What is the meaning of quaternion interpolation?

Question

Suppose I take the average between two quaternions, how does one see the meaning of the resulting rotation to make sure it is sensible, unlike interpolating Euler angles?

I'm looking for an argument that would show that rotating a point by the new quaternion is equivalent to rotating it by the original two and then going half way on the great circle arc in between.

Answer 1

You need to assume that both are unit quaternions. Rotate everything so that the first quaternion is 1. Then the second is $\cos α+n\sin α$ with $n$ being a purely imaginary unit quaternion. The midpoint is now at $$ \frac12 [ (1+\cosα)+n\sin α]=\cos \tfracα2[\cos \tfracα2+n\sin\tfracα2] $$ so that indeed if $\cos \tfracα2\ne 0$ the midpoint after normalization gives the rotation with same rotation axis and half the angle.

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For general interpolation between two quaternions with constant rotation velocity, see my answer to a similar question, or any other source on quaternions and rotation. In short,

• compute the decomposition $q_1^{-1}q_2=\cosα+\vec n\sinα$, or more specifically, the angle $α$ between the two quaternions, as before,
• then the interpolation is given as $$q(t)=\frac{\sin((1-t)α)}{\sinα}q_1+\frac{\sin(tα)}{\sinα}q_2$$