Geographical distance or geodetic distance is the distance measured along the surface of the Earth. 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.
Nomenclature
Distance, is calculated between two points, and . The geographical coordinates of the two points, as (latitude, longitude) 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.
"Mean latitude" is labeled and calculated as follows:
Colatitude is labeled and calculated as follows: For latitudes expressed in radians:
For latitudes expressed in degrees:
Unless specified otherwise, the radius of the Earth for the calculations below is: = 6,371.009 kilometers = 3,958.761 statute miles = 3,440.069 nautical miles.
= 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
Longitude has singularities at the Poles (longitude is undefined) 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 mean latitude () may not give the expected answer for positions near the Poles or the ±180° meridian. Consider e.g. the value of ("east displacement") when and are on either side of the ±180° meridian, or the value of ("mean latitude") for the two positions (=89°, =45°) and (=89°, =−135°).
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 formulae
A planar approximation for the surface of the Earth may be useful over small distances. The accuracy of distance calculations using this approximation become increasingly inaccurate as:
The separation between the points becomes greater; A point becomes closer to a geographic pole.
The shortest distance between two points in plane is a 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 projected to a plane
This formula takes into account the variation in distance between meridians with latitude:
where: and are in radians; must be in units compatible with the method used for determining