Graphics Reference
In-Depth Information
which, when substituted in Eq. (7.10), produce
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
+
c
1
d
2
−
d
1
c
2
i
+
d
1
b
2
−
b
1
d
2
j
+
b
1
c
2
−
c
1
b
2
k
(7.11)
Equation (7.11) has an obvious structure, which probably did not emerge immediately. However,
with our knowledge of vectors, it is possible to recognise the dot product:
b
1
b
2
+
c
1
c
2
+
d
1
d
2
a vector multiplied by a scalar
a
1
b
2
i
+
c
2
j
+
d
2
k
and
a
2
b
1
i
+
c
1
j
+
d
1
k
and the cross product
c
1
b
2
k
Hamilton gave the name
quaternion
to this form and the name
vector
to its imaginary component.
He had also given birth to non-commutative algebra, as the quaternion product is non-
commutative, i.e.,
ij
c
1
d
2
−
d
1
c
2
i
+
d
1
b
2
−
b
1
d
2
j
+
b
1
c
2
−
ji
. Hyper-complex numbers also became a new set for describing numbers.
Today, the set of hyper-complex numbers embraces real numbers, which have a rank 1, complex
numbers, which have a rank 2, and quaternions, which have a rank 4.
Thus, the product of two quaternions
=
q
1
=
s
1
+
v
1
and q
2
=
s
2
+
v
2
equals
v
2
Hamilton was so pleased with his discovery that he immediately wrote to his mathematician,
friend John Graves [1806-1870], who had encouraged him in the first place to pursue this
avenue of research. By December of the same year, Graves had discovered the next algebra of
rank 8, which he called
octaves
. He also proposed that other algebras might exist of the form
“2
m
-ions”, where 2
0
are real numbers, 2
1
are complex numbers, 2
2
are quaternions, and 2
3
are
octaves. He tried, in vain, to find an algebra of the form 2
4
.
Unfortunately for Graves, his idea of octaves was published after Arthur Cayley [1821-1895]
published his own discovery of
octonions
in 1845, and Cayley's publishing priority ensured
that he was credited with their discovery. To learn more about quaternions and octonions,
John Conway's and Derek Smith's topic,
On Quaternions and Octonions
, is highly recommended
[Conway, 2003].
Although some mathematicians of the day supported quaternions, the newly discovered vector
had even more supporters and by 1900 vector analysis was the preferred analytical tool. By the
1960s quaternions were being used in flight simulators, and in the 1980s Shoemake [Shoemake,
1985] found a use for them in computer graphics. Today, they are used in computer animation
systems and computer-game development systems.
q
1
q
2
=
s
1
s
2
−
v
1
·
v
2
+
s
1
v
2
+
s
2
v
1
+
v
1
×
