Graphics Reference
In-Depth Information
7.4 Quaternions
Knowing that a complex number rotates a position vector in R
2
, it then seems natural that
there should be a R
3
equivalent, but the search for such a number proved extremely difficult.
Nevertheless, after many years of toil, Sir William Rowan Hamilton discovered its form on
October 16, 1843.
Prior to this discovery, Sir William had toyed with the idea that a hyper-complex number
would take the form
z
=
a
+
b
i
+
c
j
where
i
2
j
2
=
=−
1
However, a problem occurs when two such numbers are multiplied together:
a
1
+
+
c
1
j
a
2
+
+
=
a
1
a
2
+
+
+
+
b
1
b
2
i
2
+
+
+
+
c
1
c
2
j
2
b
1
i
b
2
i
c
2
j
a
1
b
2
i
a
1
c
2
j
b
1
a
2
i
b
1
c
2
ij
c
1
a
2
j
c
1
b
2
ji
a
1
+
b
1
i
+
c
1
j
a
2
+
b
2
i
+
c
2
j
=
a
1
a
2
−
b
1
b
2
−
c
1
c
2
+
a
1
b
2
+
b
1
a
2
i
+
a
1
c
2
+
c
1
a
2
j
+
b
1
c
2
ij
+
c
1
b
2
ji
The new number contains a real part, two imaginary parts, and two further parts containing
the products
ij
and
ji
, for which no meaning could be found.
Sir William then tried the next obvious form:
z
=
a
+
b
i
+
c
j
+
d
k
and when two such numbers are multiplied together, they create a large number of terms:
q
1
=
a
1
+
b
1
i
+
c
1
j
+
d
1
k
q
2
=
a
2
+
b
2
i
+
c
2
j
+
d
2
k
q
1
q
2
=
a
1
a
2
+
a
1
b
2
i
+
c
2
j
+
d
2
k
+
a
2
b
1
i
+
c
1
j
+
d
1
k
b
1
b
2
i
2
+
+
b
1
c
2
ij
+
b
1
d
2
ik
c
1
c
2
j
2
+
c
1
b
2
ji
+
+
c
1
d
2
jk
d
1
d
2
k
2
+
d
1
b
2
ki
+
d
1
c
2
kj
+
Invoking the rule
i
2
j
2
k
2
=
=
=
ijk
=−
1, we obtain
q
1
q
2
=
a
1
a
2
−
b
1
b
2
+
c
1
c
2
+
d
1
d
2
+
a
1
b
2
i
+
c
2
j
+
d
2
k
+
a
2
b
1
i
+
c
1
j
+
d
1
k
+
b
1
c
2
ij
+
b
1
d
2
ik
+
c
1
b
2
ji
+
c
1
d
2
jk
+
d
1
b
2
ki
+
d
1
c
2
kj
(7.10)
The new number contains a real part, three imaginary parts, and six further parts containing
the products
ij
,
ik
,
ji
,
jk
,
ki
, and
kj
, for which no obvious meaning could be found. And then
on that fateful day in 1843, Hamilton thought of the extra rules:
ij
=
k
jk
=
i
ki
=
j
ji
=−
k
kj
=−
i
and
ik
=−
j
