Editing Great-circle distance (section)

==Radius for spherical Earth==
[[File:WGS84_mean_Earth_radius.svg|thumb|Equatorial (''a''), polar (''b'') and mean Earth radii as defined in the 1984 [[World Geodetic System]] revision. (''Not to scale''.)]]
{{main|Earth radius}}
The [[Geoid|shape of the Earth]] closely resembles a flattened sphere (a [[spheroid]]) with equatorial radius <math>a</math> of 6378.137&nbsp;km; distance <math>b</math> from the center of the spheroid to each pole is 6356.7523142&nbsp;km. When calculating the length of a short north-south line at the equator, the circle that best approximates that line has a radius of <math display="inline">\frac{b^2}{a}</math> (which equals the meridian's [[Conic section#Features|semi-latus rectum]]), or 6335.439&nbsp;km, while the spheroid at the poles is best approximated by a sphere of radius <math display="inline">\frac{a^2}{b}</math>, or 6399.594&nbsp;km, a 1% difference. So long as a spherical Earth is assumed, any single formula for distance on the Earth is only guaranteed correct within 0.5% (though better accuracy is possible if the formula is only intended to apply to a limited area). Using the [[Earth radius#Arithmetic mean radius|mean Earth radius]], <math display="inline">R_1 = \frac{1}{3}(2a + b) \approx 6371.009\text{ km}</math> (for the [[WGS84]] ellipsoid) means that in the limit of small flattening, the mean square [[Relative error#Definitions|relative error]] in the estimates for distance is minimized.<ref name=mccaw32>
{{cite journal |last = McCaw |first = G. T. |title = Long lines on the Earth |journal = Empire Survey Review |volume = 1 |number = 6 |pages = 259&ndash;263 |doi = 10.1179/sre.1932.1.6.259 |year = 1932}}</ref>

For distances smaller than 500 kilometers and outside of the poles, a Euclidean approximation of an ellipsoidal Earth ([[Geographical_distance#FCC's formula|Federal Communications Commission's (FCC)'s formula]]) is both simpler and more accurate (to 0.1%).<ref>{{multiref|1={{cite web |first1=Vladimir|last1=Agafonkin|title=Fast geodesic approximations with Cheap Ruler |url=https://blog.mapbox.com/fast-geodesic-approximations-with-cheap-ruler-106f229ad016 |website=Mapbox |language=en |date=30 August 2017}}|2={{cite web |title=mapbox/cheap-ruler |url=https://github.com/mapbox/cheap-ruler |publisher=Mapbox |date=10 May 2024}}}}</ref>
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