Graphics Reference
In-Depth Information
These examples show that rotors behave just like quaternions. Rotors not only
rotate vectors, but they can be used to rotate any multivector, irrespective of their
dimension.
12.5 Rotors as Matrices
Although rotors can be computed using geometric algebra, there is a one-to-one
correspondence with matrix algebra, which we will now demonstrate.
12.5.1 2D Rotor
To begin with we will show that a 2D rotor is nothing more that a 2
×
2matrixin
disguise for rotating a point 2 θ about the origin.
Given
m
ˆ
=
m 1 e 1 +
m 2 e 2
n
ˆ
=
n 1 e 1 +
n 2 e 2
p
=
p 1 e 1 +
p 2 e 2
ˆ
ˆ
and θ is the angle between
m and
n . Therefore, we can write
ˆ
ˆ
= ˆ
· ˆ
− ˆ
∧ ˆ
n
m
n
m
m
n
m n
= n
· m
+ m
n
where
n
ˆ
· ˆ
m
=
cos θ
m
ˆ
∧ ˆ
n
=
sin θ e 12 .
Therefore, using the definition of a rotor
p = ˆ
n
mp
ˆ
m
ˆ
n
ˆ
=
( cos θ
sin θ e 12 )(p 1 e 1 +
p 2 e 2 )( cos θ
+
sin θ e 12 )
=
(p 1 cos θ e 1 +
p 2 cos θ e 2 +
p 1 sin θ e 2
p 2 sin θ e 1 )( cos θ
+
sin θ e 12 )
= (p 1 cos θ
p 2 cos θ) e 2 ( cos θ
p 2 sin θ) e 1 +
(p 1 sin θ
+
+
sin θ e 12 )
= cos 2 θ
2 cos θ sin θp 2 e 1
+ 2 cos θ sin θp 1 + cos 2 θ
sin 2 θ p 1
sin 2 θ p 2 e 2
=
(p 1 cos 2 θ
p 2 sin 2 θ) e 1 +
(p 1 sin 2 θ
+
p 2 cos 2 θ) e 2
or in matrix form
p 1
p 2
cos 2 θ
p 1
p 2
sin 2 θ
=
sin 2 θ
cos 2 θ
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