I am trying to get the ratio of intersection area of two circles and area of one of the circles, but as simply as possible.
I know I can calculate the ratio as an area of intersection divided by area of single circle:
$ z_x = \frac{A'}{A_x} $
where (based on wolfram.com)
$A' = r^2\cos^{-1}\left(\frac{d^2+r^2-R^2}{2dr}\right) + R^2\cos^{-1}\left(\frac{d^2+R^2-r^2}{2dR}\right) - \frac{1}{2}\sqrt{(-d+r+R)(d+r-R)(d-r+R)(d+r+R)}$
$A_x = {\pi}r^2$
But how to simplify this as much as possible, when I only need the ratio in the end. Is it possible to remove trigonometric functions at all? Is it perhaps somehow solely correlating to relations between $d, r, R$?
It cannot be analytically simplified any further otherwise it would have been given already in the Wolfram reference. Only special cases can be further simplified. Also notational simplification is possible but that is about all. Even a common area computation between a circle and a cutting secant line cannot in general avoid trig. Numerical calculation is the next step.