Geographical distance

Geographical distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographic coordinates in terms of latitude and longitude.

An abstraction

Calculating the distance between geographic 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.[1] Common abstractions for the surface between two geographic points are:

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, $D,\,\!$ is calculated between two points, $P_1\,\!$ and $P_2\,\!$. The geographical coordinates of the two points, as (latitude, longitude) pairs, are $(\phi_1,\lambda_1)\,\!$ and $(\phi_2,\lambda_2),\,\!$ respectively. Which of the two points is designated as $P_1\,\!$ 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:

\begin{align} \Delta\phi&=\phi_2-\phi_1;\\ \Delta\lambda&=\lambda_2-\lambda_1. \end{align} \,\!

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:

$\phi_m=\frac{\phi_1+\phi_2}{2}.\,\!$

Colatitude is labeled and calculated as follows:

$\theta=\frac{\pi}{2}-\phi;\,\!$
For latitudes expressed in degrees:
$\theta=90^\circ-\phi.\,\!$

Unless specified otherwise, the radius of the earth for the calculations below is:

$R\,\!$ = 6,371.009 kilometers = 3,958.761 statute miles = 3,440.069 nautical miles.

$D_\,\!$ = 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.

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 global latitude and longitude coordinates onto a plane is the realm of cartography.

The formulae presented in this section provide varying degrees of accuracy.

Pythagorean formula with parallel meridians

$D=R\sqrt{(\Delta\phi)^2+(\Delta\lambda)^2};{\color{white}\frac{\big|}{.}}\,\!$
where $\Delta\phi\,\!$ and $\Delta\lambda\,\!$ are in radians.
To change latitude or longitude to radians use following formulae:
• latitude in radians = (latitude in degrees)*PI/180
• longitude in radians = (longitude in degrees)*PI/180
where PI is Ludolph's number (3.14159...)
Maximum error if the distance is less than about 20 km (12 mi):[2]
• 30 meters (100 ft) error for latitudes less than 70 degrees;
• 20 meters ( 66 ft) error for latitudes less than 50 degrees;
• 9 meters ( 30 ft) error for latitudes less than 30 degrees.

Pythagorean formula with converging meridians

Spherical Earth projected to a plane

The accuracy of the previous formula may be improved by accounting for the variation in distance between meridians with latitude:

$D=R\sqrt{(\Delta\phi)^2+(\cos(\phi_m)\Delta\lambda)^2};{\color{white}\frac{\big|}{.}}\,\!$
where:
$\Delta\phi\,\!$ and $\Delta\lambda\,\!$ are in radians;
$\phi_m\,\!$ must be in units compatible with the method used for determining $\cos(\phi_m).\,\!$

Ellipsoidal Earth projected to a plane

The FCC prescribes essentially the following formulae in 47 CFR 73.208 for distances not exceeding 475 km /295 miles:[3]

$D=\sqrt{(K_1\Delta\phi)^2+(K_2\Delta\lambda)^2};{\color{white}\frac{\big|}{.}}\,\!$
where
$D\,\!$ = Distance in kilometers;
$\Delta\phi\,\!$ and $\Delta\lambda\,\!$ are in degrees;
$\phi_m\,\!$ must be in units compatible with the method used for determining $\cos(\phi_m);\,\!$
\begin{align} K_1&=111.13209-0.56605\cos(2\phi_m)+0.00120\cos(4\phi_m);\\ K_2&=111.41513\cos(\phi_m)-0.09455\cos(3\phi_m)+0.00012\cos(5\phi_m).\end{align}\,\!
It may be interesting to note that:
$K_1=M\frac{\pi}{180}\,\!$ = kilometers per degree of latitude difference;
$K_2=\cos(\phi_m)N\frac{\pi}{180}\,\!$ = kilometers per degree of longitude difference;
where $M\,\!$ and $N\,\!$ are the meridional and its perpendicular, or "normal", radii of curvature (the expressions in the FCC formula are derived from the binomial series expansion form of $M\,\!$ and $N\,\!$, set to the Clarke 1866 reference ellipsoid).

Polar coordinate flat-Earth formula

$D=R\sqrt{\theta^2_1\;\boldsymbol{+}\;\theta^2_2\;\mathbf{-}\;2\theta_1\theta_2\cos(\Delta\lambda)};{\color{white}\frac{\big|}{.}}\,\!$
where the colatitude values are in radians. For a latitude measured in degrees, the colatitude in radians may be calculated as follows: $\theta=\frac{\pi}{180}(90^\circ-\phi).\,\!$

Spherical-surface formulae

A spherical approximation for the surface of the earth may be useful over great distances.

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.

Great-circle distance

The great-circle distance article presents formulae for calculating the exact distance along a great-circle. The great-circle distance article includes a worked example for calculating distances by this method.

Ellipsoidal-surface formulae

An ellipsoidal approximation for the surface of the earth may be useful over great distances. The shortest distance along the surface of an ellipsoid between two points on the surface is along the geodesic.

Vincenty's formulae

The Vincenty's formulae article presents an algorithm for calculating the geodesic distance between two points on an ellipsoid. The results are accurate to about 0.5 mm; however, the algorithm fails to converge for points that are nearly antipodal.