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))
which simplifies to (see forum discussion):
(- 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:
-vp (vq•vq)
= vp*(vqx*vqx+vqy*vqy+vqz*vqz) |
| -vqx2-vqy2-vqz2 |
0 |
0 |
| 0 |
-vqx2-vqy2- vqz2 |
0 |
| 0 |
0 |
-vqx2- vqy2- vqz2 |
|
|
|
2*vq (vq•vp)
= 2*(vqx+vqy+vqz)(vqx*vpx + vqy*vpy + vqz*vpz)
= 2*vpx*vqx*(vqx+vqy+vqz)+
2*vpy*vqy*(vqx+vqy+vqz)+
2*vpz*vqz*(vqx+vqy+vqz) |
| 2*vqx*vqx |
2*vqx*vqy |
2*vqx*vqz |
| 2*vqy*vqx |
2*vqy*vqy |
2*vqy*vqz |
| 2*vqz*vqx |
2*vqz*vqy |
2*vqz*vqz |
|
|
|
| - sq*2*(vp x vq) |
| 0 |
-2*sq*vqz |
2*sq*vqy |
| 2*sq*vqz |
0 |
-2*sq*vqx |
| -2*sq*vqy |
2*sq*vqx |
0 |
|
|
|
| +sq*sq*vp |
|
| adding these |
————————————————————— |
| total: |
| -vqx2+vqy2+vqz2 + sq2 |
2*vqx*vqy-2*sq*vqz |
2*vqx*vqz+2*sq*vqy |
| 2*vqy*vqx + 2*sq*vqz |
vqx2- vqy2+vqz2+ sq2 |
2*vqy*vqz-2*sq*vqx |
| 2*vqz*vqx - 2*sq*vqy |
2*vqz*vqy + 2*sq*vqx |
vqx2+vqy2- vqz2+ sq2 |
|
This is the expected result (derived here):
| 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 |
where
- sq = qw = cos(angle/2)
- (qx,qy,qz) = axis*sin(angle/2)
Reflection - expansion of result using scalar and vector format
We can expand out all the terms of the rotation as follows:
P2=q * P1 * q
(0,vq)(0,vp)(0,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:
(0,vq)(-vp•vq,vp x vq)
(-vq•(vp x vq),vq x (vp x vq) + (-vp•vq)*vq)
(0,vq x (vp x vq) + (-vp•vq)*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) - vq (vq•vp) - (vp•vq)*vq)
(0,vp (vq•vq) - 2*vq (vq•vp))
To try to understand this we can express it in matrix form, first lets calculate the matrix equivalent of each part:
vp (vq•vq)
= vp*(vqx*vqx + vqy*vqy + vqz*vqz) |
| vqx*vqx + vqy*vqy + vqz*vqz |
0 |
0 |
| 0 |
vqx*vqx + vqy*vqy + vqz*vqz |
0 |
| 0 |
0 |
vqx*vqx + vqy*vqy + vqz*vqz |
|
|
|
- 2*vq (vq•vp)
= -2*(vqx+vqy+vqz)(vqx*vpx + vqy*vpy + vqz*vpz)
= -2*vpx*vqx*(vqx+vqy+vqz)-
2*vpy*vqy*(vqx+vqy+vqz)-
2*vpz*vqz*(vqx+vqy+vqz) |
| -2*vqx*vqx |
-2*vqx*vqy |
-2*vqx*vqz |
| -2*vqy*vqx |
-2*vqy*vqy |
-2*vqy*vqz |
| -2*vqz*vqx |
-2*vqz*vqy |
-2*vqz*vqz |
|
|
|
| adding these |
————————————————————— |
| total: |
| vqy*vqy + vqz*vqz-vqx*vqx |
-2*vqx*vqy |
-2*vqx*vqz |
| -2*vqy*vqx |
vqx*vqx + vqz*vqz-vqy*vqy |
-2*vqy*vqz |
| -2*vqz*vqx |
-2*vqz*vqy |
vqx*vqx + vqy*vqy -vqz*vqz |
|
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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