How much do you have to look down to see the same altitude in the distance?

Question

I am wondering, if you are standing on a flat plain, looking at mountains in the distance, how much you have to look down to see exactly the same height?

Obviously with the curvature of the earth, you have to look down some... What is the ratio - maybe in a "for every x feet/miles, look down 1 degree?"

Also, does it matter if you are instead on a mountain yourself looking across at another mountain?

Answer 1

It is not a smooth linear relationship except in the near distance. There are a couple of complications. If you are on a mountain, you are further from the center of the earth and as the mountain you are looking gets further away, it is effectively tilting away from you.
Suppose the mountain was 100 kilometers away and both your eyes and the top of the mountain were 6380 km from the center [radius of Earth = 6 378 100 meters per Google] Then the three points of the center and two locations would be an isosceles triangle with two sides of 6380 and 100. This angle would be the same as the one formed by the tangent to the earth (your eyes looking level) and the line from your eyes to the top of the mountain.
The sine of the half angle of the isosceles is 50/6380 [0.00783699] and for small angles the sine and the angle are the same, so the full angle is 0.01567398 degrees or 0.940439 minutes or 56.42633 seconds or roughly one minute per 100 kilometers

Answer 2

Your question is a bit broad - I can tell you that standing on perfectly flat plane, the curvature of the earth is 8 inches to a mile

Answer 3

This is related to map reading question that if there is no barrier land feature between the altitude and the look down distance we have a chance to see it. I think in every 167 meters 1degree altitude will increase or decrease according to the starting point.