Is there a way to find a function $F:\mathbb S^2 \rightarrow \mathbb R^3$ of class $C^1$, minimizing
$$\int_{\mathbb S^2\times\mathbb S^2}(d(F(x),F(y))−\delta(x,y))^2 dx dy$$
, where $d$ stands for the euclidean distance in $\mathbb R^3$ and $\delta$ the geodesic distance on the sphere $\mathbb S^2$?
Or $d$ could stand for the squared euclidean distance, and $\delta$ the square geodesic distance, if this makes the problem simpler. The goal is thus to approximate geodesic distances by euclidean distances of transformed points.
I tried to perform a Multi-Dimensional Scaling to get this least square solution for a finite set of point, but it seems that the solution was just the identity (or a uniform scaling)... is that right?
Thanks!
Hints in order for you to solve this problem:
(1) Let $x, y \in S^2$. The segment from $x$ to $y$ that minimizes the distance on $S^2$ must be contained on a great circle.
(2) Look at the plane that contains $x, y$ and the origin of $\mathbb{R}^3$. There the problem is reduced in dimension (as Anton Petrunin pointed out in MO).
(3) Draw the circle and mark $x$ and $y$. The euclidean distance is the length of the straight segment connecting then. Use the law of cosines to relate this distance to the length of the circle arc connecting $x$ to $y$.
I realize this doesn't give the function you're asking for, but this gives you a formula for $d(x,y)$ in terms of $\delta(x,y)$. Is that the point? If it is, you should also check this. Otherwise, I have to think a bit more about it.