0

I have two quaternions $p$ and $q$ that represent orientations of an object about its center of mass. I want to find a rotation (also in quaternion form) $r$ that rotates the object from orientation $p$ to orientation $q$. I don't care about the "path" that the rotation takes to get from $p$ to $q$ (e.g. the short way around vs. the long way around).

What is the difference between the following?:

$r_1 = q p^{-1}$

and

$r_2 = p^{-1} q$

I think that rotations are supposed to be premultiplied in the order they are applied, therefore I assume that only $r_1$ is correct, since $r_1 p = q p^{-1} p = q$, whereas $r_2 p = p^{-1} q p$, which is a case of conjugating $q$ by $p^{-1}$ (although I don't know what that even represents -- can anyone explain this?).

I have also heard that there are different ways to represent rotations, and (even though I have seen claims that there are no such thing as "right-handed quaternions" vs. "left-handed quaternions") different representations may result in a sequence of rotations being applied using either preorder multiplication or postorder multiplication. I assume this is referring to rotations being "applicative" or "corrective" (i.e. forwards vs. reverse)?

I have asked related questions before but couldn't seem to get an answer that made sense to me. Can somebody please explain the different ways quaternions can be used to represent rotations, and then the way to calculate the difference in rotations using either $r_1$ or $r_2$ above?

Luke Hutchison
  • 361
  • 1
    One is the inverse of the other, so you need to pick a convention for which side to multiply on to single one out. – Qiaochu Yuan Sep 13 '20 at 17:39
  • 1
    There is a difference between "right/left-handed quarternions" and a "right/left-handed representation of a rotation by a quaternion". That is, there is nothing inherent to the quarternion that is right or left handed. But given a particular rotation, there is one quarternion that represents it by multiplication on the right, and another that represents it by multiplication on the left. – Paul Sinclair Sep 13 '20 at 22:16
  • I addressed this i comments to your previous question: https://math.stackexchange.com/q/3779413/11323 If the material there and on other questions still isn't making sense, you should probably read a text/watch lectures on quaternions and rotations. – Kimball Sep 13 '20 at 23:43
  • @Kimball I appreciated your answer, but I was still missing one fundamental thing that your answer didn't cover, hence this different variant of the question. The comment by Paul Sinclair does a good job of explaining what I was missing. Thanks Paul! – Luke Hutchison Sep 15 '20 at 04:13

0 Answers0