- Great-circle distance
The

**great-circle distance**is the shortestdistance between any two points on the surface of asphere measured along a path on the surface of the sphere (as opposed to going through the sphere's interior). Becausespherical geometry is rather different from ordinaryEuclidean geometry , the equations for distance take on a different form. The distance between two points inEuclidean space is the length of a straight line from one point to the other. On the sphere, however, there are no straight lines. Innon-Euclidean geometry , straight lines are replaced withGeodesic s. Geodesics on the sphere are the "great circle s" (circles on the sphere whose centers are coincident with the center of the sphere).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. A great circle endowed with such a distance is the

Riemannian circle .Between two points which are directly opposite each other, called "

antipodal point s", there are infinitely many great circles, but all great circle arcs between antipodal points have the same length, i.e. half thecircumference of the circle, or $pi\; r$, where "r" is theradius of thesphere .Because the Earth is approximately spherical (see

Earth radius ), 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 innavigation .**The geographical formula**Let $phi\_s,lambda\_s;\; phi\_f,lambda\_f;!$ be the geographical

latitude andlongitude of two points (a base "standpoint" and the destination "forepoint"), respectively, $Deltalambda;!$ the longitude difference and $Deltawidehat\{sigma\};!$ the (spherical) angular difference/distance, orcentral angle , which can be constituted from thespherical law of cosines ::$\{color\{white\}BigDeltawidehat\{sigma\}=arccosig(cosphi\_scosphi\_fcosDeltalambda+sinphi\_ssinphi\_fig).;!$

The distance "d", i.e. the

arc length , for a sphere of radius "r" and $Deltawidehat\{sigma\}!$ given inradian s, is then::$d\; =\; r\; Deltawidehat\{sigma\}.$

This arccosine formula above can have large

rounding error s for the common case where the distance is small, however, so it is not normally used. Instead, an equation known historically as thehaversine formula was preferred, which is much more accurate for small distances: [*R.W. Sinnott, "Virtues of the Haversine", Sky and Telescope, vol. 68, no. 2, 1984, p. 159*]:$\{color\{white\}frac\{igg\{Deltawidehat\{sigma\}=arctanleft(frac\{sqrt\{left(cosphi\_fsinDeltalambda\; ight)^2+left(cosphi\_ssinphi\_f-sinphi\_scosphi\_fcosDeltalambda\; ight)^2\{sinphi\_ssinphi\_f+cosphi\_scosphi\_fcosDeltalambda\}\; ight);;!$

(When programming a computer, one should use the

function rather than the ordinary arctangent function (atan2 ()`atan()`

), in order to simplify handling of the case where the denominator is zero.)If "r" is the great-circle radius of the sphere, then the great-circle distance is $r,Deltawidehat\{sigma\};!$.

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

**pherical distance on the Earth**The shape of the Earth more closely resembles a flattened

spheroid with extreme values for the radius of arc, or "arcradius"—theradius of curvature , of 6335.437 km at the equator (vertically) and 6399.592 km at the poles, and having an of 6372.795 km (3438.461nautical mile s). (Note that "arcradius" or "radius of curvature" is "not" the distance from the center of the earth to the surface. The distance from the center to the surface is smaller at the poles than at the equator; the arcradius is larger at the poles than at the equator.) 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 andlongitude 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'First, convert these coordinates to decimal degrees (Sign × (Deg + (Min + Sec / 60) / 60)) and

radian s (× π / 180) before you can use them effectively in a formula. After conversion, the coordinates become:* BNA: $phi\_s=\; 36.12^circapprox\; 0.6304mbox\{\; rad\};;;lambda\_s=-86.67^circapprox\; -1.5127mbox\{\; rad\};;!$

* LAX: $phi\_f=\; 33.94^circapprox\; 0.5924mbox\{\; rad\};;;lambda\_f=-118.40^circapprox\; -2.0665mbox\{\; rad\};;!$Using these values in the angular difference/distance equation:

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

Thus the distance between LAX and BNA is about 2887 km or 1794 miles (× 0.62137) or 1558 nautical miles (× 0.539553).

**ee also***

Air navigation

*Flight planning

*Spherical geometry

*Spherical trigonometry

*Great-circle navigation

*Central angle

*Haversine formula

*Geodesy

*SIGI

*Vincenty's formulae **References****External links*** [

*http://www.elisanet.fi/master.navigator/xCalc/ Earth navigation calculator for Windows (free)*]

* [*http://mathworld.wolfram.com/GreatCircle.html GreatCircle*] at MathWorld

* U. S. Census Bureau Geographic Information Systems FAQ, [*http://www.census.gov/cgi-bin/geo/gisfaq?Q5.1 What is the best way to calculate the distance between 2 points?*] (broken link; content has been [*http://www.movable-type.co.uk/scripts/GIS-FAQ-5.1.html mirrored here*] ) As of 30 May 2007, there was an unbroken link to an updated version of this article, which deals with the same topic as this wiki article, at http://www.usenet-replayer.com/faq/comp.infosystems.gis.html.

* [*http://www.infoplease.com/atlas/calculate-distance.html Global Distance Calculator*] Tool finds distances between over 7.5 million place names, with dynamic text suggestion while typing to help find locations.

* [*http://www.daftlogic.com/projects-google-maps-distance-calculator.htm Google Maps Distance Calculator*] Tool to measure distance on a map using Great-circle distance.

* [*http://www.mapcrow.info Distance between countries*] Shows estimated distance between country mid-points.

* [*http://www.milecalc.com Great Circle Distance Airline Mileage Calculator*] A tool for computing frequent flyer miles using great circle distances.

* Ed Williams's [*http://williams.best.vwh.net/avform.htm "Aviation Formulary"*] :* [*http://williams.best.vwh.net/gccalc.htm Distance calculator*]

* [*http://www.airport-accommodation.co.uk/distance-calculator.php Great Circle Distance Calculator*] Distances generated by site users showing maps and travel times

* [*http://www.movable-type.co.uk/scripts/LatLong.html Haversine formula in JavaScript*] Haversine and other formulae for calculating distances, bearings, etc.

* [*http://www.freemaptools.com/how-far-is-it-between.htm How Far is it Between*] Find the distance between two named points and see the result on a map.

* [*http://www.acscdg.com/ Great Circle Distance*] Graphical tool for drawing great circles over maps. Also shows distance and azimuth in a table.

* [*http://gc.kls2.com/ Great Circle Mapper*] Interactive tool for plotting great circle routes

* [*http://aviador.es/ Meteo·Mobile Route between airports*] Interactive tool for plotting aviation routes between two airports

* [*http://www.apsalin.com/great-circle-distance-bearing.aspx Great Circle Distance and Bearing Calculator*] for radio/TV broadcast industry

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