Graphics Reference
In-Depth Information
Chapter 11
Quaternion Transforms in Space
11.1 Introduction
Quaternions were introduced in Chap. 5 as a mathematical object that combines a
scalar with a vector, in the same way a complex number combines a scalar with an
imaginary quantity. Quaternions, like complex numbers, possess rotational qualities,
but work in four dimensions rather than on the complex plane.
Hamilton invented quaternions in October 1843, and by December of the same
year his friend, John T. Graves, had invented octonions. Arthur Cayley had also
been intrigued by Hamilton's quaternions, and independently discovered octonions
in 1845.
There are four such composition algebras : real
R
, complex
C
, quaternion
H
,
and octonion
that obey an n -square identity used to compute their magnitudes.
Adolf Hurwitz (1859-1919) proved that the product of the sum of n squares by
the sum of n squares is the sum of n squares only when n is equal to 1, 2, 4 and
8, which are represented by reals, complex, quaternions and octonions. No other
system is possible, which shows how important quaternions are within the realm of
mathematics. Appendix C provides further information on this topic.
In this chapter we investigate how quaternions are used to rotate 3D vectors about
an arbitrary axis.
O
11.2 Definition
A quaternion q is the union of a scalar and a vector:
v
where s is a scalar and v is a 3D vector. If we express the vector v in terms of its
components, we have
q
=
s
+
q
=
s
+
x i
+
y j
+
z k
where s,x,y and z are all real numbers.
 
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