Hull-down

The globe and a flat Earth

On the sphere, once the ship is past the horizon of the eye, the sea between them hides the lower part. The hidden fraction is a height on the ship. Magnification multiplies angular size, including the size of the part that is already above the horizon. It does not move the cut, and it does not bring the hull back.

A flat-Earth prediction for this page is the plane column and the right-hand drawing. On a plane the waterline stays geometrically visible at every finite distance. The ship subtends a smaller angle as it gets farther away, and magnification brings that angle back. There is no hidden band for magnification to restore.

Both drawings use the same magnification, framed on the ship. Only the sphere drawing raises the waterline to the horizon on the hull. The angular sizes under the form are the small-angle estimate, height divided by surface distance, before and after magnification.

Assumptions

• Hidden height is the same function as on the curvature page, on the GRS80 mean radius ….
• Refraction off uses R. Refraction on uses R / (1 − κ), starting at Normal, 0.14, the same setting as the curvature page. The drawing follows the control. Both sphere columns stay filled.
• The silhouette is a schematic. Its length is not a length of any ship. Only the two heights are inputs. “Hull” means waterline to the deck height you enter. “Height overall” means waterline to the masthead.
• Hull-down, in the wording of the summary, means the deck is at or below the horizon while some of the mast is still above it.
• The angular sizes ignore the difference between surface distance and slant range, and they are not a ray-traced image. They are there to show that magnification scales the angle and leaves the hidden fraction alone.
• No waves or mirage. Either one can move a real hull-down appearance away from this geometric cut.

Formulas

The concealed height up from the waterline is the shared hidden height at the entered distance and eye height. The fraction of the ship below the horizon is that height divided by the height overall, and it is not more than the whole ship.

Small-angle angular height, in arcminutes, is (metres of height / metres of distance) × (180 / π) × 60. Magnification multiplies that angle.

Sources

• Moritz, H. (1980). Geodetic Reference System 1980. Bulletin Géodésique 54(3), 395–405. The radius is (2a + b) / 3 from a = 6,378,137 m and 1/f = 298.257222101.
• Young, A. T. Dip of the horizon. A single refraction coefficient is only a stand-in for the temperature gradient. aty.sdsu.edu/explain/atmos_refr/dip.html
• The hidden-height construction is the one stated on the curvature page: beyond the horizon, concealed height equals the drop of the remaining distance. That function is shared, not rewritten here.