Graphics Reference
In-Depth Information
Given the definition of the geometric product, let's evaluate
i
2
:
ii
=
i
·
i
+
i
∧
i
.
Using the definition for the inner product (
6.4
)wehave
i
·
i
=
1
×
1
×
cos 0°
=
1
whereas using the definition of the outer product (
6.5
)wehave
i
∧
i
=
1
×
1
×
sin 0°
i
∧
i
=
0
.
Thus
i
2
1 and
j
2
2
:
=
=
1, and
aa
=|
a
|
aa
=
a
·
a
+
a
∧
a
=|
||
|
+|
||
|
∧
a
a
cos 0°
a
a
sin 0°
i
j
2
.
aa
=|
a
|
This result is much more satisfying than the square of a pure quaternion
q
:
2
.
qq
=−|
q
|
Now let's evaluate
ij
:
ij
=
i
·
j
+
i
∧
j
.
Using the definition for the inner product (
6.4
)wehave
i
·
j
=
1
×
1
×
cos 90°
=
0
whereas using the definition of the outer product (
6.5
)wehave
i
∧
j
=
1
×
1
×
sin 90°
i
∧
j
=
i
∧
j
.
Thus
ij
j
? Well, it is a new object called a
bivector
and defines
the orientation of the plane containing
i
and
j
. As the order of the vectors is from
i
to
j
, the angle is
=
i
∧
j
. But what is
i
∧
+
90° and sin
(
+
90
)
°
=
1. Whereas, if the order is from
j
to
i
the
angle is
−
90° and sin
(
−
90°
)
=−
1. Consequently,
ji
=
j
·
i
+
j
∧
i
=
0
+
1
×
1
×
sin
(
−
90°
)
i
∧
j
=−
i
∧
j
.
j
is to imagine moving along the
vector
i
and then along the vector
j
, which creates an anticlockwise rotation. Con-
versely, for the bivector
j
A useful way of visualising the bivector
i
∧
i
, imagine moving along the vector
j
followed by vector
i
,
which creates a clockwise rotation. Another useful picture is to sweep vector
j
along
vector
i
to create an anticlockwise rotation, and vice versa for
j
∧
∧
i
. These ideas are
shown in Fig.
6.2
.
The following equation
ab
=
9
+
12
i
∧
j
