For context, my goal is to interpolate aircraft position data at a relatively low sample rate, and as planes do not turn (change their heading) on a dime, I would like to use something more accurate than linear interpolation to interpolate the said turns.
The best idea I could come up with is using a cubic Hermite spline or similar where I can use the heading of the aircraft at the two points to compute the first derivatives. The problem is that I need to do this in spherical coordinates as opposed to Cartesian ones. Slerp is a well known algorithm for linear spherical interpolation, which is trivial to implement, but I scoured the internet for an algorithm that would let me do such nonlinear spline interpolation but came out emptyhanded.
Here's a diagram of what I want to achieve. I drew it in terms of a Cartesian plane (which is a good approximation of what I'm doing at short distances and low latitudes anyway) but I of course want to achieve the same on a sphere. $f(t)$ represents a linear interpolation, $f(t) = \mathrm{Slerp}(p_1, p_2; t)$ in this case, while $g(t)$ is a spline with control points $p_1$ and $p_2$ which are tuples of spherical coordinates and their respective derivatives/tangents $p'_1$ and $p'_2$, $t \in [0, 1]$. $g(t)$ is what I am looking for.
Therefore, I would like to ask if anyone is aware of such an algorithm and can provide more information on how to implement it.