Graphics Programs Reference
In-Depth Information
An appropriate measure of the size of a quaternion is its
norm
, defined as
q
=
w
2
+
x
2
+
y
2
+
z
2
=
s
2
+
x
2
+
y
2
+
z
2
.
q
∗
=
q
∗
·
|
q
|
=
q
·
It is easy to verify that the norm is multiplicative,
(i.e., the norm of a
product equals the product of the two individual norms). A
unit quaternion
is one for
which
|
q
|
=1.
The inverse of a quaternion is given by
|
q
1
q
2
|
=
|
q
1
||
q
2
|
q
∗
(
qq
∗
)
=
q
∗
|
q
∗
w
2
+
x
2
+
y
2
+
z
2
,
q
−
1
=
=
2
q
|
so quaternion division
q
1
/
q
2
(except by zero) is performed by multiplying
q
1
by the
inverse
q
−
2
. It'seasytoverifythat
qq
−
1
=[1
,
(0
,
0
,
0)]=[0
,
0
].
[Mathworld 05] and [WikiQuaternion 05] are basic references for quaternions.
Exercise B.1:
(If 4, why not more?) Quaternions are an extension of vectors. Are
there extensions of quaternions?
Every morning in the early part of the above-cited month [Oct. 1843]
on my coming down to breakfast, your brother William Edwin
and yourself used to ask me, “Well, Papa, can you multiply
triplets?” Whereto I was always obliged to reply, with a sad
shake of the head, “No, I can only add and subtract them.”
—William Rowan Hamilton
