Due to the curvature of the earth, altitude measurements must be corrected accordingly.
With an average earth radius of 6371 km (see
) the ideal earth surface deviates from a tangential plane downwards (radial, direction of the earth) as follows:
0.8 mm at 100 m
20 mm at 500 m
78 mm at 1000 m
1.96 m at 5000 m
7.85 m at 10,000 m
As a simple approximation, exactly for small distances L, the formula [math-displaystyle y=-tfrac -L-2-2-2R-[-math) can be used, where [math-displaystyle L-[/math the distance, [math-displaystyle R-[/math the Earth radius of 6,371,000 meters and [math-displaystyle y”[/math is the deviation in meters.
To illustrate, two people are 10,000 m apart (= [math-displaystyle 2L-[/math)In order for them to have visual contact, either both must be at an eye level of 1.96 m (each [math-displaystyle L-[/math = 5000 m to the plane in the middle at 0 m height) or a person in the plane at an eye level of zero meters and the 10,000 m distant Person at an eye level of 7.85 m.
Graphic illustrating the curvature of the earth; the two black marked points are 1000 km apart and are both at an altitude of 19.6 km
With a slightly more accurate approximation formula [math-displaystyle y=-sqrt “L-2″+R-R-R-R-R-[/math with [math-displaystyle R-[/math = Earth radius, [math-displaystyle L-[/math = Distance and [math-displaystyle y-[/math = Humiliation, which is the height that disappears under the tangential plane at “straight view” (see also Geodetic viewrange), the following values of [math-displaystyle y”[/math at a predetermined [math-displaystyle L”[/math :
1.96 m at 5 km
7.84 m at 10 km
196 m at 50 km
784 m at 100 km
1,764 m at 150 km
3,135 m at 200 km
4,898 m at 250 km
The correction of altitude measurements due to the curvature of the earth is therefore indispensable even on short distances and grows squarewith the distance.
In a practical example, the elevation angle determination of mountains in the mountains, the earth’s curvature, for example for Mont Blanc at 4810 m in height, results in the following angles of altitude depending on the distance (assuming a view point on the altitude, in brackets the values without curvature of the earth):
at 50 km 5.27掳 (5.49掳)
at 100 km 2.30掳 (2.75掳)
at 150 km 1.16掳 (1.83掳)
at 200 km 0.48掳 (1.38掳)
at 250 km (0.02掳 )
The 250 km value indicates that at this distance the tip of Mont Blanc is below the “horizon line”.
For observation points above the sea level, the computational angle of altitude increases, because the “horizon line” is removed from the observer and only the proportion of the earth’s curvature beyond it becomes effective. In practice, terrestrial refraction also plays a role.They break the light rays in the direction of the earth’s curvature, so that the elevation angles are slightly enlarged. They can be interpreted as reducing the humiliationcaused by the curvature of the earth by five to 15%, depending on meteorological conditions. For example, if the influence of the refraction were 15%, the last case would be 0.04掳.