Maths - Quaternion Notations - As a quantity similar to axis-angle

It is quite difficult to give a physical meaning to a quaternion, it is just a quantity which represents a rotation. If you need a physical meaning then this is probably the best way to think of it:

q = cos(a/2) + i ( x * sin(a/2)) + j (y * sin(a/2)) + k ( z * sin(a/2))


So it is closely related to the axis angle representation of rotations.

In the following diagram:

Note: we are talking about rotations about the origin so both the rotation axis and the rotation plane go through the origin.

axis angle

Imagine a point P1 which is a unit distance from the origin, we will be rotating it through an angle (a) to P3 through midpoint P2.

axis angle

So the point P1 is transformed to P3, if it travels in a straight line it passes through a point cos(a/2) from the rotation axis and sin(a/2) from both P1 and P3.

metadata block
see also:


Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.


cover us uk de jp fr ca Quaternions and Rotation Sequences.

Terminology and Notation

Specific to this page here:


This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2021 Martin John Baker - All rights reserved - privacy policy.