I am working on an engineering task where we have a flying object which knows its height above a curved surface as well as how close it is to other flying objects. If I want to interpolate between two sets of gathered data points, where the respective frames of reference are not collocated, what would an interpolated point mean?
I'll give more detail below, as well as a pretty picture.
Background
Suppose that a continuous, smooth surface surrounds the origin of coordinates. This will define an inertial and nonmoving frame of reference that I will talk about later.
Around this surface, $N\geq1$ objects moves continuously over time $t\in (t_i, t_f)$. At a time $t$, the locations of these objects are defined in a frame whose origin is defined to be on the surface, one of whose axes is normal to the surface (it doesn't matter if it's inward or outward), and whose other two axes span a plane tangent to the surface. The location of an arbitrary object $n$ at some time is $\vec{r} = (x_n(t), y_n(t), z_n(t))$, where $\hat{z}$ is normal to the surface. The last constraint on the coordinate system definition is that for object $0$, its position at any time is $(0, 0, z_0(t))$, meaning that the coordinate frame is always beneath object $0$ on the surface
Question
Suppose that the coordinates of all objects are gathered in some initial frame at a time $t_i$ and at some final frame at a time $t_f$, but not at a time $t$ between them. If we use this set of coordinates for linear interpolation in order to generate a set of positions for a time $t$, it would look like this for object $n$:
$$\vec{r}_n(t) = \vec{r}_n(t_i) + (\vec{r}_n(t_f) - \vec{r}_n(t_i))\rho,\ \ \rho = \frac{t-t_i}{t_f-t_i}\in [0,1]$$
The question is this: Since the frames from which the coordinates came was rotated and translated, what does this new coordinate $\vec{r}_n(t)$ mean? If it means something valid, what is it?
Some colleagues think this is incorrect to do. One should use the inertial nonmoving frame, perform interpolations there, then transform to the frames desired. The interpolated values will certainly be different between the two approaches, and I'm not looking to see which one is more or less correct, but more attempting to understand what $\vec{r}_n(t)$ means in the way I have presented it.
Intuitive Answer
My intuition was that since the frame is defined as being under object $0$ at any point in time, and the frame is on the surface, then an interpolated value will be a frame on the surface under the object $0$. The complexity starts to arise when I ask "where along the surface is this frame?"
If we imagine an observer under object $0$ for a surface which is flat, this seems clear. Object $0$ starts at $z_0(t_i)$ and ends at $z_0(t_f)$, and if we wish to see where the object might be with linear interpolation after a fraction $\rho$ of its motion, then the observer must also move a fraction $\rho$ of the total path length from their starting point to ending point.
Again, intuitively I think this argument should hold for curved surfaces, multiple objects, and have a simplified example in 2D on an ellipse with only two objects to illustrate my point a bit more.
Formal Answer
How to prove this is where I'm getting a bit stuck. I think I effectively want to prove that in order to interpolate by $\rho$, the frame must also move along its path length by $\rho$, and that they are the same $\rho$. But I can't tell if I'll just be making a claim and not proving it, or successfully illustrating that my intuition was correct. I want to know what the new coordinate $\vec{r}_n$ means and in what frame, regardless of where my intuition is guiding me.
Any guidance on how to proceed with this proof, figure out what needs to be proven, or even referencing a book where this is done would be great.
Thank you.
Simplified 2D example
