# Index Ellipsoid

Another, entirely equivalent, method of finding the eigen-indices of refraction for a given direction of propagation in a medium is the index ellipsoid. Starting with our surfaces of constant energy density, we have:

$$\frac{D_x^2}{\epsilon_x} + \frac{D_y^2}{\epsilon_y} + \frac{D_z^2}{\epsilon_z} = 2U_e$$

where we can then make a few substitutions to obtain:

$$\frac{x^2}{n_x^2} + \frac{y^2}{n_y^2} + \frac{z^2}{n_z^2} = 1$$