Maths - Vector Expansion

So why do we use an equation like: Pout = q * Pin * conj(q) to rotate a point?

We can demonstrate it works by expanding out the parts of the complex number using:

• q = (sq,vq)
• q' = (sq,-vq)
• Pin = (0,vp)

(sq,vq)(0,vp)(sq,-vq)

As explained on this page quaternion multipication can be defined using scalar and vector as: (sa,va) * (sb,vb) = (sa*sb-va•vb,va x vb + sa*vb + sb*va) so multiplying the first two gives:

(sq,vq)(-vp•vq,-vp x vq + sq*vp)

now expand out the second operand:

(-sq*(vp•vq) + vq•(-vp x vq + sq*vp) , - vq x (-vp x vq + sq*vp) + sq*(-vp x vq + sq*vp) - vq*(vp•vq))

(- vq•(vp x vq) , vq x (vp x vq) - sq*2*(vp x vq) + sq*sq*vp - vq*(vp•vq))

Since (vq x vp) gives a vector which is mutually perpendicular to vq and vp then (vq x vp)•vp=0, therefore the scalar part is zero which is what we want:

(0 , - vq x (vp x vq) + sq*2*(vp x vq) + sq*sq*vp - vq*(vp•vq))

using the triple product for cross product: A x (B x C) = B (A•C) - C (A•B) we get:

(0 , -vp (vq•vq) + 2*vq (vq•vp) - sq*2*(vp x vq) + sq*sq*vp)

So the result of the rotation is a vector:

-vp (vq•vq) + 2*vq (vq•vp) - sq*2*(vp x vq) + sq*sq*vp

where:

• vp = input vector
• sq = cos(rotation angle/2)
• vq = rotation axis * sin(rotation angle/2)

To try to understand this we can express it in matrix form, first lets calculate the matrix equivalent of each part: