Graphics Reference
In-Depth Information
It should be no surprise that post-multiplying a vector
p
by the product
mn
rotates
it
β
.
The above results are summarised as follows:
nmp
=
pmn
rotates
p
,β
pnm
=
mnp
rotates
p
,
−
β.
7.5.3 Rotate a Point About the Origin
In Chap. 9 on 3D rotations we show the origins of rotors in geometric algebra using
double reflections. The plane containing the vectors
n
and
m
is defined by the wedge
product
m
∧
n
, which means we can write the product
mn
as
mn
=
m
·
n
+
m
∧
n
and the product
nm
in the same plane as
nm
=
m
·
n
−
m
∧
n
which accounts for the negative sign in the following bivector term
sin
β
e
12
) .
Furthermore, if we make
n
and
m
unit vectors we can replace them by a rotor
R
β
whose magnitude is 1 because
nm
=|
n
||
m
|
(
cos
β
−
cos
2
β
sin
2
β
|
R
β
|=
+
=
1
therefore no scaling occurs, which means that
R
β
p
rotates
p
,β
i.e. anticlockwise, and
pR
β
rotates
p
,
−
β
i.e. clockwise
.
The effect of this rotor is illustrated as follows:
=
e
1
+
p
e
2
R
45
°
=
cos 45°
−
sin 45°
e
12
√
2
2
−
√
2
2
=
e
12
√
2
2
−
√
2
2
e
12
(
e
1
+
p
=
R
45
°
p
=
e
2
)
√
2
2
√
2
2
√
2
2
√
2
2
=
e
1
+
e
2
+
e
2
−
e
1
√
2
e
2
=
asshowninFig.
7.7
.