If I have a cubic Bezier spline that interpolates the points $p_1 ... p_n$, I can extend it to another point $p_{n+1}$ by just adding a new segment with the first two control points determined by the existing spline (to maintain continuity and smoothness), and two control points that I'm free to choose (the last being $p_{n+1}$).
I'd like to do the same with a SQUAD quaternion spline (or have confirmation that I can't). That is, suppose I have some arbitrary quaternion spline that is smooth and continuous and interpolates the quaternions $q_1 ... q_n$, with the control quaternions $s_1 ... s_n$, which I'd like to extend to add $q_{n+1}$. Is it possible to choose $s_{n+1}$ to maintain smoothness at $q_n$? And, better yet, leave me so that when $q_{n+2}$ comes through the door, I can continue extending the spline and maintaining smoothness?
The derivation of $s_i$ in my go-to reference depends on $q_{i-1}$, $q_i$, and $q_{n+1}$, but I'm in a situation where I need to fix the spline up to $q_i$ before I can determine what $q_{n+1}$ will be.