Horizon geometry, calculated.
These pages give the horizon distance and the height of a distant object that sits below it. The same sphere is used for a curvature table, a radar horizon, and a hull-down ship. The English Channel page shows the miss when a gun is laid without compensating for the Earth’s rotation. Three more pages are the altitude of the celestial pole, the turn of a Foucault pendulum, and an Earth radius from two sticks. Eight more are the midnight sun, a great circle, the Earth’s umbra at the Moon, a geostationary dish, sunset from a height, the Schuler period, the plumb line, and a sextant noon sight.
The curvature, radar horizon, and hull-down pages show the sphere and a plane side by side. On a plane these horizon effects are zero. That comparison is stated as geometry, with the assumptions and the sources beside it.
Curvature calculator
Bulge above the chord, drop below the horizontal, and hidden height, with refraction off and on.
Radar horizon
Antenna height and target height. The geometric horizon, and the longer standard-atmosphere horizon.
Hull-down
How much of a ship is below the horizon. Magnification changes the angular size and leaves that fraction as it is.
English Channel
If the gun is laid without compensating for the Earth’s rotation, the round does not fall where it was aimed. Wind is part of that same fall. The line of sight, the published range, and a chosen aim point stay on the page.
Polaris and latitude
The altitude of the north celestial pole is the latitude. Polaris is a small offset from that pole, not a substitute for it.
Foucault pendulum
The swing plane turns at the Earth’s rate times the sine of the latitude. One turn per sidereal day at the poles, and none of that turning at the equator.
Eratosthenes
Two equal sticks, their noon shadows, and the distance between them. The preset is Syene and Alexandria at the June solstice, in metres.
Midnight sun
Latitude and day of the year. Geometric midnight sun, geometric polar night, and a separate line for refraction and the solar semidiameter.
Great circle
Distance and initial bearing on the mean sphere, and the rhumb line beside them. Two places, typed or clicked.
Lunar eclipse
The Earth’s umbra at the Moon, and the Moon’s angular size beside it. Geometry of the cone, not the date of the next eclipse.
Satellite dish
Azimuth and elevation of a geostationary longitude. Below the horizon is said plainly.
Sunset from a height
The dip of the horizon, and how much later the Sun sets. Refraction off and on, labelled apart.
Inertial navigation
The Schuler period, the walk of an uncorrected bias, and Earth rate times the sine of the latitude.
Plumb line
The angle between the ellipsoid normal and the gravity vector. Astronomical latitude and geodetic latitude differ by that deflection.
Sextant
A noon sight for latitude and longitude, and a line of position from a second altitude. No ephemeris.
One geometry module
Earth radius, refraction, horizon distance, and hidden height are implemented once, in assets/js/earth-geometry.js. The curvature, radar horizon, and hull-down pages import that file. The Eratosthenes page imports the same mean radius when it places the stick result next to the modern Earth. The great-circle distance, the lunar umbra, the geostationary dish, sunset from a height, and the Schuler period use that same radius. They do not keep another copy. The sextant dip is that same horizon distance divided by the radius. The English Channel page keeps its own 6,371 km sphere for distance and bearing, with no refraction. The pages are a PHP shell. The arithmetic runs in your browser. There is no database.
The radius in use is the GRS80 mean radius, …, from a = … and 1/f = ….