MarkCC has a post about quaternions and the fact that they can be used for rotation, so I thought I’d chime in with exactly how they represent rotations.

A 2D rotation by an angle θ can be represented by the complex number on the unit circle

cos θ + *i* sin θ

Then multiplying complex numbers is the same as multiplying rotations.

A 3D rotation by an angle θ about a line defined by a unit vector (b,c,d) can be represented by the quaternion on the unit hypersphere

cos θ/2 + sin θ/2(*bi* + *cj* + *dk*)

Then multiplying quaternions is the same as multiplying rotations.

But why is it θ/2 instead of θ?

Well, in 3D, in some sense you have to rotate through 720° to really get back where you started from. There’s a nice animation of the Dirac Belt Trick that shows how you can remove a 720° twist from a belt without moving the ends. However, you can’t remove a 360° twist. The quaternions q and -q are different but both represent the same 3D rotation.

Andrew Hanson has some demonstration programs that show how quaternions do rotations. These accompany his book Visualizing Quaternions.