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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!

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    The radius of the sphere mentioned in Rahul Narain's comment would be $(6\pi-8)/9\doteq 1.20551$. The square root of the mean quadratic deviation would be $0.168261$, resulting mainly from pairs $(x,y)$ with large $\delta(x,y)$.2012-07-15

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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.

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    I understand the idea, but my intuition tells me that if you deform the sphere so that some points behave like a plane you'll end up increasing the difference of the metrics for others points (and maybe this explains why you are getting the identity as a solution). Btw, maybe you want $F$ to be a diffeomorphism, that would be a more natural hypothesis from the geometric viewpoint. Then again, I'd guess that if this was a better option somebody would have pointed this already.2011-06-09