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Derivation of Angle to Target Top Equation

The angle α to the target top is measured from the eye level line of the observer to the top of the nearest target. If the target top is below eye level, the angle is negative, else positive.

(1)
width
where'
' =' ' vertical angle from eye level to target top; negative if target is below eye level
' =' '(refracted) radius of the earth
' =' 'observer height
' =' 'target height
' =' 'distance from observer to target along the surface

Note: this equation is robust and gives positive and negative angles α correctly without the need of handling multiple cases.

Derivation

Figure Calculation of Drop Angle to Target Top

We know R, T, Z and D. From this we can calculate a = a1 + a2 = R + Z, b = R + T and γ = D / R.

In the first step I calculate h using Pythagoras:

(2)
and

Combining this 2 equations we can eliminate h2 and solve for c2:

(3)
(4)
(5)

We can calculate a2 using trigonometry and a1 = aa2:

(6)
(7)

We can insert (7) into (5) to get c with all values known:

(8)
(9)
(10)

Our unknown drop angle is according to trigonometry α = −asin( a1 / c ). Now we have all we need to calculate the target top angle α:

(11)
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Created Tuesday, September 28, 2021
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Changed Tuesday, September 28, 2021