Consider I have a set of $N$ points $\mathbf{p}_{1,\ldots,N} \in \mathbb{R}^2$ where I know their coordinates and the Euclidean distance between each pair of them $||\mathbf{p}_i -\mathbf{p}_j ||$ on the plane.
I want to map those points on an ellipsoid with known center and axes such that on the $3D$ ellipsoid, the geodetic distances between pairs of points are preserved. What kind of transformation is this?
Addditionally, the transformation should be somewhat fixed as I know that the barycenter of the points in 2D is mapped to a fixed known point on the ellipsoid and also one of the points $\mathbf{p}$ has a known correspondent $\mathbf{p}'$ on the ellipsoid, that basically determines the direction of the axes.
My question is, can I determine the form of the transformation so that I'm able to map all the other points in $\mathbb{R}^2$ on the ellipsoid while maintaining their distances on the geodesics?
In the figure, the barycenter of the points is $\mathbf{x}$ and is mapped to $\mathbf{x}'$ which I know. I also map $\mathbf{y}$ to a known 3D coordinate $\mathbf{y}'$. I need to compute the mapping to preserve the distances of every pair of points $\mathbf{p}_{1,...,N}$ in geodetic terms.