Horizon distance along ground

In the previous post we used the Pythagorean theorem to estimate the visual distance to the horizon. However, the distance along the ground is slightly different, as it follows the curve of the Earth instead of a straight line. Given the diagram from the previous post,

we're now trying to find s rather than d, which is trickier due to the trigonometry, but it has a couple interesting techniques and confirms an intuition from the previous post.

When θ is in radians, s = Rθ, and θ is arccosine of the length of the adjacent side divided by the length of the hypotenuse, or, symbolically,

To approximate cos(θ), this Math StackExchange post demonstrates the relationship between cosine and the infinite Taylor series for small angles:

Choosing θ equal to the square root of 2y gives us

Now we can conveniently restructure our original equation to match:

Which means

and

When R is much greater than h,

so

which matches the estimate for d in the previous post, confirming our intuition that when when h is much smaller than R, the ground distance to the horizon is approximately the visual distance to the horizon.