Editing Geographic coordinate system (section)

==Length of a degree==
{{Main|Length of a degree of latitude|Length of a degree of longitude}}
{{See also|Arc length#Great circles on Earth}}

On the [[Geodetic Reference System 1980|GRS{{nbsp}}80]] or [[World Geodetic System#WGS84|WGS{{nbsp}}84]] spheroid at [[sea level]] at the Equator, one latitudinal second measures 30.715 [[metre|m]], one latitudinal minute is 1843 m and one latitudinal degree is 110.6&nbsp;km. The circles of longitude, meridians, meet at the geographical poles, with the west–east width of a second naturally decreasing as latitude increases. On the [[Equator]] at sea level, one longitudinal second measures 30.92&nbsp;m, a longitudinal minute is 1855&nbsp;m and a longitudinal degree is 111.3&nbsp;km. At 30° a longitudinal second is 26.76&nbsp;m, at Greenwich (51°28′38″N) 19.22&nbsp;m, and at 60° it is 15.42 m.

On the WGS{{nbsp}}84 spheroid, the length in meters of a degree of latitude at latitude {{mvar|ϕ}} (that is, the number of meters you would have to travel along a north–south line to move 1 degree in latitude, when at latitude {{mvar|ϕ}}), is about

{{block indent|1=
<math>111132.92 - 559.82\, \cos 2\phi + 1.175\, \cos 4\phi - 0.0023\, \cos 6\phi</math><ref name=GISS>[http://gis.stackexchange.com/questions/75528/length-of-a-degree-where-do-the-terms-in-this-formula-come-from] {{Webarchive|url=https://web.archive.org/web/20160629203521/http://gis.stackexchange.com/questions/75528/length-of-a-degree-where-do-the-terms-in-this-formula-come-from |date=29 June 2016 }} Geographic Information Systems – Stackexchange</ref>
}}

The returned measure of meters per degree latitude varies continuously with latitude.

Similarly, the length in meters of a degree of longitude can be calculated as

{{block indent|1=
<math>111412.84\, \cos \phi - 93.5\, \cos 3\phi + 0.118\, \cos 5\phi</math><ref name=GISS/>
}}

(Those coefficients can be improved, but as they stand the distance they give is correct within a centimeter.)

The formulae both return units of meters per degree.

An alternative method to estimate the length of a longitudinal degree at latitude <math>\phi</math> is to assume a spherical Earth (to get the width per minute and second, divide by 60 and 3600, respectively):

{{block indent|1=
<math> \frac{\pi}{180}M_r\cos \phi \!</math>
}}

where [[Earth radius#Meridional Earth radius|Earth's average meridional radius]] <math>\textstyle{M_r}\,\!</math> is {{nowrap|6,367,449 m}}. Since the Earth is an [[Spheroid#Oblate spheroids|oblate spheroid]], not spherical, that result can be off by several tenths of a percent; a better approximation of a longitudinal degree at latitude <math>\phi</math> is

{{block indent|1=
<math>\frac{\pi}{180}a \cos \beta \,\!</math>
}}

where Earth's equatorial radius <math>a</math> equals 6,378,137 m and <math>\textstyle{\tan \beta = \frac{b}{a}\tan\phi}\,\!</math>; for the GRS{{nbsp}}80 and WGS{{nbsp}}84 spheroids, <math display="inline">\tfrac{b}{a}=0.99664719</math>. (<math>\textstyle{\beta}\,\!</math> is known as the [[Latitude#Parametric (or reduced) latitude|reduced (or parametric) latitude]]). Aside from rounding, this is the exact distance along a parallel of latitude; getting the distance along the shortest route will be more work, but those two distances are always within 0.6 m of each other if the two points are one degree of longitude apart.

{| class="wikitable"
|+ Longitudinal length equivalents at selected latitudes
|-
! style="width:100px;" | Latitude
! style="width:150px;" | City
! style="width:100px;" | Degree
! style="width:100px;" | Minute
! style="width:100px;" | Second
! style="width:100px;" | 0.0001°
|-
| 60°
| [[Saint Petersburg]]
| style="text-align:center;" | 55.80&nbsp;km
| style="text-align:center;" | 0.930&nbsp;km
| style="text-align:center;" | 15.50&nbsp;m
| style="text-align:center;" | 5.58&nbsp;m
|-
| 51° 28′ 38″ N
| [[Greenwich]]
| style="text-align:center;" | 69.47&nbsp;km
| style="text-align:center;" | 1.158&nbsp;km
| style="text-align:center;" | 19.30&nbsp;m
| style="text-align:center;" | 6.95&nbsp;m
|-
| 45°
| [[Bordeaux]]
| style="text-align:center;" | 78.85&nbsp;km
| style="text-align:center;" | 1.31&nbsp;km
| style="text-align:center;" | 21.90&nbsp;m
| style="text-align:center;" | 7.89&nbsp;m
|-
| 30°
| [[New Orleans]]
| style="text-align:center;" | 96.49&nbsp;km
| style="text-align:center;" | 1.61&nbsp;km
| style="text-align:center;" | 26.80&nbsp;m
| style="text-align:center;" | 9.65&nbsp;m
|-
| 0°
| [[Quito]]
| style="text-align:center;" | 111.3&nbsp;km
| style="text-align:center;" | 1.855&nbsp;km
| style="text-align:center;" | 30.92&nbsp;m
| style="text-align:center;" | 11.13&nbsp;m
|}
<!--The Equator is the [[fundamental plane (spherical coordinates)|fundamental plane]] of all geographic coordinate systems. All spherical coordinate systems define such a fundamental plane.-->