Geographical distance


Geographical distance or geodetic distance is the distance measured along the surface of the Earth, or the shortest arch length.
The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. This distance is an element in solving the second (inverse) geodetic problem.

Introduction

Calculating the distance between geographical coordinates is based on some level of abstraction; it does not provide an exact distance, which is unattainable if one attempted to account for every irregularity in the surface of the Earth. Common abstractions for the surface between two geographic points are:
  • Flat surface;
  • Spherical surface;
  • Ellipsoidal surface.
All abstractions above ignore changes in elevation. Calculation of distances which account for changes in elevation relative to the idealized surface are not discussed in this article.

Classification of Formulae based on Approximation

  • short-range approximations: Flat surface, Gauss-mid-latitude;
  • * Bowring's method for short lines improved by Karney using reduces latitude and mid-latitude;
  • long-range approximations; on the closed hemisphere
  • * -order approximation method: Spherical surface
  • * higher-order approximations based on Ellipsoid: : Andoyer; Andoyer-Lambert, : Andoyer-Lambert-Thomas, : Vincenty, : Karney
The theoretical estimations of error are added in above and is the flattening of the Earth.

Nomenclature

Arc distance, is the minimum distance along the surface of sphere/ellipsoid calculated between two points, and. Whereas, the [|tunnel distance], or chord length,, is measured along Cartesian straight line. The geographical coordinates of the two points, as pairs, are and respectively. Which of the two points is designated as is not important for the calculation of distance.
Latitude and longitude coordinates on maps are usually expressed in degrees. In the given forms of the formulae below, one or more values must be expressed in the specified units to obtain the correct result. Where geographic coordinates are used as the argument of a trigonometric function, the values may be expressed in any angular units compatible with the method used to determine the value of the trigonometric function. Many electronic calculators allow calculations of trigonometric functions in either degrees or radians. The calculator mode must be compatible with the units used for geometric coordinates.
Differences in latitude and longitude are labeled and calculated as follows:
It is not important whether the result is positive or negative when used in the formulae below.
"Mid-latitude" is labeled and calculated as follows:
Unless specified otherwise, the radius of the Earth for the calculations below is:
= Distance between the two points, as measured along the surface of the Earth and in the same units as the value used for radius unless specified otherwise.

Singularities and discontinuity of latitude/longitude

The approximation of sinusoidal functions of, appearing in some flat-surface formulae below, may induce singularity and discontinuity. It may also degrade the accuracy in the case of higher latitude.
Longitude has singularities at the Poles and a discontinuity at the ±180° meridian. Also, planar projections of the circles of constant latitude are highly curved near the Poles. Hence, the above equations for delta latitude/longitude and mid-latitude may not give the expected answer for positions near the Poles or the ±180° meridian. Consider e.g. the value of when and are on either side of the ±180° meridian, or the value of for the two positions and.
If a calculation based on latitude/longitude should be valid for all Earth positions, it should be verified that the discontinuity and the Poles are handled correctly. Another solution is to use n-vector instead of latitude/longitude, since this representation does not have discontinuities or singularities.

Flat-surface approximation formulae for very short distance

A planar approximation for the surface of the Earth may be useful over very small distances. It approximates the arc length,, to the tunnel distance,, or omits the conversion between arc and chord lengths shown below.
The shortest distance between two points in plane is a Cartesian straight line. The Pythagorean theorem is used to calculate the distance between points in a plane.
Even over short distances, the accuracy of geographic distance calculations which assume a flat Earth depend on the method by which the latitude and longitude coordinates have been projected onto the plane. The projection of latitude and longitude coordinates onto a plane is the realm of cartography.
The formulae presented in this section provide varying degrees of accuracy.

Spherical Earth approximation formulae

The tunnel distance,, is calculated on Spherical Earth. This formula takes into account the variation in distance between meridians with latitude, assuming :
The square root appearing above can be eliminated for such applications as ordering locations by distance in a database query. On the other hand, some methods for computing nearest neighbors, such as the vantage-point tree, require that the distance metric obey the triangle inequality, in which case the square root must be retained.

In the case of medium or low latitude

Although not being universal, the above is furthermore simplified by approximating sinusoidal functions of, justified except for high latitude:

Ellipsoidal Earth approximation formulae

The above formula is extended for ellipsoidal Earth:
where and are the meridional and its perpendicular, or "normal", radii of curvature of Earth.
It is derived by the approximation of in the square root.
This approximation can be viewed simply as the 3D Cartesian chord distance between two points on the ellipsoid, and equivalently as a chordal simplification of the Gauss mid-latitude method. Although we have not found this explicit formula in classical sources, the Gauss mid-latitude method itself is described in Rapp.

In the case of medium or low latitude

Although not being universal, the above is furthermore simplified by approximating sinusoidal functions of, justified except for high latitude as above:

FCC's formula

The Federal Communications Commission prescribes the following formulae for distances not exceeding :

Spherical-surface formulae

If one is willing to accept a possible error of 0.5%, one can use formulas of spherical trigonometry on the sphere that best approximates the surface of the Earth.
The shortest distance along the surface of a sphere between two points on the surface is along the great-circle which contains the two points.
The great-circle distance article gives the formula for calculating the shortest arch length on a sphere about the size of the Earth. That article includes an example of the calculation. For example, from tunnel distance,
For short distances,

Tunnel distance

A tunnel between points on Earth is defined by a Cartesian line through three-dimensional space between the points of interest.
The tunnel distance is the great-circle chord length and may be calculated as follows for the corresponding unit sphere:

Ellipsoidal-surface formulae

An ellipsoid approximates the surface of the Earth much better than a
sphere or a flat surface does. The shortest distance along the surface
of an ellipsoid between two points on the surface is along the
geodesic. Geodesics follow more complicated paths than great
circles and in particular, they usually don't return to their starting
positions after one circuit of the Earth. This is illustrated in the
figure on the right where f is taken to be 1/50 to accentuate the
effect. Finding the geodesic between two points on the Earth, the
so-called inverse geodetic problem, was the focus of many
mathematicians and geodesists over the course of the 18th and 19th
centuries with major contributions by
Clairaut,
Legendre,
Bessel,
and Helmert English translation of . .
Rapp
provides a good summary of this work.
Methods for computing the geodesic distance are widely available in
geographical information systems, software libraries, standalone
utilities, and online tools. The most widely used algorithm is by
Vincenty,
who uses a series which is accurate to third order in the flattening of
the ellipsoid, i.e., about 0.5 mm; however, the algorithm fails to
converge for points that are nearly antipodal. This defect is cured in the
algorithm given by
Karney,
who employs series which are accurate to sixth order in the flattening.
This results in an algorithm which is accurate to full double precision
and which converges for arbitrary pairs of points on the Earth. This
algorithm is implemented in GeographicLib.
The exact methods above are feasible when carrying out calculations on a
computer. They are intended to give millimeter accuracy on lines of any
length; one can use simpler formulas if one doesn't need millimeter
accuracy, or if one does need millimeter accuracy but the line is short.
The short-line methods have been studied by several researchers.
Rapp, Chap. 6, describes the Puissant method,
the Gauss mid-latitude method, and the Bowring method. Karl Hubeny got the expanded series of Gauss mid-latitude one represented as the correction to flat-surface one.

Andoyer-Lambert formula for long lines

Historically, the long-line formulae were derived in the form of expansion series with regard to flattening.
Andoyer-Lambert formulae
use the first-order correction and reduced latitude,, for better accuracy. They give accuracy on the order of 10 meters over thousands of kilometers.
First convert the latitudes, of the two points to reduced latitudes, .
Then calculate the central angle in radians between two points and on a sphere using the Great-circle distance method, with longitudes and being the same on the sphere as on the spheroid.
where is the equatorial radius of the chosen spheroid.
On the GRS 80 spheroid Lambert's formula is off by

Gauss mid-latitude method for short lines

It has the similar form of the arc length converted from tunnel distance. Detailed formulas are given by Rapp, §6.4. It is consistent with the above-mentioned flat-surface formulae apparently.

Bowring's method for short lines

Bowring maps the points to a sphere of radius R′, with latitude and longitude represented as φ′ and λ′. Define
where the second eccentricity squared is
The spherical radius is
The spherical coordinates are given by
where,,
,. The resulting problem on the sphere may be solved using the techniques for great-circle navigation to give approximations for the spheroidal distance and bearing. Detailed formulas are given by Rapp §6.5 and Bowring. The use of mid-latitude,, improves the accuracy, shown by Karney.

Altitude correction

The variation in altitude from the topographical or ground level down to the sphere's or ellipsoid's surface, also changes the scale of distance measurements.
The slant distance s between two points can be reduced to the arc length on the ellipsoid surface S as:
where R is evaluated from Earth's azimuthal radius of curvature and h are ellipsoidal heights are each point.
The first term on the right-hand side of the equation accounts for the mean elevation and the second term for the inclination.
A further reduction of the above Earth normal section length to the ellipsoidal geodesic length is often negligible.