Graphics Reference
In-Depth Information
However, this notation is not generally adopted by the geometric algebra commu-
nity. The reason being that
i
is normally only associated with a scalar, with which it
commutes. Whereas in 2D,
e
12
is associated with scalars and vectors, and although
scalars present no problem, under some conditions, it anticommutes with vectors.
Consequently, an upper-case
I
is used so that there is no confusion between the two
elements. Thus (
6.15
) is written as
ab
=
(a
1
b
1
+
a
2
b
2
)
+
(a
1
b
2
−
a
2
b
1
)I
where
I
2
=−
1
.
It goes without saying that the 3D unit basis bivectors are also imaginary quantities,
so too, is
e
123
.
Multiplying a complex number by
i
rotates it 90° on the complex plane. There-
fore, it should be no surprise that multiplying a 2D vector by
e
12
rotates it by 90°.
However, because vectors are sensitive to their product partners, we must remember
that pre-multiplying a vector by
e
12
rotates a vector clockwise and post-multiplying
rotates a vector anticlockwise. For instance, post-multiplying
e
1
by
e
12
creates
e
2
,
which is an anticlockwise rotation, whereas, pre-multiplying
e
1
by
e
12
creates
−
e
2
,
which is a clockwise rotation.
Whilst on the subject of rotations, let's consider what happens in 3D. We begin
with a 3D vector
a
=
a
1
e
1
+
a
2
e
2
+
a
3
e
3
and the unit basis bivector
e
12
asshowninFig.
6.5
. Next we construct their geomet-
ric product
e
12
a
=
a
1
e
12
e
1
+
a
2
e
12
e
2
+
a
3
e
12
e
3
=
a
1
e
121
+
a
2
e
122
+
a
3
e
123
=−
a
1
e
2
+
a
2
e
1
+
a
3
e
123
=
a
2
e
1
−
a
1
e
2
+
a
3
e
123
which contains two parts: a vector
(a
2
e
1
−
a
1
e
2
)
and a volume
a
3
e
123
.
Figure
6.5
shows how the projection of vector
a
is rotated anticlockwise on the
bivector
e
12
. A volume is also created perpendicular to the bivector. This enables us
to predict that if the vector is coplanar with the bivector, the entire vector is rotated
90° and the volume component is zero.
Post-multiplying
a
by
e
12
creates
ae
12
=−
a
2
e
1
+
a
1
e
2
+
a
3
e
123
which shows that while the volumetric element has remained the same, the projected
vector is rotated anticlockwise. You may wish to show that the same happens with
the other two bivectors.
