Graphics Programs Reference
In-Depth Information
B
Quaternions
Complex numbers can be interpreted as points in the
xy
plane. The complex number
(
a, b
) can be interpreted as the point with coordinates (
a, b
). Is it possible to define
hy-
percomplex
numbers of the form (
a, b, c
) that could be interpreted as three-dimensional
points? This question bothered the Irish mathematician William Rowan Hamilton for a
long time. The problem was that multiplying complex numbers could be interpreted as
a rotation in two dimensions (Section 1.4.5), so it made sense to require that multiplying
the new hypercomplex numbers would be equivalent to a rotation in three dimensions.
Readers of this topic know (from Section 1.4.3) that a general rotation in three dimen-
sions is fully defined by four numbers: one for the rotation angle and three for the
rotation axis. Three numbers are not enough to fully specify such a rotation.
Hamilton could not come up with a reasonable rule for multiplying hypercomplex
numbers that are triplets, and he eventually discovered, in October 1843, that he needed
to add a fourth component to his triplets (i.e., turn them into 4-tuples) in order to
multiply them in a way that made sense. He called these new entities
quaternions
.
Using modern notation, a quaternion
q
can be represented as a 2
×
2 matrix of complex
numbers
q
=
z
=
a
+
ib
,
w
c
+
id
w
∗
z
∗
−
−
c
+
id
a
−
ib
where
z
and
w
are complex numbers and
a
,
b
,
c
,and
d
are real. This can also be written
(by analogy with the complex numbers
a
·
1+
b
·
i
)as
q
=
a
U
+
b
I
+
c
J
+
d
K
,where
U
=
10
01
,
I
=
i
,
J
=
01
−
,
and
K
=
0
.
0
i
0
−
i
10
i
0
(Note that
U
,not
I
, is used here to denote the identity matrix. These matrices are
closely related to the Pauli spin matrices used in particle physics.) From the definitions
above, it follows that
I
2
=
U
,
J
2
=
U
,and
K
2
=
−
−
−
U
. We therefore conclude that
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