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Between any two points on a sphere which are not Directly Opposite Each Other , there is a unique great circle. The two points separate the great circle into two arcs. The length of the shorter arc is the great-circle distance between the points. Between two points which are directly opposite each other (called '' Antipodal Point s'') there infinitely many great circles, but all have the same length, equal to half the Circumference of the circle, or \pi r, where ''r'' is the radius of the sphere.

Because the Earth is approximately spherical (see Spherical Earth ), the equations for great-circle distance are important for finding the shortest distance between points on the surface of the Earth, and so have important applications in Navigation .


THE FORMULA


Let \phi_1,\lambda_1;\ \phi_2,\lambda_2\,\! be the Latitude and Longitude of two points, respectively, \Delta\lambda\,\! the longitude difference and \Delta\sigma\,\! the angular difference/distance can be determined from the Spherical Law Of Cosines as:

:\Delta\sigma=\arccos\left\{\sin\phi_1\sin\phi_2+\cos\phi_1\cos\phi_2\cos\Delta\lambda ight\}

This arccosine form can have large Rounding Error s for the common case where the distance is small, however, so it is not normally used. Instead, a simpler equation known historically as the Haversine Formula was preferred, which is much more accurate for small distances:

:\Delta\sigma
=2\arcsin\left\{\sqrt{\sin^2\left( rac{\phi_2-\phi_1}{2} ight)+\cos{\phi_1}\cos{\phi_2}\sin^2\left( rac{\Delta\lambda}{2} ight)} ight\}.\!

(Historically, the use of this formula was simplified by the availability of tables for the s (on opposite ends of the sphere). A more complicated formula that is accurate for all distances is:

:\Delta\sigma=\arctan\left\{ rac{\sqrt{\left[\cos\phi_2\sin\Delta\lambda ight]^2+\left[\cos\phi_1\sin\phi_2-\sin\phi_1\cos\phi_2\cos\Delta\lambda ight]^2}}{\sin\phi_1\sin\phi_2+\cos\phi_1\cos\phi_2\cos\Delta\lambda} ight\}

(On a computer, one should use Atan2 rather than the ordinary arctangent function, in order to correctly handle the case where the denominator is zero.)

If ''r'' is the great-circle radius of the sphere, then the great-circle distance is r\,\Delta\sigma\,\!.

Note: above, accuracy refers to rounding errors only; all formulas themselves are exact (for a sphere).


SPHERICAL DISTANCE ON THE EARTH


The of 6372.795 km.

Using a sphere with a radius of 6372.795 km thus results in an error of up to about 0.5%.


A WORKED EXAMPLE


In order to use this formula for anything practical you will need two sets of coordinates. For example, the Latitude and Longitude of two airports:

  • Nashville International Airport (BNA) in Nashville, TN, USA: N 36°7.2', W 86°40.2

  • Los Angeles International Airport (LAX) in Los Angeles, CA, USA: N 33°56.4', W 118°24.0'


You will first have to convert these coordinates to Whole Degrees and Radian s before you can use them effectively in a formula. After conversion, the coordinates become:

  • BNA:\phi_1= 36.12^\circ\approx 0.6304\mbox{ rad},\ \ \lambda_1=-86.67^\circ\approx -1.5127\mbox{ rad};\,\!

  • LAX:\phi_2= 33.94^\circ\approx 0.5924\mbox{ rad},\ \ \lambda_2=-118.40^\circ\approx -2.0665\mbox{ rad};\,\!


Using these values in the angular distance equation:

::r\,\Delta\sigma\approx 6372.795 imes0.45306 \approx 2887.259\mbox{ km}.\,\!

Thus the distance between LAX and BNA is about 2887 km or 1794 miles.


SPHERICAL COORDINATES


In the Spherical Coordinates used by mathematicians and physicists, usually when considering other spheres than the Earth's surface, the great-circle distance is found as follows. If arphi\,\! is the azimuthal angle and heta\,\! the colatitude, then the spherical distance is given by

:r\,\Delta\sigma
=2r\arcsin\left\{\sqrt{\sin^2\left( rac{ heta_2- heta_1}{2} ight)
+\sin{ heta_1}\sin{ heta_2}\sin^2\left( rac{\Delta arphi}{2} ight)} ight\}.\!
::=r\arctan\left\{ rac{\sqrt{\left[\sin heta_2\sin\Delta arphi ight]^2+\left[\sin heta_1\cos heta_2-\cos heta_1\sin heta_2\cos\Delta arphi ight]^2}}{\cos heta_1\cos heta_2+\sin heta_1\sin heta_2\cos\Delta arphi} ight\};\,\!

  :<math>r\,\Delta\sigma r heta_2- heta_1\,\!</math>