Maths - Expansion of q * P1 * q'

Rotation - Expanding out all the terms

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:

We can expand out all the terms of the rotation as follows:

P2=q * P1 * q'

gives the following, first substitute in P1 and q':

P2=q * (0 + i x + j y + k z) * (qw - i qx - j qy - k qz)

P2=q *

0*qw +x*qx +y*qy +z*qz
+ i (x*qw - 0*qx - y*qz+z*qy)
+ j (-0*qy +x*qz+ y*qw - z*qx)
+ k (0*-qz - x*qy +y*qx + z*qw)

now substitute terms in q and multiply out terms:

P2=

qw*(0*qw +x*qx +y*qy +z*qz) - qx*(x*qw - 0*qx - y*qz+z*qy) - qy*(-0*qy +x*qz+ y*qw - z*qx) - qz* (0*-qz - x*qy +y*qx + z*qw)
+ i (qx*(0*qw +x*qx +y*qy +z*qz) + qw*(x*qw - 0*qx - y*qz+z*qy) + qy* (0*-qz - x*qy +y*qx + z*qw)- qz*(-0*qy +x*qz+ y*qw - z*qx))
+ j (qw*(-0*qy +x*qz+ y*qw - z*qx) - qx* (0*-qz - x*qy +y*qx + z*qw)+ qy*(0*qw +x*qx +y*qy +z*qz) + qz*(x*qw - 0*qx - y*qz+z*qy))
+ k (qw* (0*-qz - x*qy +y*qx + z*qw) + qx*(-0*qy +x*qz+ y*qw - z*qx) - qy*(x*qw - 0*qx - y*qz+z*qy) + qz*(0*qw +x*qx +y*qy +z*qz))

so expanding out gives (padding with 0):

real part = qw*(0*qw +x*qx +y*qy +z*qz) - qx*(x*qw - 0*qx - y*qz+z*qy) - qy*(-0*qy +x*qz+ y*qw - z*qx) - qz* (0*-qz - x*qy +y*qx + z*qw)
real part = qw(x*qx+y*qy +z*qz) - qx*(x*qw - y*qz+z*qy) - qy*(x*qz+ y*qw - z*qx) - qz* (- x*qy +y*qx + z*qw)
real part = x*(qw*qx - qx*qw- qy*qz + qz*qy) + y*(qw*qy+qx*qz- qy*qw- qz*qx) + z*(qw*qz - qx*qy+ qy*qx- qz*qw)
real part = 0

i part = qx*(0*qw +x*qx +y*qy +z*qz) + qw*(x*qw - 0*qx - y*qz+z*qy) + qy* (0*-qz - x*qy +y*qx + z*qw)- qz*(-0*qy +x*qz+ y*qw - z*qx)
i part = qx*(x*qx +y*qy +z*qz) + qw*(x*qw - y*qz+z*qy) + qy* (- x*qy +y*qx + z*qw)- qz*(x*qz+ y*qw - z*qx)
i part = x*(qx*qx+qw*qw-qy*qy- qz*qz) + y*(qx*qy- qw*qz+ qy*qx- qz*qw) + z*(qx*qz+ qw*qy+ qy*qw+ qz*qx)
i part = x*(qx*qx+qw*qw-qy*qy- qz*qz) + y*(2*qx*qy- 2*qw*qz) + z*(2*qx*qz+ 2*qw*qy)

j part = qw*(-0*qy +x*qz+ y*qw - z*qx) - qx* (0*-qz - x*qy +y*qx + z*qw)+ qy*(0*qw +x*qx +y*qy +z*qz) + qz*(x*qw - 0*qx - y*qz+z*qy)
j part = qw*(x*qz+ y*qw - z*qx) + qx* (x*qy - y*qx - z*qw)+ qy*(x*qx +y*qy +z*qz) + qz*(x*qw - y*qz+z*qy)
j part = x*(qw*qz + qx*qy+ qy*qx+ qz*qw) + y*(qw*qw - qx*qx+ qy*qy - qz*qz)+ z*(-qw*qx- qx*qw+ qy*qz+ qz*qy)
j part = x*(2*qw*qz + 2*qx*qy) + y*(qw*qw - qx*qx+ qy*qy - qz*qz)+ z*(-2*qw*qx+ 2*qy*qz)

k part = qw* (0*-qz - x*qy +y*qx + z*qw) + qx*(-0*qy +x*qz+ y*qw - z*qx) - qy*(x*qw - 0*qx - y*qz+z*qy) + qz*(0*qw +x*qx +y*qy +z*qz)
k part = qw* (-x*qy +y*qx + z*qw) + qx*(x*qz+ y*qw - z*qx) - qy*(x*qw -y*qz+z*qy) + qz*(x*qx +y*qy +z*qz)
k part = x*(-qw*qy+ qx*qz- qy*qw+ qz*qx) + y*(qw*qx+ qx*qw+ qy*qz+ qz*qy)+ z*(qw*qw - qx*qx- qy*qy+ qz*qz)
k part = x*(-2*qw*qy+ 2*qx*qz) + y*(2*qw*qx+ 2*qy*qz)+ z*(qw*qw - qx*qx- qy*qy+ qz*qz)

So combining these gives:

P2.x = x*(qx*qx+qw*qw-qy*qy- qz*qz) + y*(2*qx*qy- 2*qw*qz) + z*(2*qx*qz+ 2*qw*qy)
P2.y = x*(2*qw*qz + 2*qx*qy) + y*(qw*qw - qx*qx+ qy*qy - qz*qz)+ z*(-2*qw*qx+ 2*qy*qz)
P2.z = x*(-2*qw*qy+ 2*qx*qz) + y*(2*qw*qx+ 2*qy*qz)+ z*(qw*qw - qx*qx- qy*qy+ qz*qz)

This can be written in the form of a matrix:

P2.x
P2.y
P2.z
=
qx*qx+qw*qw-qy*qy- qz*qz 2*qx*qy- 2*qw*qz 2*qx*qz+ 2*qw*qy
2*qw*qz + 2*qx*qy qw*qw - qx*qx+ qy*qy - qz*qz -2*qw*qx+ 2*qy*qz
-2*qw*qy+ 2*qx*qz 2*qw*qx+ 2*qy*qz qw*qw - qx*qx- qy*qy+ qz*qz
P1.x
P1.y
P1.z

Reflection - expansion of result

We can expand out all the terms of the rotation as follows:

P2=q * P1 * q

gives the following, first substitute in P1 and q:

P2=q * (0 + i x + j y + k z) * (qw + i qx + j qy + k qz)

P2=q *

- x*qx - y*qy -z*qz
+ i (x*qw + y*qz - z*qy)
+ j ( - x*qz+ y*qw + z*qx)
+ k ( x*qy - y*qx + z*qw)

now substitute terms in q and multiply out terms:

P2=

qw*(- x*qx - y*qy -z*qz) - qx*(x*qw + y*qz - z*qy) - qy*( - x*qz+ y*qw + z*qx)- qz*( x*qy - y*qx + z*qw) +
i (qx*(- x*qx - y*qy -z*qz) + qw*(x*qw + y*qz - z*qy) + qy*( x*qy - y*qx + z*qw) - qz*( - x*qz+ y*qw + z*qx)) +
j (qw*( - x*qz+ y*qw + z*qx) - qx*( x*qy - y*qx + z*qw) + qy*(- x*qx - y*qy -z*qz) + qz*(x*qw + y*qz - z*qy)) +
k (qw*( x*qy - y*qx + z*qw) + qx*( - x*qz+ y*qw + z*qx) - qy*(x*qw + y*qz - z*qy) + qz*(- x*qx - y*qy -z*qz))

For reflections qw = 0 so we can simplify to:

P2=

- qx*(y*qz - z*qy) - qy*( - x*qz + z*qx)- qz*( x*qy - y*qx) +
i (qx*(- x*qx - y*qy -z*qz) + qy*( x*qy - y*qx ) - qz*( - x*qz + z*qx)) +
j (- qx*( x*qy - y*qx ) + qy*(- x*qx - y*qy -z*qz) + qz*(y*qz - z*qy)) +
k (qx*( - x*qz + z*qx) - qy*( + y*qz - z*qy) + qz*(- x*qx - y*qy -z*qz))

combining terms gives:

P2=

- qx*y*qz + qx*z*qy + qy*x*qz - qy*z*qx -qz*x*qy + qz*y*qx) +
i (-qx*x*qx - qx*y*qy -qx*z*qz + qy*x*qy - qy*y*qx + qz*x*qz - qz*z*qx) +
j (-qx*x*qy + qx*y*qx - qy*x*qx - qy*y*qy -qy*z*qz + qz*y*qz - qz*z*qy) +
k (- qx*x*qz + qx*z*qx - qy*y*qz + qy*z*qy - qz*x*qx - qz*y*qy -qz*z*qz)

cancelling gives:

P2=

0+
i (-qx*x*qx - qx*y*qy -qx*z*qz + qy*x*qy - qy*y*qx + qz*x*qz - qz*z*qx) +
j (-qx*x*qy + qx*y*qx - qy*x*qx - qy*y*qy -qy*z*qz + qz*y*qz - qz*z*qy) +
k (- qx*x*qz + qx*z*qx - qy*y*qz + qy*z*qy - qz*x*qx - qz*y*qy -qz*z*qz)

grouping together into matrix form:

Refl = 1 / (Px2 + Py2 + Pz2)*
-qx2 + qy2+ qz2 -2*qx*qy -2*qx*qz
-2*qx*qy qx2 - qy2+ qz2 -2*qy*qz
-2*qx*qz -2*qy*qz qx2 + qy2- qz2
[P1]

Which agrees with the result on this page:

Refl = 1 / (Px2 + Py2 + Pz2)*
-Px2 + Pz* Pz + Py* Py - 2 * Px * Py - 2 * Px * Pz
- 2 * Py * Px -Py2 + Px*Px + Pz*Pz - 2 * Py * Pz
- 2 * Pz * Px -2 * Pz * Py -Pz2 + Py*Py + Px*Px
[Va]

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