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?

mapping an ellipsoid

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.

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    $\begingroup$ You cannot preserve all distances, in general. You can see that clearly in the case of a sphere. $\endgroup$ – Aretino Oct 18 '16 at 16:41
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    $\begingroup$ I'm confused by your notation $\langle \mathbf p_i,\mathbf p_j\rangle$, because this looks like an inner (dot) product. But the fact that the plane is flat and the ellipsoid is curved tells you that there can be no such distance-preserving map. This requires differential geometry to prove, but is a fact nonetheless. $\endgroup$ – Ted Shifrin Oct 18 '16 at 17:05
  • $\begingroup$ Sure there is no such perfect mapping but if I allow some distortions, is there a mapping in least squares sense, at least? $\endgroup$ – linello Oct 19 '16 at 7:33
  • $\begingroup$ Let's take an example: a sphere of radius $R$ on which you want to map five points in a plane, the center $O$ of a square and its four vertices $ABCD$. If diagonals $AC$ and $BD$ have a length of $\pi R$, how do you propose to map these points on the sphere? $\endgroup$ – Aretino Oct 19 '16 at 12:06
  • $\begingroup$ First I map the center of the five points $O$ on the sphere, then I select another point $A$ on the sphere such that the geodesic distance on the sphere is the same as in the plane. This fixes the "direction". Based on that I try to map all the remaining points on the sphere, trying to keep as much as possible their relative distances over the geodesics. If the diagonal $AC$ measures $\pi R$ the mapping is over the complete sphere. I'd like instead to focus onto a locally flat part (no more than a given curvature) $\endgroup$ – linello Oct 19 '16 at 13:18

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