# Projection from an ellipsoid onto a sphere that preserves geodesy?

I am currently taking an ellipsoid $E$ and a sphere $S$ of radius equal to the semi-major axis of $E$, both centered at the origin. I take a point on the surface of the ellipsoid and project it onto the surface of $S$ by following the line from the origin onto the point.

In other words if the point $A$ is on $E$ then I get the semi-line $\vec{OA}$ and follow it until it intersects $S$, and the point of intersection of these 2 objctes is the projected point.

This mapping doesn't preserve geodesy. I want to know if there is an invertible map from $E$ to $S$ that preserves geodesy (i.e if a set of points $P$ is contained in a geodesic line of $E$ their projection is also contained in a geodesic line of $S$ and vice versa).

If no such mapping exists, is there one that can get close to it?