## Projections from the Sphere and Pseudosphere and their Indices of Refraction

Based on the eikonal equation of geometrical optics and the Legendre transform, Luneburg was able to associate an elliptical orbit created by a Coulomb field with the line element of a sphere. This meant that to any sphere of a given radius, $r_0$, there belongs a conjugate sphere of radius $r_1=1/r_0$, which represents a perfect undistorted optical image of the original sphere. Its image, however, is inverted, and its magnification is the ratio of the radii, $r_1/r_0$.

The line element,

$$ds^2=4\frac{dx^2+dy^2}{(1+r^2)^2},$$