Graphics Reference
In-Depth Information
by
()
=
()
=
re
q
r
and
pu
q
v
.
The direct sum decomposition of a quaternion into a real and pure part is the
analog of the real and imaginary parts of a complex number. Next, we introduce a
few standard definitions associated with quaternions and list some simple facts in
Propositions 20.2.2-20.2.6. The proofs are left as exercises. They are trivial and simply
involve expressing quaternions in terms of
i
,
j
, and
k
and then computing the appro-
priate expressions using the relevant definitions.
20.2.2
Proposition.
A quaternion is real if and only if it commutes with every
quaternion.
Proof.
Exercise 20.2.2.
Definition.
If
q
= r + a
i
+ b
j
+ c
k
is a quaternion, then the
conjugate
of
q
,
q
, is defined
by
()
-
()
=- - -
qq q
=
re
pu
r
a
i
b
j
c .
k
The map
¯
:
H
Æ
H
, which sends
q
to , is called the
conjugation map
of
H
. (Note that
this map restricts to the usual conjugation map of the complex numbers.)
q
20.2.3
Proposition.
Let
a
,
b
Œ
H
and r Œ
R
. The conjugation map of
H
has the fol-
lowing properties:
(1)
(2)
(3)
(4)
(5)
a
Œ
R
if and only if
ab
a
=+
b
+
r
aa
=
a
=
a
ab
=
ba
a
=
a
.
(6)
a
Œ
R
3
if and
only if
a
=-
a
.
(7) re (
a
) = (
a
+
)
/2
(8) pu (
a
) = (
a
- )/2
(9) If • is the dot product in
R
4
, then
a
•
b
= re (
b
). In particular,
a
•
a
=
a
a
a
a
a
.
Proof.
This is straightforward and left as Exercise 20.2.3.
It follo
w
s from Proposition 20.2.3 (
9
) tha
t
a
is a nonnegative real. Note that the
conjugate commutes with
a
, that is,
a
=
a
.
a
a
a
a
Definition.
The
norm
or
absolute value
of a quaternion
a
, |
a
|, is defined by
a a
=
.
It is easy to check that
2
2
2
2
2
rv v v
+ + +
i
j
k
=+++
rvvv
(20.2a)
1
2
3
