** 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 pdf 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 and are both real numbers and . The quaternions are just an extension of this complex number form.

A quaternion is usually written as

This is clearly an extension of the complex number system - where the complex numbers are those quaternions that have and the real numbers are those that have .

** 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

We can rewrite the addition formula for two quaternions and 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 can be associated with the real numbers, and the quaternions of the form 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.

- Addition - The sum of
and
is
- Negation - The additive inverse
of
is
a
- Subtraction - The difference of
and
is
- Multiplication - The product of
and
is
- Identity -
The multiplicative identity is
. This can be directly
checked by

- Multiplicative Inverse - The inverse
of
is given
by

and so . - Division - The quotient of
and
is

** Notation**

Quaternions of the form are normally denoted in their real number form - as . this allows a scalar multiplication property to be given by

- Scalar Multiplication - If
is a scalar, then

- Multiplicative Inverse - The inverse
of
is given
by

** 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

** Length of a Quaternion, Unit Quaternions**

We define the length of a quaternion to be

**Return to
the Graphics Notes Home Page**

**Return
to the Geometric Modeling Notes Home Page**

**Return
to the UC Davis Visualization and Graphics Group Home Page**

**This document maintained by Ken
Joy**

All contents copyright (c) 1996, 1997, 1998,
1999

Computer Science Department

University of California, Davis

All rights reserved.

1999-12-06