Rotating a sphere

[Johnny5]

I have a question about the following article:

Hamiltonian Quaternions

 

In the article you will see the following quote:

 

Rotations in 3-space

 

Let us write

 

[math] U = \mathcal{f} q \in \mathbb{H}\text{ such that }||q|| = 1 \mathcal{g} [/math]

 

With multiplication, U is a group. Let us briefly sketch the relation between U and the group SO3 of rotations (about the origin) in 3-space.

An arbitrary element q of U can be expressed

 

[math] cos(\frac{\theta}{2}) + sin(\frac{\theta}{2}) (ai+bj+ck) [/math], for some real numbers q,a,b,c such that a^2+b^2+c^2=1 . The permutation v--> qv of U thus gives rise to a permutation of the real sphere. It turns out that that permutation is a rotation. Its axis is the line through (0,0,0) and (a,b,c), and the angle through which it rotates the sphere is q. If rotations F and G correspond to quaternions q and r respectively, then clearly the permutation v-->qrv corresponds to the composite rotation F o G .

 

My question is this:

 

If you look up you will see w,x,y,z expressed for the product of two quaternions. There are 12 components.

 

does anyone understand how you go from those 12 components, to a rotation of a sphere?