Graphics Reference
In-Depth Information
Ta b l e 6 . 2
3D bivector
product rules
e
23
e
31
e
12
e
23
−
1
−
e
12
e
31
e
31
e
12
−
1
−
e
23
e
12
−
e
31
e
23
−
1
Ta b l e 6 . 3
Left-handed 3D
bivector product rules
e
32
e
13
e
21
e
32
−
1
e
21
−
e
13
e
13
−
e
21
−
1
e
32
e
21
e
13
−
e
32
−
1
6.18 Reverse of a Multivector
You will have noticed how sensitive geometric algebra is to the sequence of vectors,
and it should not be too much of a surprise to learn that a special function exists to
reverse sequences of elements. For example, given
ab
the reverse of
A
is denoted using the dagger symbol
A
†
A
†
A
=
=
ba
A
or the tilde symbol
A
=
ba
.
The dagger symbol is used in this topic.
Obviously, scalars are unaffected by reversion, neither are vectors. However,
bivectors and trivectors flip their signs:
(
e
1
e
2
)
†
=
e
2
e
1
=−
e
1
e
2
and
(
e
1
e
2
e
3
)
†
e
1
e
2
e
3
.
When reversing a multivector containing terms up to a trivector, it's only the bivector
and trivector terms that are reversed. For example, given a multivector
A
=
e
3
e
2
e
1
=−
A
=
λ
+
v
+
B
+
T
where
λ
is a scalar
v
is a vector
B
is a bivector, and
T
is a trivector
