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Longitude
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==Length of a degree of longitude== {{see also|Length of a degree of latitude}} The length of a degree of longitude (east–west distance) depends only on the radius of a circle of latitude. For a sphere of radius {{mvar|a}} that radius at latitude {{mvar|φ}} is {{math|''a'' [[cosine|cos]] ''φ''}}, and the length of a one-degree (or {{sfrac|{{pi}}|180}} [[radian]]) arc along a circle of latitude is <math display="block">\Delta^1_{\rm long}= \frac{\pi}{180}a \cos \phi </math> {| class="wikitable" style="float: right; margin-left:1em; text-align:right;" ! {{mvar|φ}} || {{math|Δ{{su|p=1|b=lat}}}} || {{math|Δ{{su|p=1|b=long}}}} |- | 0° || 110.574 km || 111.320 km |- | 15° || 110.649 km || 107.551 km |- | 30° || 110.852 km || 96.486 km |- | 45° || 111.133 km || 78.847 km |- | 60° || 111.412 km || 55.800 km |- | 75° || 111.618 km || 28.902 km |- | 90° || 111.694 km || 0.000 km |} {{WGS84_angle_to_distance_conversion.svg}} When the Earth is modelled by an [[ellipsoid]] this arc length becomes<ref name=osborne>{{Cite book |last=Osborne |first=Peter |year=2013 |url=http://www.mercator99.webspace.virginmedia.com/mercator.pdf |doi=10.5281/zenodo.35392 |title=The Mercator Projections: The Normal and Transverse Mercator Projections on the Sphere and the Ellipsoid with Full Derivations of all Formulae |chapter=Chapter 5: The geometry of the ellipsoid |location=Edinburgh |access-date=2016-01-24 |archive-url=https://web.archive.org/web/20160509180529/http://www.mercator99.webspace.virginmedia.com/mercator.pdf |archive-date=2016-05-09 |url-status=dead }}</ref><ref name=rapp>{{cite book |last=Rapp |first=Richard H. |date=April 1991 |title=Geometric Geodesy Part I |chapter=Chapter 3: Properties of the Ellipsoid |publisher=Department of Geodetic Science and Surveying, Ohio State University |location=Columbus, Ohio. |hdl=1811/24333 }}</ref> <math display="block">\Delta^1_{\rm long}=\frac{\pi a\cos\phi}{180 \sqrt{1 - e^2 \sin^2 \phi}}</math> where {{mvar|e}}, the eccentricity of the ellipsoid, is related to the major and minor axes (the equatorial and polar radii respectively) by <math display="block">e^2=\frac{a^2-b^2}{a^2}</math> An alternative formula is <math display="block">\Delta^1_{\rm long}= \frac{\pi}{180}a \cos \beta \quad \mbox{where }\tan \beta = \frac{b}{a} \tan \phi</math> Here <math>\beta</math> is the so-called [[Parametric latitude|''parametric'' or ''reduced'' latitude]]. cos {{mvar|φ}} decreases from 1 at the equator to 0 at the poles, which measures how circles of latitude shrink from the equator to a point at the pole, so the length of a degree of longitude decreases likewise. This contrasts with the small (1%) increase in the [[length of a degree of latitude]] (north–south distance), equator to pole. The table shows both for the [[WGS84]] ellipsoid with {{mvar|a}} = {{val|6378137.0|u=m}} and {{mvar|b}} = {{val|6356752.3142|u=m}}. The distance between two points 1 degree apart on the same circle of latitude, measured along that circle of latitude, is slightly more than the shortest ([[geodesic]]) distance between those points (unless on the equator, where these are equal); the difference is less than {{convert|0.6|m|ft|0|abbr=on}}. A [[geographical mile]] is defined to be the length of one [[minute of arc]] along the equator (one equatorial minute of longitude) therefore a degree of longitude along the equator is exactly 60 geographical miles or 111.3 kilometers, as there are 60 minutes in a degree. The length of 1 minute of longitude along the equator is 1 geographical mile or {{convert|1.855|km|mi|disp=or|abbr=in}}, while the length of 1 second of it is 0.016 geographical mile or {{convert|30.916|m|ft|disp=or|abbr=in}}.
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