The following sketch shows how the earth's horizon (red arc) appears to curve down left and rigth in the frame of view with a horizontal viewing angle α. The downward curve has to do with the perspective transformation of the horizon onto a flat display. If we look down from a certain height to the horizon, the horizon at the edges of the frame appears lower than at the center of the frame.
We can calculate the anglular drop δ between the center of the horizon and the edges and using the distance to the center of the horizon v we can calculate from this angle the apparent drop left to right p as it were measured by a scale at the horizon.
To look at the horizon on a spherical earth we have to look down from eye level a certain angle. This angle is called the drop angle φ. This angle is the same as the angle between the lines AZ and CZ, where Z is the center of the earth. Because the triangle ACZ is a right angled triangle with the right angle at C and the angle φ at Z, we can use trigonometry to calculate the angle:
(1)  
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The distance v from the observer's eye to the horizon can be calculated using Pythagoras, because we have a right angle triangle ACZ, where Z is the center of the earth and the right angle is at C. So AZ = R + h and CZ = R and v = AC:
(2)  
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To calculate the line x = BE = BC we can use similar triangles. The triangles ACZ and ABC are both right angle triangles and share the same angle φ at Z and at C respectively. We can equate the ratio adjacent : hypotenuse = CZ : AZ = BC : AC and solve for BC = x:
(3)  
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To calculate y we look at the right angle triangle AED, where y = ED and the angle at A is half the horizontal field of view angle α. Using trigonometry we get:
(4)  
where^{'} 

The field of view angle θ is commonly refered to the diagonal of the viewing frame with aspect ratio r. So we have to calculate the horizontal field of view angle as follows:
(5)  
where^{'} 

The rest of the calculations can only be executed if y < x, i.e. the earth does not fit as a whole in the frame of view, so there is an intersection between the leftright border of the frame and the horizon at all.
Now we need the length of the line AB = m + h. We can calculate using Pythagoras the line AB from AE and BE:
(6) 
And we need the line BD = t from the right angled triangle BED with the right angle at D:
(7) 
Now we can see what happens if the earth fits as a whole in the frame. In this case y > x so the argument of the square root gets negative and we have no real solution for t.
Now we can calculate the angle β of the right angled triangle ABD with the right angle at B and the angle β at A:
(8) 
Finally we can calculate the horizon leftright drop angle δ:
(9) 
and the horizon leftright drop as measured on a scale at the horizon in the center of view:
(10) 
In this case the drop p is an arc with center at the observers eye. If you want the drop as a perpendicular to the line of sight, use tan(δ) instead of δ. If you want the drop as a perpendicular to the line AD use sin(δ).