Graphics Reference
In-Depth Information
Fig. 7.5
e
12
p
rotates
p
,
−
90°
However, the reverse product
e
12
p
rotates
p
,
−
90°:
p
=
p
1
e
1
+
p
2
e
2
pe
12
=
e
12
(p
1
e
1
+
p
2
e
2
)
=−
p
2
e
1
=
p
2
e
1
−
p
1
e
2
p
1
e
2
+
asshowninFig.
7.5
.
We also discovered in Chap. 2 that a complex number
z
=
a
+
bi
can be repre-
sented in exponential form as
z
e
iβ
=|
|
=|
|
+
i
sin
β)
which, if used to multiply another complex number, scales it by
z
z
(
cos
β
|
z
|
and rotates it
β
.
Figure
7.6
shows a plane defined by
m
∧
n
and the vectors
n
and
m
such that
n
is rotated
β
further than
m
:
n
=
n
1
e
1
+
n
2
e
2
m
=
m
1
e
1
+
m
2
e
2
nm
=
n
·
m
−
m
∧
n
=|
n
||
m
|
cos
β
−|
m
||
n
|
sin
β
e
12
=|
n
||
m
|
(
cos
β
−
sin
β
e
12
) .
Fig. 7.6
The bivector
m
∧
n
