Was doing some research this morning in relation to light in FET when I remembered this formula existing in our wiki on this page:

http://wiki.tfes.org/Electromagnetic_AcceleratorI searched the old forums for where this simple formula came from, but all I could find was a thread where Euclid derived the following equation for how light from polaris bends moving rimwards from the north pole:

Success! I have derived an equation for the path of light from the north star in the north south direction that exhibits the above assumptions.

y(x) = h - x Cot[r/h] - (x^2 (3 h - 2 r Cot[r/h] - r Tan[Pi/2 (1 - r/R)]))/r^2 - (x^3 (-2 h + r Cot[r/h] + r Tan[Pi/2 (1 - r/R)]))/r^3

y is the height of the light beam as a function of x, the distance from the north pole. h is the height of the Sun. r is distance of a ground observer of the light beam from the north pole. R is the distance from the equator to the north pole.

This is a cubic equation. Further degrees of polynomials could be used up to an infinite Taylor series, but they would require more unknown parameters. Perhaps a theory for cause of bendy light could provide values for these unknown parameters. Quadratic and lower polynomials are unable to satisfy the assumptions.

The thread goes on for a while, but ends with Parsifal:

I have recently come to the realisation that for any function y = f(x) that models the curvature of light, its derivative function f'(x) must be an injective function. Otherwise, the action of Dark Energy on rays of light at a particular gradient will be ambiguous. Euclid's equation does not fit with this requirement, as its derivative function is a quadratic whose value therefore approaches positive infinity as x approaches either positive or negative infinity.

I enjoy maths, so I'd like to continue the discussion here. Parsifal, was any more work done beyond this, and how does that thread relate to the EA equation that is found in the Wiki?