Local level is extended across the world
The claim is that water seeks its level, so a large body of water must form a single flat surface. Its immediate evidence is familiar: the sea does not look like a mound from the beach. The step requiring examination is the extension of a local horizontal into a plane that stays horizontal everywhere.
Level depends on the direction of gravity where it is measured. A local tangent plane can be horizontal at one location and cease to be horizontal elsewhere on a curved surface. Keeping this distinction allows water to be locally level all along a surface whose direction changes gradually with position.
The observer stands above the surface being viewed
Surveyors describe the gravity-related reference for global mean sea level as the geoid. It is curved and gently irregular because Earth’s shape and mass distribution are irregular. Remove tides and currents from the ocean and its equilibrium surface follows that gravity field. The geoid is also extended through land to provide a reference for elevations.
For the horizon calculation, replace that detailed surface with a sphere. The last unobstructed straight ray from an elevated eye just touches it. The touching point is the geometric horizon. An eye above the water therefore looks slightly downward toward that boundary, even though the nearby surface is locally level.
A distant target can also extend above its own horizon. Seeing the upper part of a lighthouse or a mast beyond the distance to the observer’s water horizon is not an exception to the geometry. Both endpoint heights matter. A claim based only on the camera height has omitted a measurable part of the sightline.
Two heights determine the available sightline
A radius drawn to a tangent point meets the sightline at a right angle. For Earth radius R and observer height h, the right triangle gives the straight distance d = √[h(2R + h)]. The same calculation applies to the target’s height. At the limiting shared sightline, those two tangent distances add together.
The opening assumptions place the eye 2 metres above the water and the target’s top 10 metres above it. With JPL’s mean Earth radius, the observer’s tangent distance is about 5.05 kilometres. The target contributes about 11.29 kilometres. The maximum unobstructed straight sightline between those heights is therefore about 16.34 kilometres.
The larger combined result is not a claim that the observer can see water itself that far away. The target stands above the water. Set its height to zero to recover the distance to a point on the surface. Raise either endpoint and more distance becomes available; at heights small compared with Earth’s radius, quadrupling one height approximately doubles that endpoint’s horizon distance.
These are straight-ray distances over a smooth sphere. For an actual observation, use the height above the intervening water, including the local water level, and account for any terrain. Atmospheric refraction bends the light path and can change the visible range. A photograph taken through an unknown near-water temperature profile does not supply a measured straight ray.
The boundary is close because the observer is low
For an eye 2 metres above the water, the geometric water horizon is about 5.05 kilometres away. A target standing 10 metres above the same surface extends the available straight sightline to about 16.34 kilometres. Its height accounts for the extra range beyond the observer’s own water horizon.
The nearby horizon gives a low observer a small view of a large curved surface. Raising either endpoint extends that view while the water remains locally level under gravity. Comparing an observation with these distances requires both endpoint heights and the light path: refraction changes the path, while a taller target changes the geometry.
Observe one target from several heights
Keep the target and distance fixed while changing camera height, and record which named parts remain visible. Choose a stable shore target or fixed structure with surveyed feature heights, and measure the camera centre above the same surface datum used for the target. Record the water level rather than the tripod’s height above the pavement. Photograph from at least three camera heights without changing the target, keeping full uncropped frames, focal length and exposure metadata, and identify a feature’s lowest visible point, not merely whether the object appears.
Run the calculator for each height at the same distance, first with straight rays, then with an explicitly entered refraction scenario. Do not tune a coefficient merely to force one image to match. Repeat at different times with temperature and weather notes: a height-dependent change that persists across conditions is different from a transient stretched or inverted image.
Why people believe it
Standing on a beach, the sea looks flat, and a flat-looking surface is easy to extend in the mind to the whole world. The error is one of scale: a horizon from 2 m eye height lies about 5 km away, and over that distance the surface drops by only 2 m, too little to see. Camera zoom strengthens the impression, because it enlarges what is visible without revealing what a lower sight line has hidden behind the horizon.
