Calcualting minimu LOS altitude at specifi distance (function)

I need to be able to calculate the line-of-sight minimum altitude between two objects on the earth taking into account the objects between them. Calculating LOS is pretty easy and there are numerous online calculators for that, but none of them take into account objects between them.

For example, say I have two communicating sides - a RC pilot and an RC plane. Supposing it can fly 100km (D) away, then what is it's minimum flight altitude H_ant2 to remain in LOS when there is a known obstacle with height H_obs at a distance L away from RC operator who holds the radio at height H_ant1. I would need to have an expression that let's me calculate the minimum flight altitude so I could draw a function graph which shows flight altitude in respect to distance from the RC pilot.

The exercise seemed pretty trivial at first, but after doing little math I found it to be out of my scope of knowledge because there are a lot of variable and constraints to take into account.

If someone could provide me an formula for calculating minimum altitude I would be very grateful. However if someone can just point me into right direction and help me with an explanation then that would be of help to. I know there are probably numerous ways to solve this, but I would like to stick with the most easiest to grasp and implement (that means no dif. equations please).

Best regards!

As I got no answers from here I went out and annoyied some of the mathematicians I know and got to a result myself. I want share it here in case somebody runs into a similar problem.

So, to calculate the minimum height for LOS at a specific height you would use cosine formula:

  c^2 = a^2 + b^2 - 2cos(aplha)ab


Here, a is the earths radius plus operators height (don't forget to convert units!), b is the distance between UAV and the oprator, c is the earths radius plus UAVs height above earth and alpha is the angle between the line originating from earths center (basically radius line) and between the obstructing object. You would obtain the minimum height by calculating the c and then subtracting earths radius from it. Angle alpha can be calculated with a simple pythagoras theorem provided that the distances are reasonable for the error to be marginable.

This of course gives some error which can be significant when goign into extremes.

Cheers!

• Your question requires only a trivial usage of the Pythagoras' Formula. It was neglected probably because somehow no one has seen it. Typically, such questions get a lot of top-level answers very quickly. Now I have to go to work; I will give you a better answer answer that. Sorry for it - next time you will have more luck, it is sure. – peterh says reinstate Monica Jun 19 '18 at 6:42
• Pleas explain to me how you can solve the problem with trivial use of pythagoras. At first I thought the problem is that simple but then, when I had to account for earths curvature and obscuring object height at some arbitrary distance, the problem quickly escaped the scope of simple pyhtagoras – rongard Jun 19 '18 at 13:04