Quote:
Originally Posted by Fortis
I had thought that we were talking about position coordinates in spacetime. You are quite correct about rotations, but you started this discussion by taking the product of two quaternions describing position, i.e.
P1 = ct1 + ix1 + jy1 + kz1
and
P2 = ct2 + ix2 + jy2 + ky2
Am I missing something?
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Generally multiplying can be seen as a subset of rotation. if the magnitude of the multiplier is 1, then it is a rotation, if not it is a rotation and an extension or reduction of size. This is not inconsistent with the product of two positions. The quaternion can represent a position and a rotation.
For example: p1= r1 + ix1 + jy1 + kz1= r1 + v1, p2=r2 + v2
Then the product is p3= p1p2= (r1r2 - v1.v2) + r1v2 + r2v1 +v1xv2
This is a new position with real art r3=(r1r2 -v1.v2) and vector
v3=r1v2 + r2v1 + v1xv2
This could also be the the rotation of cos(p1+p2) + sin(p1 + p2) with
cos(p1 + p2) = (r2r2 - v1.v2) and sin(p1+p2)= r1v2 + r2v1 + v1xv2
Rotation is the generalization of multiplication or vice versa as you like.
You are not missing anything. The same thing can be called by different names product or rotation. The talk we had earlier. Some things can be a matter of taste.