Reverse refraction solver

Built for the arguments that recur: can you really see the Chicago skyline across Lake Michigan, Milwaukee from Muskegon, Corsica from Genoa, the Rampion wind farm's yellow turbine bases from a Sussex beach, or a distant island from the waterline? Load a case above, or enter any sighting; share the link and it opens pre-solved. To embed this solver in your own page, iframe this URL with ?embed=1.

How it solves. Hidden = target − visible. On the globe, k is found so that (D − √(2·R·h/(1−k)))² · (1−k)/2R equals the hidden height. On the flat plane, the same hiding must come entirely from ray curvature: the equal-hiding ray radius r gives k = −R/r (negative: light bending away from the surface). Why negative is forced: a flat surface never curves away from the light, so a downward-bent ray (any k ≥ 0) can always arc over and reach every point on the surface — standard refraction occludes nothing on a plane (at worst it folds grazing images, superior-mirage style). Hiding on a plane can only come from rays bending away from the surface.

Sign and limits of k. Light bends toward denser air; the pressure and temperature gradients compete and cancel at the autoconvective lapse rate, ≈3.4 °C per 100 m (AMS Glossary of Meteorology) — the 0.0343 in the formula. Cooling steeper than that turns k negative (inferior mirage), but air in that state is denser above than below — absolutely unstable — so it overturns and survives only as a thin forced layer against a hot surface: measured and simulated inferior-mirage layers over water are ≈10–20 cm deep (Am. J. Phys. 91, 999, 2023), and the zone they erase spans roughly an arcminute at the apparent horizon (Young, inferior-mirage simulations) — about 17 m at 60 km. Whole low islands can vanish this way for a camera held inside the layer, and reappear on standing up. Inversions, by contrast, can be deep and stable, so large positive k (looming, ducts) is well documented at sea, while a deep path-averaged negative k of the size a flat plane needs here has never been observed — and cannot persist, because the required lapse is several times past the overturning threshold through the full depth of the sightline. Temperature from k = 503·(P/T²)·(0.0343 + dT/dh) — Bislin's formulation; T in kelvin, P in millibars. This tool is inspired by, and runs in the opposite direction to, Walter Bislin's Advanced Earth Curvature Calculator — his forward solver (pick a k, see the view) is the standard of the genre and the right tool for prediction; this one answers the inverse question (see a view, solve the k). Thanks, Walter. Regime bands on the scale are indicative descriptors, not sharp physical lines: normal marine k 0.10–0.20 (Gauss survey standard 0.13; ISA sea-level lapse gives 0.17), minor looming 0.20–0.35, major looming 0.35–0.60, and duct / Fata Morgana above ≈0.60 (at k = 1 light follows the surface and the horizon is unlimited). The upper regimes are documented as locally routine: Ives measured 10–15′ false-sea-horizon lifts as "a standard midday condition" on the Gulf of California (J. Franklin Inst. 252, 285–295, 1951) — equivalent to k ≈ 0.6–0.95 over a ~55 km path. Near-surface k is measured to swing far outside the survey band (Hirt et al., J. Geophys. Res. 2010). The parabolic hiding formula is good to well under 1% at these distances; single-k is a whole-path average and the last metre above the water can deviate from it (mirage layer).

Where this tool could be wrong. It models refraction as one coefficient averaged over the whole path; real air is layered, and a sightline grazing the last metre above the water crosses the mirage skin where the local gradient can differ wildly from the path average — read results for very low observer heights as effective values, not point measurements. The Ives band converts measured horizon lifts into the k giving the same angular lift over the entered path; it is an equivalence, not a direct k measurement. The temperature line assumes the entered pressure and temperature hold along the path. And the answer is only as good as the visible-height input — measure it against surveyed target topography where possible. The measurement-± field expresses that precision: the solver re-runs at visible ± that amount, and the whiskers on both markers and the “k range” ledger rows show how far the answer moves.

Why solve both columns. On geometry alone, a flat plane with k′ = k − 1 is exactly degenerate with a globe at k — both corrections go as D²/2R — so no single photograph separates curvature from refraction. What separates them is the thermodynamic price of each branch, which is what this tool computes: ordinary air sits near k ≈ 0.13–0.17, while the flat branch demands gradients several times past the autoconvective overturning limit, sustained along the whole path. The method, in short: concede that the geometry cannot decide, then price the flat branch. More at funwithscience.net/globe-deconstruction.