My question is similar to Quaternion barycentric interpolation, but since that question is 6+ years old, I figured it could be asked again. I also will eventually be trying to do this with unit octonions (essentially replace every usage of quaternion with an octonion), but I think starting with the better known quaternion is best.
I have a set of $N$ unit quaternions, $Q = {q_0, q_1, ... q_N}$, and a test quaternion, $q_t$. For each quaternion in $Q$, there is an associated scalar value, $v_0 ... v_N$. I want to find an appropriate value for the test point, $v_t$, which is an interpolation between the nearest quaternions in $Q$ to $q_t$.
I recognize this question has multiple parts, and help on any part of would be beneficial. I have found solutions that would work in a euclidian space, but the fact that the unit quaternions all "live" on a hypersphere has been confusing me. The parts that I can think of are:
- How can I find an appropriate set of the 4 nearest quaternions in $Q$? (If the space were euclidean, I would probably mesh $Q$ using a delaunay triangulation and find the tetrahedron surrounding $q_t$, but I'm not sure if this valid for quaternions on the 3-sphere)
- How can I interpolate between those 4 quaternions to obtain the test value $v_t$? (In euclidean space you can use the barycentric coordinates, but I don't think that is defined for points on a hypersphere. My thought is to do something similar to the linear interpolation in interpolation between 3 points, but I am not sure if it is valid for quaternions)
Hopefully that makes sense, thank you.