Distance to the horizon?

The distance to the horizon can be approximated fairly accurately by Euclidean geometry. Earth is assumed to be a sphere of radius r = 6378 kilometers. Let h be the height above ground of the eye of the observer. Any ray of light that hits the eye represents one of infinitely many potential lines of sight. Now, of interest to us is the one line of sight that is a tangent to the Earth’s surface. Tangent line l intersects with the surface in exactly one point, and any of the Earth’s surface beyond that point is invisible at this height:

By definition, tangent l is perpendicular to the Earth’s radius. Pythagoras’ theorem states

2 2 2 (r+ h) = r + l ,

and by solving for l the distance is obtained:

∘ -------- l = 2hr + h2.

The actual distance to the horizon depends on h. Consequently, a tall person can see farther than small person. The following graphs shows how l develops in relation to h:

Interestingly enough, the first diagram shows another right triangle spanned with with hypotenuse l. Pythagoras’ theorem states that the following equality holds:

∘ ---------------- ∘ -------- l = x02 +(r +h - y0)2 = 2hr+ h2

In order to verify this assertion, however, the intersection point (x₀,y₀) has to be determined. Earth’s surface is a function

At the point where f and l intersect, the slope of l equals the slope of f. Using the derivative

f′(x) = - √-x----, r2 - x2

an explicit representation of tangent l is the function

l(x ) = f(x0)+ f′(x0)(x- x0).

By definition, l(0) = r + h must hold, because the eye of the observer is another point of l. Solving the equation for x₀ is fairly straightforward:

	f(x₀) + f′(x₀)(0 - x₀)	= r + h
⇐⇒		= r + h
⇐⇒	x₀	= .

The expected result l = √2hr-+-h2-

follows if x₀ is substituted into the equation given by Pythagoras’ theorem, which in turn allows further simplification of the point of intersection to

--rl- x0 = r + h.

So far, the distance to the horizon was regarded as the length of a line of sight, but that length is not the same as the distance that a person walking Earth’s surface from (0,r) to (x₀,y₀) would travel. That distance would be rα. The angle α is defined as

x0 r sin (α) = r- or cos(α) = r+-h-,

and solving for α gives:

( ) ( ) α = arcsin --l-- = arccos --r-- . r+ h r + h

It turns out that the difference between l and rα is negligible. Even if the eye of the observer is as high as 10,000 meters above the ground, l and rα differ by less than 400 meters — which is hardly significant, given the magnitude of error involved in the computation to begin with just because Earth is not actually a sphere. The following graph uses a logarithmic scale for h to illustrate the magnitude of the “error” even for large values.

Still, there is a difference between l and rα. If h grows infinitely large, then l will grow infinitely large, too. An eye that is arbitrarily far away from Earth has to have an arbitrarily long line of sight to reach back to Earth. Still, no eye can see past the point (r,0), so the actual distance spanned on the surface is limited by the angle 0 ≤ α ≤.