1
)
=
e
12121
−
e
12221
p
=−
e
2
.
Let's now define a rotor in terms of its bivector and the actual angle a vector is
rotated as follows. The bivector defining the plane is
e
1
+
ˆ
∧
ˆ
m
n
and
θ
is the rotor angle,
which means that the bivector angle is
θ/
2. Let
R
θ
= ˆ
n
m
ˆ
R
θ
= ˆ
m
n
ˆ
where
n
ˆ
m
ˆ
= ˆ
n
· ˆ
m
− ˆ
m
∧ ˆ
n
m
ˆ
n
ˆ
= ˆ
n
· ˆ
m
+ ˆ
m
∧ ˆ
n
n
ˆ
· ˆ
m
=
cos
(θ/
2
)
sin
(θ/
2
)
B
.
m
ˆ
∧ ˆ
n
=
Therefore,
sin
(θ/
2
)
B
R
θ
=
−
cos
(θ/
2
)
R
θ
=
sin
(θ/
2
)
B
.
cos
(θ/
2
)
+
We now have an equation that rotates a vector
p
through an angle
θ
about an axis
defined by
B
:
R
θ
pR
θ
p
=
