I'm not sure that this is a very practical or useful way to define quaternions? However in might be interesting to investigate.

p_{2}=q * p_{1} * conj(q)

Why can't we have a simple transform, similar to matrix transforms, where a point is transformed just by multiplying by the quaternion:

p_{2}=q * p_{1}

where:

- p
_{2}= is a vector representing a point after being rotated - q = is a quaternion representing a rotation.
- p
_{1}= is a vector representing a point before being rotated

This almost works, but not quite, so it might be instructive to try and work out the reasons.

If we rearrange the equation we get:

q = p_{1} / p_{2}

The division of two vectors does not have a solution, as explained here, by the usual types of multiplication used with vectors (cross multiplication). However, using quaternions, we can define a way to do this division.

We can define a vector using quaternions by setting the real part to zero, and the imaginary pars to the vector coordinates:

p_{1} = 0 + a i + b j + c k

p_{2} = 0 + e i + f j + g k

Dividing these vectors gives:

q = (a*e + b*f +c*g) + (g*b - f*c) i + (e*c - a*g) j + (f*a - e*b) k

So we get a quaternion where the real part is the dot product and the complex parts are the cross product of the vectors.

This quaternion can be used to represent the rotation of one vector to another, see angle between vectors.

Can this quaternion, used in this way, be combined and used to represent subsequent rotations? It turns out that this will only work if the axies of both rotations are aligned. So this can't be used to concatenate general rotations, to do that we must use:

p_{2}=q * p_{1} * conj(q)

where q is defined in terms of angle/2 in the normal way.

It seems like we have to do half the rotation by pre-multiplying by q and then the other half of the rotation by post-multiplying by conj(q). Its almost like doing the rotation in two parts like this cancels out the errors?

Is this true?

I can't think how to carry this argument further. Any ideas?