Overview
The quaternion number system was discovered by Hamilton, a physicist who was looking for an extension of the complex number system to use in geometric optics. Quaternions have developed a wide-spread use in computer graphics and robotics research because they can be used to control rotations in three dimensional space. In these notes we define and review the basic properties of quaternions.
For a postscript version of these notes look here
What are Quaternions?
Remember complex numbers? These numbers are an extension of the real
number system and can be written in the form 
, where a
and b are both real numbers and 
. The quaternions are
just an extension of this complex number form.
A quaternion is usually written as
where a, b, and c are scalar values, and 
, 
and 
are
the unique quaternions with the properties that
and
This is clearly an extension of the complex number system - where the complex numbers are those quaternions that have c = d = 0 and the real numbers are those that have b = c = d = 0.
Adding and Multiplying Quaternions
Addition of quaternions is very straightforward: We just add the
coefficients. That is, if
and
,
then the sum of the two quaternions is
Multiplication is somewhat more complicated, as we must first multiply
componentwise, and then use the
product formulas for 
, 
, and 
to simplify the resulting
expression. So the product of 
and 
is
An Alternate Representation for Quaternions
The expression for multiplication of quaternions, given above, is quite complex - and results in even worse complexity for the division and inverse formulas. The quaternions can be written in an different form - one which involves vectors - which dramatically simplifies the formulas. These expressions have become the preferred form for representing quaternions.
In this form,
the quaternion 
is written as
where 
is the vector < b, c, d >.
We can rewrite
the addition formula for
two quaternions 
and 
as
and the product formula as
With some algebraic manipulation, these formulas can be shown to be
identical with those of the 
, 
, 
representation. We
note that the quaternions of the form (a,<0,0,0>) can be associated
with the real numbers, and the quaternions of the form (a, <b,0,0>)
can be associated with the complex numbers.
Properties of Quaternions
With this new representation, it is straightforward to develop a complete set of properties of quaternions.
Given the quaternions 
, 
,
and 
, we can verify the following
properties.
and 
is
of 
is
a
and 
is
and 
is
. This can be directly
checked by
of 
is given
by
.
and 
is
Notation
Quaternions of the form 
are normally denoted in their
real number form - as a. this allows a scalar multiplication
property to be given by
of 
is given
by
, and the additive identity
as 0.
The Quaternions are not Commutative under Multiplication
Whereas we can add, subtract, multiply and divide quaternions, we must
always be aware of the order in which these operations are made. This
is because
quaternions do not commute under multiplication - in general
.
To give an example of this consider the two quaternions 
and 
. Multiplying these we obtain
or
and they are not equal. This is because the vector cross products give
different results depending on the order of the vectors - in
general,
.
Length of a Quaternion, Unit Quaternions
We define the length of a quaternion 
to be
where 
is the length of the vector 
.
The unit quaternions are those that have length one.
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This document maintained by
Ken Joy
All contents copyright (c) 1996, 1997 |
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