Instead of computing quaternion interpolation between $2$ orientations $p_1$ and $p_2$, can I calculate the inverse of $p_1$ and apply this to $p_2$ so that $p_1$ becomes coordinate axes and $p_2$ transform to $p_2^\prime$. Now I can calculate the interpolation by just interpolating the angle $t$ with rotation axis of $p_2^\prime$. Now I can invert all the transformation and I get an interpolation between $p_1$ and $p_2$, I think this gives a geodesic path or am I wrong?