'Difference' between two quaternions

Question

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?

Answer 1

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}.

Answer 2

Think of it this way

QInitial * QTransition = QFinal

solve for QTransition by multiplying both sides by QInitial^{-1} (^{-1} being the quaternion conjugate)

QTransition = QFinal * QInitial^{-1}

It's just that easy.

• note to @Dan Park - if you disagree with my math, please post a response to my answer, don't change the math. As far as I know, it's right.