Curvature calculator

The globe and a flat Earth

On the sphere, the drop and the bulge grow with distance. Raising the eye moves the horizon outward and reduces the hidden height. With the eye on the surface, the hidden height and the drop are the same number.

A flat-Earth prediction for this page is the plane column. On a plane, a horizontal line stays at one height above the surface, so the bulge, the drop, and the hidden height are zero at every finite distance. Distance still makes an object look smaller. It does not hide a lower slice of it.

The preset “1 statute mile” puts the eye on the surface and turns refraction off. Over one international statute mile the geometric drop is 8.00 inches and the bulge is 2.00 inches. Those are the well-known short-distance figures d² / (2R) and d² / (8R). The page uses the exact expressions below; at one mile the two agree to a hundredth of an inch.

Assumptions

• The surface is a smooth sphere of the GRS80 mean radius, …. There is no terrain.
• Distance is measured along the surface. It is not the chord and not the slant range.
• Eye height is height above that sphere.
• Refraction off uses the radius R. Refraction on uses R / (1 − κ). The page starts at Normal, 0.14, a round figure near one seventh. Surveyors have long used values around 0.13 to 0.17. A sunny afternoon can make the value negative. The number field accepts that.
• The solid Earth is not larger when κ is larger. The larger radius is the usual stand-in that gives a curved ray the same shape as a straight ray over a bigger sphere.
• Hidden height reuses the drop. From the eye, the line of sight touches the sphere at the horizon. Beyond that point it is the horizontal tangent, so the concealed height is the drop for the remaining distance.
• No waves, spray, or mirage. A wave in front of the line of sight can hide more than the curve. A strong inversion can hide less.
• The drawing exaggerates vertical distances when it says so. The cards, not the drawing, are the result. The drawing follows the “Drawing” control; both sphere columns stay on the page.

Formulas

The defining constants are a = … and 1/f = …, so the polar radius b is …. The mean radius is (2a + b) / 3.

Horizon distance along the surface, from height h: R · atan2(√(h (2R + h)), R).

With θ = d / R, the drop is R · (1 − cos θ) / cos θ. The bulge is R · (1 − cos(θ / 2)).

Let s be the horizon distance for the eye. If d ≤ s, the hidden height is zero. Otherwise it is the drop at distance d − s, on the same radius.

Sources

• Moritz, H. (1980). Geodetic Reference System 1980. Bulletin Géodésique 54(3), 395–405. Defining constants a = 6,378,137 m and 1/f = 298.257222101.
• Young, A. T. Dip of the horizon. The coefficient κ depends on the temperature gradient; under a standard atmosphere it is about 1/6 to 1/7, and values near 0.13 have long been used in surveying. aty.sdsu.edu/explain/atmos_refr/dip.html
• The international statute mile of 1,609.344 m and the inch of 25.4 mm are the units of the one-mile check. The international nautical mile used in the secondary distance figures is 1,852 m.