Reciprocal Zenith Angles

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About this visualization

The principle. The zenith is the direction exactly opposite gravity — what a plumb line points away from. A theodolite measures the zenith angle: how far you tilt down from zenith to aim at a target. When two stations aim at each other ("reciprocal" observation), the two angles must sum to exactly 180° if their plumb lines are parallel — basic geometry of a straight line crossing two parallels. On a curved Earth the plumb lines are not parallel: they converge at the planet's center, tilted by the central angle θ between the stations. The sum then comes out to 180° + θ, every time. Measure the overshoot, divide the distance by it (in radians), and you get the radius of the Earth: R = d ÷ θ.

Doesn't air bend the light?

Yes — and that's the genius of the reciprocal method. Atmospheric refraction bends both lines of sight downward (air gets denser lower down), which makes each measured zenith angle slightly smaller. Refraction therefore shrinks the overshoot — it works against the curvature signal, never for it. Surveyors observe both directions simultaneously so the bending is the same on each side and can be solved for and removed. A flat earth would need light bending upward, by exactly the right amount at every distance, in every climate, to fake these numbers. Air does the opposite.

See it yourself: the atmospheric refraction slider in the controls bends the sight lines live. The value k is the surveyor's refraction coefficient — how sharply the ray curves compared with the planet itself (k = R_Earth ÷ R_ray). k = 0 is vacuum; k ≈ 0.13 is typical air, a curve about 8× gentler than the Earth's; k = 1 is light hugging the surface — a duct that only ever forms for rays grazing metres above cold water or ice, never along an elevated survey line. Each measured angle loses δ ≈ k·θ/2, so the sum loses k·θ: at any k the globe's overshoot only ever shrinks toward 180°, while on the flat earth the same air drags the sum below 180°. Either way, refraction is bookkeeping — never an escape hatch.

Standard values, real conditions. Surveying references tabulate k by scenario — the preset list in the controls: 0 vacuum; 0.13–0.15 between hilltops on a mild day; 0.17 over land with the line well above the ground; 0.33 low over land or above water under overcast; 0.5 low over water late in the day; 0.75–1.0 skimming much-warmer air over cold water or ice — mirage territory. The wild numbers exist only for rays grazing within a few metres of the surface, where the temperature gradient is steep: research lines 1–5 m over grass or water have logged freak moments from −4 to +16. Elevated reciprocal-zenith lines between towers and summits live at k ≈ 0.13–0.17, and observing both ends simultaneously measures the bending so it can be removed. (Scenario table and research roundup: mctoon.net/refraction.)

Is this really used?

Constantly. It's called trigonometric leveling with reciprocal vertical angles — standard practice in geodetic surveys since the 1800s (the great triangulations of France, Britain and India all relied on it). It's why long bridges and tunnels must budget for curvature: the two ends of the 35 km Channel Tunnel survey would miss by meters if Earth were modeled flat.

What's exaggerated here?

By default you're looking at a toy planet ~1,000 km in radius so the angles are big enough to see. Tower heights auto-scale so the sight line clears the ground. Switch the radius slider to 6,371 km (the real Earth) and the numbers become exactly what real surveyors measure — the excess gets small (about 0.9° per 100 km), but it never becomes zero, and it always implies the same radius. On a flat earth the excess is 0.000° at every distance. That difference is the whole point.

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