Derivation of a Formula for Obstructed Height

August 21, 2016

Figure 1: Diagram showing the geometric elements used in the derivation. We want to derive a formula that expresses h2 in terms of R (the circle’s _ radius), s (the length of the arc AB), and h1 . We’ll use the formula for the cosine of the difference of two angles, cos (θ − α) = cos θ cos α + sin θ sin α,

(1)

R . R + h2 s The angle θ, in radians, is equal to . From Fig. 1, R

and the fact that cos (θ − α) =

cos α =

R R + h1

p and sin α = 1 − cos2 α =

1

q (R + h1 )2 − R2 R + h1

.

(2)

Making all of these substitutions, Eq. (1) becomes  s R = cos R + h2 R



R R + h1



s + sin R 

q (R + h1 )2 − R2 R + h1

,

(3)

from which

h2 = R

     R cos

  

R + h1 −1 .  s i q   + sin (R + h1 )2 − R2 R R

s

h

2

(4)