Editing Latitude (section)

==Meridian distance{{anchor|Length of a degree of latitude}}==
{{main|Meridian arc}}
{{see also|Length of a degree of longitude}}
The length of a degree of latitude depends on the [[figure of the Earth]] assumed.

===Meridian distance on the sphere===
On the sphere the normal passes through the centre and the latitude ({{mvar|ϕ}}) is
therefore equal to the angle subtended at the centre by the meridian arc from the equator to the point concerned. If the [[Meridian arc|meridian distance]] is denoted by {{math|''m''(''ϕ'')}} then
<math display="block"> m(\phi)=\frac{\pi}{180^\circ}R\phi_\mathrm{degrees} = R\phi_\mathrm{radians}</math>
where {{mvar|R}} denotes the [[Earth radius#Mean radii|mean radius]] of the Earth. {{mvar|R}} is equal to {{convert|6371|km|mi|disp=or|abbr=in}}. No higher accuracy is appropriate for {{mvar|R}} since higher-precision results necessitate an ellipsoid model. With this value for {{mvar|R}} the meridian length of 1 degree of latitude on the sphere is {{convert|111.2|km|mi|abbr=in|disp=preunit||statute }} (60.0 nautical miles). The length of one minute of latitude is {{convert|1.853|km|mi|abbr=in|disp=preunit||statute }} (1.00 nautical miles), while the length of 1 second of latitude is {{convert|30.8|m|ft|disp=or|abbr=in}} (see [[nautical mile]]).

===Meridian distance on the ellipsoid===

In [[Meridian arc]] and standard texts<ref name=torge>{{cite book|last=Torge |first=W. |date=2001 |title=Geodesy |edition=3rd |publisher=De Gruyter |isbn=3-11-017072-8}}</ref><ref name=osborne/><ref name=rapp/> it is shown that the distance along a meridian from latitude {{mvar|ϕ}} to the equator is given by ({{mvar|ϕ}} in radians)

<math display="block">m(\phi) = \int_0^\phi M(\phi')\, d\phi' = a\left(1 - e^2\right)\int_0^\phi \left(1 - e^2 \sin^2\phi'\right)^{-\frac{3}{2}}\, d\phi'</math>

where {{math|''M''(''ϕ'')}} is the meridional [[radius of curvature (applications)|radius of curvature]].

The ''[[quarter meridian]]'' distance from the equator to the pole is

<math display="block">m_\mathrm{p} = m\left(\frac{\pi}{2}\right)\,</math>

For [[WGS84]] this distance is {{val|10001.965729|u=km}}.

The evaluation of the meridian distance integral is central to many studies in geodesy and map projection. It can be evaluated by expanding the integral by the binomial series and integrating term by term: see [[Meridian arc]] for details. The length of the meridian arc between two given latitudes is given by replacing the limits of the integral by the latitudes concerned. The length of a ''small'' meridian arc is given by<ref name=osborne>{{cite book|first=Peter |last=Osborne |title=The Mercator Projections |year=2013 |doi=10.5281/zenodo.35392 |chapter=Chapters&nbsp;5,6}} for LaTeX code and figures.</ref><ref name=rapp>{{cite book|last=Rapp |first=Richard H. |date=1991 |title=Geometric Geodesy, Part I |publisher=Dept. of Geodetic Science and Surveying, Ohio State Univ. |location=Columbus, OH|chapter=Chapter 3|hdl=1811/24333 }}</ref>

<math display="block">\delta m(\phi) = M(\phi)\, \delta\phi = a\left(1 - e^2\right) \left(1 - e^2 \sin^2\phi\right)^{-\frac{3}{2}}\, \delta\phi</math>

{|class="wikitable" style="margin:1em auto 1em auto;float:right;clear:right;"
!<math>\phi</math>||{{math|Δ{{su|p=1|b=lat}}}}||{{math|Δ{{su|p=1|b=long}}}}
|- style="text-align:right;"
|  0° || 110.574&nbsp;km || 111.320&nbsp;km
|- style="text-align:right;"
| 15° || 110.649&nbsp;km || 107.550&nbsp;km
|- style="text-align:right;"
| 30° || 110.852&nbsp;km || 96.486&nbsp;km
|- style="text-align:right;"
| 45° || 111.132&nbsp;km || 78.847&nbsp;km
|- style="text-align:right;"
| 60° || 111.412&nbsp;km || 55.800&nbsp;km
|- style="text-align:right;"
| 75° || 111.618&nbsp;km || 28.902&nbsp;km
|- style="text-align:right;"
| 90° || 111.694&nbsp;km || 0.000&nbsp;km
|}

When the latitude difference is 1 degree, corresponding to {{sfrac|{{pi}}|180}} radians, the arc distance is about
<math display="block">\Delta^1_\text{lat} = \frac{\pi a\left(1 - e^2\right)}{180^\circ\left(1 - e^2 \sin^2\phi\right)^\frac{3}{2}}</math>

The distance in metres (correct to 0.01 metre) between latitudes <math>\phi</math>&nbsp;−&nbsp;0.5 degrees and <math>\phi</math>&nbsp;+&nbsp;0.5 degrees on the WGS84 spheroid is
<math display="block">\Delta^1_\text{lat} = 111\,132.954 - 559.822\cos 2\phi + 1.175\cos 4\phi</math>

The variation of this distance with latitude (on [[WGS84]]) is shown in the table along with the [[length of a degree of longitude]] (east-west distance):
<math display="block">\Delta^1_\text{long} = \frac{\pi a\cos\phi}{180^\circ\sqrt{1 - e^2 \sin^2\phi}}\,</math>
<!-- 
The more recent but little used [[IERS]] 2003 ellipsoid provides equatorial and polar semi-axes of {{val|6378136.6}} and {{val|6356751.9|u=m}} and an inverse flattening of {{val|298.25642}}.<ref>{{cite web|url=http://www.iers.org/MainDisp.csl?pid=46-25776|title=IERS Conventions |date=2003 |page=1-12}}</ref> Lengths of degrees on the WGS84 and IERS 2003 ellipsoids are the same when rounded to six [[significant digit]]s.
-->
A calculator for any latitude is provided by the U.S. Government's [[National Geospatial-Intelligence Agency]] (NGA).<ref>{{cite web|url=https://msi.nga.mil/api/publications/download?key=16920949/SFH00000/Calculators/degree.html&type=view|title=Length of degree calculator|publisher=National Geospatial-Intelligence Agency|access-date=2011-02-08|archive-url=https://web.archive.org/web/20121211031023/http://msi.nga.mil/MSISiteContent/StaticFiles/Calculators/degree.html|archive-date=2012-12-11|url-status=dead}}</ref>