Graphics Reference
In-Depth Information
CHAPTER 20
Quaternions
Prerequisites:
Some linear algebra and facts about motions from Chapters 1 and 2 in
[AgoM05]
20.1
Introduction
In a broad sense the topic of this chapter is to determine if and how one can gener-
alize the following two observations about the plane to
R
n
:
(1) The Euclidean space
R
2
admits an algebraic structure that makes it into a two-
dimensional division algebra over the reals. (This structure comes of course
from the complex numbers
C
because one usually identifies
C
with
R
2
.)
(2) The unit sphere (circle)
S
1
in
R
2
can be identified in a natural way with the
special orthogonal group
SO
(2) and the rotations about the origin in
R
2
.
Section 20.2 will begin with some general comments about the generalization of
observation (1) and this will lead us to the quaternions, a less-known, but neverthe-
less very useful, generalization of the complex numbers. We shall define them and
summarizes some of their basic properties. Section 20.3 will discuss the important
connection of quaternions to transformations and this connection can be thought of
as a generalization of observation (2).
20.2
Basic Facts
To be a division algebra basically means is that we have a vector space with the usual
vector addition along with a multiplicative structure that includes multiplicative
inverses. In the case of the complex numbers the multiplication is actually commu-
tative, so that they are a field over the reals, but commutativity is not required for
division algebras in general. What about higher-dimensional Euclidean spaces?
