Can you decompose a rotation into a rotation around an axis and a quaternion?

Question

I have a rotation represented by a quaternion and I want to get a rotation angle around the forward axis together with another quaternion which will together represent the original rotation.

The reason for this is that I am interpolating the rotation along a path and I want to be able to interpolate the rotation around the path in such a way that it does not have any discontinuities or sudden flips.

Answer 1

Avoiding discontinuities is a bit tricky for reasons outlined in this Q&A - if we try to strictly control, say, the up direction to align as closely as possible with world up, then we'll have to flip somewhere if our rotation does a vertical loop-de-loop, reversing its relationship with the world up axis suddenly.

Instead, let's attach a reference "up" vector to each interpolated point, perpendicular to current local forward vector, and we'll try to align it with world up when we can.

You can choose your first point's reference "up" vector arbitrarily (eg. the local up vector of the initial rotation, so your twist angle starts at zero)

For each subsequent point, we can choose our reference vector as:

sourceUp = previousReferenceUp + untwist * dot(previousReferenceUp, worldUp) * worldUp;

referenceRight = normalize(cross(sourceUp, localForward));

referenceUp = cross(localForward, referenceRight);

Here I'm assuming left-handed coordinate system, with untwist being a small tuning coefficient (eg. 0.01) to to control how much our reference up seeks vertical in non-vertical sections.

Now we can compute an angular twist about the forward axis relative to this referenceUp direction as:

localUp = rotation * worldUp;
angle = atan2(-dot(referenceRight, localUp), dot(referenceUp, localUp));

Our twist quaternion can then be expressed as:

// Remember to convert to degrees if expected by your Quaternion library (eg. Unity's)
twist = Quaternion.AngleAxis(angle, worldForward);

We can compute its inverse (ie. negate the x, y, z components):

untwist = Quaternion.Inverse(twist); 

And then compute our untwisted remaining rotation as:

untwisted = rotation * untwist;

Then for any vector v, the result of rotating that vector by our twist angle then by the remaining untwisted rotation is equivalent to rotating the vector by our original quaternion:

rotation * v == untwisted * twist * v;