Graphics Reference
In-Depth Information
R
θ
vR
θ
rotates a vector anticlockwise by
θ
R
θ
vR
θ
rotates a vector clockwise by
θ.
Furthermore, maintaining our convention about rotating points and frames:
R
θ
vR
θ
rotates a frame clockwise by
θ
R
θ
vR
θ
rotates a frame anticlockwise by
θ.
12.5.3 Extracting a Rotor
Say we are presented with
b
R
θ
aR
θ
=
a
and
b
and have to discover
R
θ
. Here is one way we can undertake
the task, which is cunning, rather than obvious!
Figure
12.8
shows vectors
where we know
ˆ
b
and a third vector
a
and
ˆ
n
, mid-way between the
ˆ
a
and
b
, therefore, the product
two vectors. Vector
n
bisects the angle
θ
separating
ˆ
ˆ
b
∧
b
by
θ
, which
n
must be a rotor capable of rotating any vector in the plane
ˆ
n
ˆ
permits us to write
b
=
b
n b
n
ˆ
a
ˆ
ˆ
or
aR
†
θ
b
=
R
θ
ˆ
where
R
θ
=
b
n
ˆ
(12.12)
R
θ
= ˆ
n b
.
(12.13)
Next, to eliminate
n
we compute
ˆ
+
b
ˆ
a
ˆ
=
n
+
b
|ˆ
a
|
Vector
n
bisects
θ
Fig. 12.8
