Shoreline Hidden Height Check

Materials Needed

known distant shoreline, bridge, or tower across water
camera with zoom if available
observer height estimate
map distance or GPS distance

Setup Steps

Choose a distant target across water with a recognizable base and upper section.
Photograph it from a low viewpoint and note what portion is visible.
Log observer height estimate, target distance, and which sections are visible at each height.
If possible, repeat from a higher viewpoint and compare how much lower structure returns.
Treat small refraction or heat-shimmer effects as uncertainty, repeat on a clearer day if results are borderline.

Expected Calculation

d_obs=√2Rh + h²

d_beyond=max(d − d_obs, 0)

hidden=d_beyond²2R_eff

sagitta=R_eff (1 − cos(d2R_eff))

clearance=max(targetHeight − hidden, 0)

Curved-surface geometry predicts that lower sections disappear first and that raising the observer restores more line of sight. Pure perspective shrink does not selectively hide only the bottom behind a geometric boundary.

Graphic

Schematic horizon overlays, no copyrighted photos. Horizon dip, hidden height, and visible structure come from the same sphere math used in the Curvature Calculator. Toggle refraction to compare geometric vs standard 7/6 R lift or a custom k-factor.

Lower section hidden below the horizon line Hidden 2.9 m · Visible 5.1 m · Horizon dip 0.042° · Horizon distance 5.45 km

With refraction off, hidden height would be 3.8 m instead of 2.9 m at this distance.

Standard sphere refraction lifts the apparent horizon here by 0.003° dip and cuts hidden height by about 0.9 m at this range. Flat-Earth refraction is not a single consistent k-factor: bending strong enough to erase globe hidden height on one bridge sightline typically over-lifts dip elsewhere, fails bottom-first ship recovery when you climb, and cannot match sunset-delay geometry without per-scene retuning (Bislin-style limit, not a full ray-tracer).

Schematic preset inspired by a long low water bridge. Real sightlines vary with weather, refraction, and camera height.

Visible above horizonHidden below horizonHorizon line from lib math

Open Curvature Calculator with these values

Sunset / refraction case study

Sunset timing is a predict vs observe check, not a single-photo proof. Sphere geometry plus standard terrestrial refraction (7/6 R) lifts the apparent horizon and can delay the moment the Sun crosses it.

Geometric dip 0.045° · Refracted dip 0.042° · Lift 0.003° at 2.0 m eye height.

Horizon distance grows from 5.05 km to 5.45 km with standard refraction, compare that prediction with a timed sunset observation.

θ_dip ≈ arccos(R / (R + h)) · R_eff = (7/6)R

Compare with Local Sun, Sun Path Comparator, and the Sunset Delay experiment.

What to Observe

The lower section is hidden before the upper section disappears.
Zoom can sharpen the visible top but does not restore the missing bottom once it is blocked.
Higher observation points often reveal more of the target again.

Interpretation

The observation matches curvature and hidden-height geometry much better than a perspective-only explanation.

Sources

USGS FAQ: Curvature drop over distance: Independent reference for geometric hidden height behind the horizon line.
NWS JetStream: Atmospheric Refraction: Why standard terrestrial refraction can lift the apparent horizon over water and long sightlines.
U.S. Naval Observatory: Sunrise / Sunset Tables: Primary ephemeris tables for comparing predicted vs observed sunset timing with refraction-aware almanacs.
NOAA Solar Position Equations: Published solar-position equations used by sunset-delay and sun-path tools on this site.

Evidence Card

Claim

Distant shorelines and structures across water should lose their lower sections first as range increases, with more structure returning when the observer gains height, matching curved-surface hidden-height geometry.

Model Used

Spherical Earth hidden-height and horizon-dip geometry using mean radius R = 6371 km, with optional standard terrestrial refraction (7/6 R) or custom k-factor in the schematic overlay lab.

Formula

d_obs=√2Rh + h²

d_beyond=max(d − d_obs, 0)

hidden=d_beyond²2R_eff

sagitta=R_eff (1 − cos(d2R_eff))

clearance=max(targetHeight − hidden, 0)

Last Verified

2026-06-30

Assumptions

The overlay presets are schematic, not copyrighted photographs, but use the same lib math as the Curvature Calculator.
Observer and target heights are geometric heights above the local surface.
Refraction reduces apparent hidden height on a sphere; ad hoc flat-Earth refraction does not close dip, bottom-first recovery, and sunset-delay checks without per-case tuning.

External Sources

What Would Falsify This Page

Repeatable shoreline observations where only the bottom disappeared first, elevation restored hidden structure, and the amounts matched flat-plane predictions instead of spherical hidden-height geometry would count against this page.

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