Geographical distance

Introduction

Calculating the distance between geographical coordinates involves abstraction, as an exact distance accounting for every surface irregularity is unattainable. Common abstractions for the surface between two points include a flat surface, a spherical surface, and an ellipsoidal surface; all ignore changes in elevation. Short-range approximations include flat-surface methods, the Gauss-mid-latitude method, and Bowring's method (1981) improved by Karney using reduced latitude and mid-latitude.

Long-range approximations range from a spherical Earth model to higher-order ellipsoidal methods such as those by Andoyer (1932), Andoyer-Lambert (1942), Andoyer-Lambert-Thomas (1970), Vincenty (1975), and Karney (2011). Theoretical error estimates are added in these methods, with f representing the flattening of the Earth. Arc distance is the minimum distance along the surface of a sphere or ellipsoid, while tunnel distance (chord length) is measured along a Cartesian straight line.

The two points are given as (latitude, longitude) pairs, and which point is designated as first is not important. Latitude and longitude are usually expressed in degrees, but trigonometric functions may use any angular units compatible with the calculator. Differences in latitude and longitude are labeled and calculated, and the mid-latitude is also defined. Unless specified otherwise, the Earth's radius is taken as 6,371.009 kilometers.

Flat-surface approximation formulae for very short distance

A planar approximation for the Earth's surface may be useful over very small distances, approximating arc length to tunnel distance or omitting the conversion between them. The shortest distance in a plane is a Cartesian straight line calculated using the Pythagorean theorem. Accuracy depends on how latitude and longitude coordinates are projected onto the plane, which is the realm of cartography. The tunnel distance on a spherical Earth accounts for the variation in distance between meridians with latitude.

The square root in the formula can be eliminated for ordering locations by distance in a database query, but methods like vantage-point trees require the square root to obey the triangle inequality. A further simplification approximates sinusoidal functions of latitude, justified except at high latitudes. An ellipsoidal extension uses the meridional and normal radii of curvature, derived by approximating the cosine of latitude in the square root.

This approximation is equivalent to a chordal simplification of the Gauss mid-latitude method. The Federal Communications Commission prescribes formulae for distances not exceeding 475 km, using distances in kilometers per arc degree derived from Earth's radii of curvature.

Ellipsoidal-surface formulae

An ellipsoid approximates the Earth's surface much better than a sphere or flat surface. The shortest distance along an ellipsoid is the geodesic, which follows more complicated paths than great circles and usually does not return to its starting position after one circuit. Finding the geodesic between two points, the inverse geodetic problem, was studied by Clairaut, Legendre, Bessel, and Helmert. The most widely used algorithm is by Vincenty, accurate to about 0.5 mm but failing to converge for nearly antipodal points.

Karney's algorithm, accurate to full double precision and converging for arbitrary pairs, uses series to sixth order in flattening and is implemented in GeographicLib. For short lines, methods such as Puissant, Gauss mid-latitude, and Bowring have been studied. Historically, long-line formulae were derived as expansion series in flattening.

Did You Know?

• The variation in altitude from topographical or ground level down to the sphere's or ellipsoid's surface changes the scale of distance measurements.
• A further reduction of the Earth normal section length to the ellipsoidal geodesic length is often negligible.