Here’s my diagram and the formula I’m trying to get to.
$h_0$ = height of the observer.
$h_h$ = the hidden height.
$r$ = radius of the earth.
$d$ = distance between the two along the curved surface of the earth
If you could do this step by step I would greatly appreciate it. And I’m not sure if this is quite the right website for this so sorry if it is not.
Best Answer
I can show you why the formula you wrote works. I assume the formula you wrote is correct, and I will not derive it from first principles - these are just my thoughts upon seeing it:
Consider a set-up whereby the line extended out to the observer, of length $r+h_0$, is the hypotenuse of a right triangle, and the side adjacent to an angle at the centre of the Earth is of length $r$. This angle is given by $\cos^{-1}(r/(r+h_0))$, and the expression $d/r$ gives the angle at the centre of the real triangle in radians, so their difference tells you how much more the observer is bent away from the thing they are looking at. Cosine of that angle will give you $A/H$ in the right triangle formed by the radius of the earth and an hypotenuse of length $H_h+r$. Take $r/\cos$, you get the hypotenuse $r+H_h$, and then taking away $r$ gives you $H_h$. With pictures now!
If you excuse the use of MS Paint, we have here the angle difference $d/r-\cos^{-1}(r/(r+h_0))$ as the green angle. The red bar shows where the hidden observer is, but $H_h$ is the lateral distance along the horizontal radius extended out to the bar. The bottom triangle shows how $H_h$ is deduced - that triangle is what you get if you rotate the horizontal with $H_h$ on it by the blue angle, so that it can be compared past Earth's curvature, and then the green angle is what you get, and hopefully there's enough geometric intuition here to carry the rest of it through. I admit my explanation was hard to read, sorry.