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I'm working in Ogre, but it's a general quaternion problem.
I have an object, to which I apply a rotation quaternion Q1 initially. Later, I want to make it as if I initially rotated the object by a different quaternion Q2.
How do I calculate the quaternion which will take the object, already rotated by Q1, and align it as if all I did was apply Q2 to the initial/default orientation? I was looking at (s)lerping, but I am not sure if this only valid on orientations rather than rotations?
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You can do this using quaternion multiplication. To get the difference C between quaternions A and B you do this: C = A * Quaternion. Inverse(B);
Just to clear up some confusion: Yes usually you can add, subtract, multiply and divide quaternions.
Just apply the quaternion to a 3-D vector. If the final result from q1 and q2 is the same, then the quaternion has "same direction".
It sounds like you want the inverse of Q1 times Q2. Transforming by the inverse of Q1 will rotate the object back to its original frame (the initial orientation, as you say), and then transforming by Q2 will rotate it to its new orientation.
Note that the standard definition of a quaternion applies transformations in a right-to-left multiplication order, so you'll want to compute this as Q = Q2*Q1^{-1}.
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