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Updates

From version 1.0 - 1.1
The norm of a quaternion should be the square root of the q.q. The mistake was brought to my attention by several kind readers and upon checking the definition of the Euclidean properties for complex numbers, I realize the norm property

|| u+v || <= || u || + || v ||

is violated for the previous definition of the magnitude.

The code in the samples are updated as well.

Foreword

To me, the term 'Quaternion' sounds out of this world, like some term from quantum theory about dark matter, having dark secret powers. If you, too, are enthralled by this dark power, this article will bring enlightenment (I hope). The article will show you how to do rotations using quaternions, and bring you closer to understanding quaternions (and their powers). If you do spot a mistake please email me at robin@cyberversion.com. Also if you intend to put this on your site, please send me a mail. I like to know where this ends up.

Why use Quaternions?

To answer the question, let's first discuss some common orientation implementations.

Euler representation

This is by far the simplest method to implement orientation. For each axis, there is a value specifying the rotation around the axis. Therefore, we have 3 variables

x, y, z     <-- angle to rotate around global coordinate axis

that vary between 0 and 360 degrees (or 0 - 2pi). They are the roll, pitch, and yaw (or pitch, roll, and yaw - whatever) representation. Orientation is obtained by multiplying the 3 rotation matrices generated from the 3 angles together (in a specific order that you define).

Note: The rotations are specified with respect to the global coordinate axis frame. This means the first rotation does not change the axis of rotation for the second and third rotations. This causes a situation known as gimbal lock, which I will discuss later.

Angle Axis representation

This implementation method is better than the Euler representation as it avoids the gimbal lock problem. The representation consists of a unit vector representing an arbitrary axis of rotation, and another variable (0 - 360) representing the rotation around the vector:

x, y, z     <-- unit vector representing arbitrary axis

angle       <-- angle to rotate around above axis

Why are these representations bad?

Gimbal Lock

As rotations in the Euler representation are done with respect to the global axis, a rotation in one axis could 'override' a rotation in another, making you lose a degree of freedom. This is gimbal lock.

Say, if the rotation in the Y axis rotates a vector (parallel to the x axis) so that the vector is parallel to the z axis. Then, any rotations in the z axis would have no effect on the vector.

Later, I will show you an example of gimbal lock and how you can use quaternions to overcome it.

Interpolation Problems

Though the angle axis representation does not suffer from gimbal lock, there are problems when you need to interpolate between two rotations. The calculated interpolated orientations may not be smooth, so you will get jerky rotation movements. Euler representation suffers from this problem as well.

Next : Let's Get Started