Mapping sphere surface to a vector space such that distances are preserved? I have a unit radius sphere (say in 3D) centered on the origin. Thus the shortest distance between two points on the sphere is the geodesic. Is there a transformation (linear or non-linear) on the points on the sphere to a higher-dimensional space (or lower if it exists), such that the distance between the points on the sphere is preserved in the transformed space as well.
To make it more rigorous: let $x_1$ and $x_2$ be the two points on the sphere. Let $y_1=T(x_1)$ and $y_2=T(x_2)$ be the transformed points. Let $d_s(x_1,x_2)$ be the shortest distance between $x_1$ and $x_2$ on the sphere's surface. Is there any T(.) such that $$d_s(x_1,x_2)=||y_1-y_2||_2^2$$ (the Euclidean distance) in the transformed space holds.
Any reference to relevant research or literature is welcome as well. Any engineering approaches are welcome too. 

I think the Gnomonic projection to $\mathbb{R}^2$ would do the trick.
It maps great circles (i.e. Geodesics) on a sphere to straight lines on a plane. Therefore the shortest path is preserved.
