Editing Latitude (section)

==Latitude on the ellipsoid==

===Ellipsoids===

{{main|Ellipsoid of revolution}}
In 1687 [[Isaac Newton]] published the ''[[Philosophiæ Naturalis Principia Mathematica]]'', in which he proved that a rotating self-gravitating fluid body in equilibrium takes the form of an [[Oblate spheroid|oblate]] ellipsoid.<ref name=newton>{{cite book|first=Isaac |last=Newton|title=Philosophiæ Naturalis Principia Mathematica|chapter=Book III Proposition XIX Problem III|page= [https://archive.org/details/100878576/page/407 407] |translator-first=Andrew |translator-last=Motte |url=https://archive.org/details/100878576}}</ref> (This article uses the term ''ellipsoid'' in preference to the older term ''spheroid''.) Newton's result was confirmed by geodetic measurements in the 18th century. (See [[Meridian arc]].) An oblate ellipsoid is the three-dimensional surface generated by the rotation of an ellipse about its shorter axis (minor axis). "Oblate ellipsoid of revolution" is abbreviated to 'ellipsoid' in the remainder of this article. (Ellipsoids which do not have an axis of symmetry are termed [[triaxial ellipsoid|triaxial]].)

Many different [[Figure of the Earth|reference ellipsoids]] have been used in the history of [[geodesy]]. In pre-satellite days they were devised to give a good fit to the [[geoid]] over the limited area of a survey but, with the advent of [[GPS]], it has become natural to use reference ellipsoids (such as [[WGS84]]) with centre at the centre of mass of the Earth and minor axis aligned to the rotation axis of the Earth. These geocentric ellipsoids are usually within {{convert|100|m|ft|abbr=on}} of the geoid. Since latitude is defined with respect to an ellipsoid, the position of a given point is different on each ellipsoid: one cannot exactly specify the latitude and longitude of a geographical feature without specifying the ellipsoid used. Many maps maintained by national agencies are based on older ellipsoids, so one must know how the latitude and longitude values are transformed from one ellipsoid to another. GPS handsets include software to carry out [[Datum (geodesy)|datum transformations]] which link WGS84 to the local reference ellipsoid with its associated grid.

===The geometry of the ellipsoid===
[[File:Ellipsoid parametric euler mono.svg|thumb|A sphere of radius ''a'' compressed along the ''z'' axis to form an oblate ellipsoid of revolution.]]

The shape of an ellipsoid of revolution is determined by the shape of the [[ellipse]] which is rotated about its minor (shorter) axis. Two parameters are required. One is invariably the equatorial radius, which is the [[ellipse|semi-major axis]], {{mvar|a}}. The other parameter is usually (1)&nbsp;the polar radius or [[ellipse|semi-minor axis]], {{mvar|b}}; or (2)&nbsp;the (first) [[flattening]], {{mvar|f}}; or (3)&nbsp;the [[ellipse|eccentricity]], {{mvar|e}}. These parameters are not independent: they are related by

:<math>f=\frac{a-b}{a}, \qquad e^2=2f-f^2,\qquad b=a(1-f)=a\sqrt{1-e^2}\,.</math>
Many other parameters (see [[ellipse]], [[ellipsoid]]) appear in the study of geodesy, geophysics and map projections but they can all be expressed in terms of one or two members of the set {{mvar|a}}, {{mvar|b}}, {{mvar|f}} and {{mvar|e}}. Both {{mvar|f}} and {{mvar|e}} are small and often appear in series expansions in calculations; they are of the order {{sfrac|1|298}} and 0.0818 respectively. Values for a number of ellipsoids are given in [[Figure of the Earth]]. Reference ellipsoids are usually defined by the semi-major axis and the ''inverse'' flattening, {{math|{{sfrac|1|''f''}}}}. For example, the defining values for the [[WGS84]] ellipsoid, used by all GPS devices, are<ref>{{cite web|author=National Imagery and Mapping Agency|url=https://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf|title=Department of Defense World Geodetic System 1984|date=23 June 2004|publisher=National Imagery and Mapping Agency|id=TR8350.2 |page=3{{hyphen}}1|access-date=25 April 2020|df=dmy}}</ref>
* {{mvar|a}} (equatorial radius): {{val|6378137.0|u=m}} exactly
* {{math|{{sfrac|1|''f''}}}} (inverse flattening): {{val|298.257223563}} exactly
from which are derived
* {{mvar|b}} (polar radius): {{val|6356752.31425|u=m}}
* {{math|''e''<sup>2</sup>}} (eccentricity squared): {{val|0.00669437999014}}
The difference between the semi-major and semi-minor axes is about {{convert|21|km|0|abbr=in}} and as fraction of the semi-major axis it equals the flattening; on a computer monitor the ellipsoid could be sized as 300 by 299 pixels. This would barely be distinguishable from a 300-by-300-pixel sphere, so illustrations usually exaggerate the flattening.

===Geodetic and geocentric latitudes===
{{see also|Geodetic coordinates#Geodetic vs. geocentric coordinates}}

[[File:latitude and longitude graticule on an ellipsoid.svg|thumb|upright=0.9|right|The definition of geodetic latitude (<math>\phi</math>) and longitude (<math>\lambda</math>) on an ellipsoid. The normal to the surface does not pass through the centre, except at the equator and at the poles.]]

The graticule on the ellipsoid is constructed in exactly the same way as on the sphere. The normal at a point on the surface of an ellipsoid does not pass through the centre, except for points on the equator or at the poles, but the definition of latitude remains unchanged as the angle between the normal and the equatorial plane. The terminology for latitude must be made more precise by distinguishing:

*{{anchor|Geodetic}}''[[Geodetic latitude]]'': the angle between the normal and the equatorial plane. The standard notation in English publications is {{mvar|ϕ}}. This is the definition assumed when the word latitude is used without qualification. The definition must be accompanied with a specification of the ellipsoid.
*{{anchor|Geocentric}}''[[Geocentric latitude]]'' (also known as ''spherical latitude'', after the [[3D polar angle]]): the angle between the radius (from centre to the point on the surface) and the equatorial plane. (Figure [[#Geocentric latitude|below]]). There is no standard notation: examples from various texts include {{mvar|θ}}, {{mvar|ψ}}, {{mvar|q}}, {{mvar|ϕ′}}, {{math|''ϕ''<sub>c</sub>}}, {{math|''ϕ''<sub>g</sub>}}. This article uses {{mvar|θ}}.

'''Geographic latitude''' must be used with care, as some authors use it as a synonym for geodetic latitude whilst others use it as an alternative to the [[#Astronomical latitude|astronomical latitude]].
"Latitude" (unqualified) should normally refer to the geodetic latitude.

The importance of specifying the reference datum may be illustrated by a simple example. On the reference ellipsoid for WGS84, the centre of the [[Eiffel Tower]] has a geodetic latitude of 48°&nbsp;51′&nbsp;29″&nbsp;N, or 48.8583°&nbsp;N and longitude of 2°&nbsp;17′&nbsp;40″&nbsp;E or 2.2944°E. The same coordinates on the datum [[ED50]] define a point on the ground which is {{convert|140|m|ft|abbr=off}} distant from the tower.{{citation needed|date=December 2011}} A web search may produce several different values for the latitude of the tower; the reference ellipsoid is rarely specified.