Graphics Reference
In-Depth Information
Chapter 5
Quaternions
5.1 Introduction
As mentioned earlier, quaternions were invented by Sir William Rowan Hamilton in
1843. Sir William was looking to generalise complex numbers in higher dimensions,
and it took 14 years of toil before he stumbled upon the idea of using a 4D notation
- hence the name '
quaternion
'.
5.2 Definition
The definition and associated rules for a quaternion are:
q
dk
where
a
,
b
,
c
and
d
are scalars, and
i
,
j
and
k
are imaginary and obey the following
rules:
=
a
+
bi
+
cj
+
i
2
2
2
=−
1
,
=−
1
,
=−
1
, jk
=−
1
ij
=
k,
jk
=
i,
ki
=
j
j.
Although quaternions had some enthusiastic supporters, there were many mathe-
maticians and scientists who were suspicious of the need to involve so many imag-
inary terms. Towards the end of the nineteenth century Josiah Gibbs resolved the
problem by declaring that the three imaginary quantities could be viewed as a 3D
vector and changed the original
bi
ji
=−
k,
kj
=−
i,
ik
=−
d
k
, where
i
,
j
and
k
are unit Cartesian vectors. Today, it is convenient in computer graphics to write a
quaternion in two ways:
+
cj
+
dk
into
b
i
+
c
j
+
q
=
s,
v
(5.1)
=
+
q
s
v
(5.2)
where
s
is a scalar, and
v
is a 3D vector.
