Watchtower Horizon and Hidden Height

What is modelled

The Earth is a sphere, so the ground drops away from a straight line of sight. A lookout whose eye is at height h sees the ground out to a horizon at about the square root of twice the Earth radius times h. Beyond it, a distant object is hidden from its base up to a height that grows with the square of the extra distance. The air bends light slightly downward, which is equivalent to a slightly larger Earth and gives a longer horizon.

Equations

• Horizon distance d = √(2 Reff h) with Reff = R / (1 − k); exactly √(h (2R + h)) without refraction.
• Greatest range to see a mast top d1 + d2 with d2 = √(2 Reff ht).
• Hidden height at distance D > d: (D − d)2 / (2 Reff).
• Dip of the horizon √(2h/Reff).

Checks run in the page

• The exact tangent against the square root formula and the 3.57 sqrt(h) rule.
• A ray-tracing search for the grazing ray against the effective-radius horizon.
• A bisection for the hidden height against the formula.
• The refraction gain 1/sqrt(1 - k).

Honest simplifications

The Earth is a perfect sphere with a mean radius and the ground is level with the water, so hills, tides and waves are ignored; refraction is a single constant coefficient although real air changes by the hour (mirages and looming are not modelled); and the object is a thin mast whose lower part is taken as fully hidden. Atmospheric haze, which often limits real visibility before the horizon does, is not included.

• Press No refraction and then Cold air, long sight to see the horizon move.
• Press Low bank lookout: a boat that is not far away already loses its hull.
• Raise the platform: the horizon grows with the square root of the height, not in proportion.