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class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading">Latitude</h1> <div class="tagline"></div> </div> <ul id="p-associated-pages" class="minerva__tab-container"> <li class="minerva__tab selected"> <a class="minerva__tab-text" href="/wiki/Latitude" rel="" data-event-name="tabs.subject">Article</a> </li> <li class="minerva__tab "> <a class="minerva__tab-text" href="/wiki/Talk:Latitude" rel="discussion" data-event-name="tabs.talk">Talk</a> </li> </ul> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"> <a role="button" href="#p-lang" data-mw="interface" data-event-name="menu.languages" title="Language" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet language-selector"> Language </a> </li> <li id="page-actions-watch" class="page-actions-menu__list-item"> <a role="button" id="ca-watch" href="/w/index.php?title=Special:UserLogin&returnto=Latitude" data-event-name="menu.watch" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet menu__item--page-actions-watch"> Watch </a> </li> <li id="page-actions-edit" class="page-actions-menu__list-item"> <a role="button" id="ca-edit" href="/w/index.php?title=Latitude&action=edit" data-event-name="menu.edit" data-mw="interface" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> Edit </a> </li> </ul> </nav>  <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"><script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><section class="mf-section-0" id="mf-section-0"> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the geographical reference system. For other uses, see <a href="/wiki/Latitude_(disambiguation)" class="mw-disambig" title="Latitude (disambiguation)">Latitude (disambiguation)</a>.</div> In <a href="/wiki/Geography" title="Geography">geography</a>, latitude is a <a href="/wiki/Geographic_coordinate_system" title="Geographic coordinate system">coordinate</a> that specifies the <a href="/wiki/North" title="North">north</a>–<a href="/wiki/South" title="South">south</a> position of a point on the surface of <a href="/wiki/The_Earth" class="mw-redirect" title="The Earth">the Earth</a> or another celestial body. Latitude is given as an angle that ranges from −90° at the south pole to 90° at the north pole, with 0° at the <a href="/wiki/Equator" title="Equator">Equator</a>. <a href="/wiki/Parallel_(latitude)" class="mw-redirect" title="Parallel (latitude)">Lines of constant latitude</a>, or parallels, run east–west as circles parallel to the equator. Latitude and <a href="/wiki/Longitude" title="Longitude">longitude</a> are used together as a coordinate pair to specify a location on the surface of the Earth. <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Division_of_the_Earth_into_Gauss-Krueger_zones_-_Globe.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Division_of_the_Earth_into_Gauss-Krueger_zones_-_Globe.svg/200px-Division_of_the_Earth_into_Gauss-Krueger_zones_-_Globe.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Division_of_the_Earth_into_Gauss-Krueger_zones_-_Globe.svg/300px-Division_of_the_Earth_into_Gauss-Krueger_zones_-_Globe.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Division_of_the_Earth_into_Gauss-Krueger_zones_-_Globe.svg/400px-Division_of_the_Earth_into_Gauss-Krueger_zones_-_Globe.svg.png 2x" data-file-width="948" data-file-height="948"></a><figcaption>Earth's <a href="/wiki/Graticule_(cartography)" title="Graticule (cartography)">graticule</a>. The vertical lines from pole to pole are lines of constant <a href="/wiki/Longitude" title="Longitude">longitude</a>, or meridians. The circles parallel to the <a href="/wiki/Equator" title="Equator">equator</a> are lines of constant latitude, or parallels. The graticule shows the latitude and longitude of points on the surface. In this example meridians are spaced at 6° intervals and parallels at 4° intervals.</figcaption></figure> On its own, the term "latitude" normally refers to the geodetic latitude as defined below. Briefly, the geodetic latitude of a point is the angle formed between the vector perpendicular (or <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">normal</a>) to the ellipsoidal surface from the point, and the <a href="/wiki/Equatorial_plane" class="mw-redirect" title="Equatorial plane">plane of the equator</a>. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Background">1 Background</a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Determination">1.1 Determination</a></li> </ul> </li> <li class="toclevel-1 tocsection-3"><a href="#Latitude_on_the_sphere">2 Latitude on the sphere</a> <ul> <li class="toclevel-2 tocsection-4"><a href="#The_graticule_on_the_sphere">2.1 The graticule on the sphere</a></li> <li class="toclevel-2 tocsection-5"><a href="#Named_latitudes_on_the_Earth">2.2 Named latitudes on the Earth</a></li> </ul> </li> <li class="toclevel-1 tocsection-6"><a href="#Latitude_on_the_ellipsoid">3 Latitude on the ellipsoid</a> <ul> <li class="toclevel-2 tocsection-7"><a href="#Ellipsoids">3.1 Ellipsoids</a></li> <li class="toclevel-2 tocsection-8"><a href="#The_geometry_of_the_ellipsoid">3.2 The geometry of the ellipsoid</a></li> <li class="toclevel-2 tocsection-9"><a href="#Geodetic_and_geocentric_latitudes">3.3 Geodetic and geocentric latitudes</a></li> </ul> </li> <li class="toclevel-1 tocsection-10"><a href="#Meridian_distance">4 Meridian distance</a> <ul> <li class="toclevel-2 tocsection-11"><a href="#Meridian_distance_on_the_sphere">4.1 Meridian distance on the sphere</a></li> <li class="toclevel-2 tocsection-12"><a href="#Meridian_distance_on_the_ellipsoid">4.2 Meridian distance on the ellipsoid</a></li> </ul> </li> <li class="toclevel-1 tocsection-13"><a href="#Auxiliary_latitudes">5 Auxiliary latitudes</a> <ul> <li class="toclevel-2 tocsection-14"><a href="#Geocentric_latitude">5.1 Geocentric latitude</a></li> <li class="toclevel-2 tocsection-15"><a href="#Parametric_latitude_(or_reduced_latitude)">5.2 Parametric latitude (or reduced latitude)</a></li> <li class="toclevel-2 tocsection-16"><a href="#Rectifying_latitude">5.3 Rectifying latitude</a></li> <li class="toclevel-2 tocsection-17"><a href="#Authalic_latitude">5.4 Authalic latitude</a></li> <li class="toclevel-2 tocsection-18"><a href="#Conformal_latitude">5.5 Conformal latitude</a></li> <li class="toclevel-2 tocsection-19"><a href="#Isometric_latitude">5.6 Isometric latitude</a></li> <li class="toclevel-2 tocsection-20"><a href="#Inverse_formulae_and_series">5.7 Inverse formulae and series</a></li> <li class="toclevel-2 tocsection-21"><a href="#Numerical_comparison_of_auxiliary_latitudes">5.8 Numerical comparison of auxiliary latitudes</a></li> </ul> </li> <li class="toclevel-1 tocsection-22"><a href="#Latitude_and_coordinate_systems">6 Latitude and coordinate systems</a> <ul> <li class="toclevel-2 tocsection-23"><a href="#Geodetic_coordinates">6.1 Geodetic coordinates</a></li> <li class="toclevel-2 tocsection-24"><a href="#Spherical_polar_coordinates">6.2 Spherical polar coordinates</a></li> <li class="toclevel-2 tocsection-25"><a href="#Ellipsoidal-harmonic_coordinates">6.3 Ellipsoidal-harmonic coordinates</a></li> <li class="toclevel-2 tocsection-26"><a href="#Coordinate_conversions">6.4 Coordinate conversions</a></li> </ul> </li> <li class="toclevel-1 tocsection-27"><a href="#Astronomical_latitude">7 Astronomical latitude</a></li> <li class="toclevel-1 tocsection-28"><a href="#See_also">8 See also</a></li> <li class="toclevel-1 tocsection-29"><a href="#References">9 References</a> <ul> <li class="toclevel-2 tocsection-30"><a href="#Footnotes">9.1 Footnotes</a></li> <li class="toclevel-2 tocsection-31"><a href="#Citations">9.2 Citations</a></li> </ul> </li> <li class="toclevel-1 tocsection-32"><a href="#External_links">10 External links</a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><h2 id="Background">Background</h2> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=1" title="Edit section: Background" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> Two levels of abstraction are employed in the definitions of latitude and longitude. In the first step the physical surface is modeled by the <a href="/wiki/Geoid" title="Geoid">geoid</a>, a surface which approximates the <a href="/wiki/Sea_level" title="Sea level">mean sea level</a> over the oceans and its continuation under the land masses. The second step is to approximate the geoid by a mathematically simpler reference surface. The simplest choice for the reference surface is a <a href="/wiki/Sphere" title="Sphere">sphere</a>, but the geoid is more accurately modeled by an <a href="/wiki/Ellipsoid_of_revolution" class="mw-redirect" title="Ellipsoid of revolution">ellipsoid of revolution</a>. The definitions of latitude and longitude on such reference surfaces are detailed in the following sections. Lines of constant latitude and longitude together constitute a <a href="/wiki/Geographic_coordinate_system" title="Geographic coordinate system">graticule</a> on the reference surface. The latitude of a point on the actual surface is that of the corresponding point on the reference surface, the correspondence being along the <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">normal</a> to the reference surface, which passes through the point on the physical surface. Latitude and longitude together with some specification of <a href="/wiki/Altitude" title="Altitude">height</a> constitute a <a href="/wiki/Geographic_coordinate_system" title="Geographic coordinate system">geographic coordinate system</a> as defined in the specification of the ISO 19111 standard.<a href="#cite_note-1">[1]</a> Since there are many different <a href="/wiki/Reference_ellipsoid" class="mw-redirect" title="Reference ellipsoid">reference ellipsoids</a>, the precise latitude of a feature on the surface is not unique: this is stressed in the ISO standard which states that "without the full specification of the coordinate reference system, coordinates (that is latitude and longitude) are ambiguous at best and meaningless at worst". This is of great importance in accurate applications, such as a <a href="/wiki/Global_Positioning_System" title="Global Positioning System">Global Positioning System</a> (GPS), but in common usage, where high accuracy is not required, the reference ellipsoid is not usually stated. In English texts, the latitude angle, defined below, is usually denoted by the Greek lower-case letter <a href="/wiki/Phi_(letter)" class="mw-redirect" title="Phi (letter)">phi</a> (ϕ or φ). It is measured in <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a>, <a href="/wiki/Arcminute" class="mw-redirect" title="Arcminute">minutes and seconds</a> or <a href="/wiki/Decimal_degrees" title="Decimal degrees">decimal degrees</a>, north or south of the equator. For navigational purposes positions are given in degrees and decimal minutes. For instance, <a href="/wiki/The_Needles" title="The Needles">The Needles</a> lighthouse is at 50°39.734′ N 001°35.500′ W.<a href="#cite_note-2">[2]</a> This article relates to coordinate systems for the Earth: it may be adapted to cover the Moon, planets and other celestial objects (<a href="/wiki/Planetographic_latitude" class="mw-redirect" title="Planetographic latitude">planetographic latitude</a>). For a brief history, see <a href="/wiki/History_of_latitude" title="History of latitude">History of latitude</a>. <div class="mw-heading mw-heading3"><h3 id="Determination">Determination</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=2" title="Edit section: Determination" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Longitude_determination" class="mw-redirect" title="Longitude determination">Longitude determination</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Celestial_navigation#Latitude" title="Celestial navigation">Celestial navigation § Latitude</a></div> In <a href="/wiki/Celestial_navigation" title="Celestial navigation">celestial navigation</a>, latitude is determined with the <a href="/wiki/Meridian_altitude" title="Meridian altitude">meridian altitude</a> method. More precise measurement of latitude requires an understanding of the gravitational field of the Earth, either to set up <a href="/wiki/Theodolite" title="Theodolite">theodolites</a> or to determine GPS satellite orbits. The study of the <a href="/wiki/Figure_of_the_Earth" title="Figure of the Earth">figure of the Earth</a> together with its gravitational field is the science of <a href="/wiki/Geodesy" title="Geodesy">geodesy</a>. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><h2 id="Latitude_on_the_sphere">Latitude on the sphere</h2> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=3" title="Edit section: Latitude on the sphere" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Latitude_and_longitude_graticule_on_a_sphere.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Latitude_and_longitude_graticule_on_a_sphere.svg/200px-Latitude_and_longitude_graticule_on_a_sphere.svg.png" decoding="async" width="200" height="200" class="mw-file-element" data-file-width="825" data-file-height="825"></noscript><span class="lazy-image-placeholder" style="width: 200px;height: 200px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Latitude_and_longitude_graticule_on_a_sphere.svg/200px-Latitude_and_longitude_graticule_on_a_sphere.svg.png" data-width="200" data-height="200" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Latitude_and_longitude_graticule_on_a_sphere.svg/300px-Latitude_and_longitude_graticule_on_a_sphere.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Latitude_and_longitude_graticule_on_a_sphere.svg/400px-Latitude_and_longitude_graticule_on_a_sphere.svg.png 2x" data-class="mw-file-element"> </a><figcaption>A perspective view of the Earth showing how latitude (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></noscript><span class="lazy-image-placeholder" style="width: 1.385ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" data-alt="{\displaystyle \phi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) and longitude (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></noscript><span class="lazy-image-placeholder" style="width: 1.355ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" data-alt="{\displaystyle \lambda }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) are defined on a spherical model. The graticule spacing is 10 degrees.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="The_graticule_on_the_sphere">The graticule on the sphere</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=4" title="Edit section: The graticule on the sphere" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The graticule is formed by the lines of constant latitude and constant longitude, which are constructed with reference to the rotation axis of the Earth. The primary reference points are the <a href="/wiki/Geographical_pole" title="Geographical pole">poles</a> where the axis of rotation of the Earth intersects the reference surface. Planes which contain the rotation axis intersect the surface at the <a href="/wiki/Meridian_(geography)" title="Meridian (geography)">meridians</a>; and the angle between any one meridian plane and that through Greenwich (the <a href="/wiki/Prime_Meridian" class="mw-redirect" title="Prime Meridian">Prime Meridian</a>) defines the longitude: meridians are lines of constant longitude. The plane through the centre of the Earth and perpendicular to the rotation axis intersects the surface at a great circle called the <a href="/wiki/Equator" title="Equator">Equator</a>. Planes parallel to the equatorial plane intersect the surface in circles of constant latitude; these are the parallels. The Equator has a latitude of 0°, the <a href="/wiki/North_Pole" title="North Pole">North Pole</a> has a latitude of 90° North (written 90° N or +90°), and the <a href="/wiki/South_Pole" title="South Pole">South Pole</a> has a latitude of 90° South (written 90° S or −90°). The latitude of an arbitrary point is the angle between the equatorial plane and the normal to the surface at that point: the normal to the surface of the sphere is along the radial vector. The latitude, as defined in this way for the sphere, is often termed the spherical latitude, to avoid ambiguity with the geodetic latitude and the auxiliary latitudes defined in subsequent sections of this article. <div class="mw-heading mw-heading3"><h3 id="Named_latitudes_on_the_Earth">Named latitudes on the Earth</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=5" title="Edit section: Named latitudes on the Earth" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:December_solstice_geometry.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/December_solstice_geometry.svg/300px-December_solstice_geometry.svg.png" decoding="async" width="300" height="205" class="mw-file-element" data-file-width="367" data-file-height="251"></noscript><span class="lazy-image-placeholder" style="width: 300px;height: 205px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/December_solstice_geometry.svg/300px-December_solstice_geometry.svg.png" data-width="300" data-height="205" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/December_solstice_geometry.svg/450px-December_solstice_geometry.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/December_solstice_geometry.svg/600px-December_solstice_geometry.svg.png 2x" data-class="mw-file-element"> </a><figcaption>The orientation of the Earth at the December solstice</figcaption></figure> Besides the equator, four other parallels are of significance: <dl><dd><table class="wikitable"> <tbody><tr> <td><a href="/wiki/Arctic_Circle" title="Arctic Circle">Arctic Circle</a></td> <td>66° 34′ (66.57°) N </td></tr> <tr> <td><a href="/wiki/Tropic_of_Cancer" title="Tropic of Cancer">Tropic of Cancer</a></td> <td>23° 26′ (23.43°) N </td></tr> <tr> <td><a href="/wiki/Tropic_of_Capricorn" title="Tropic of Capricorn">Tropic of Capricorn</a></td> <td>23° 26′ (23.43°) S </td></tr> <tr> <td><a href="/wiki/Antarctic_Circle" title="Antarctic Circle">Antarctic Circle</a></td> <td>66° 34′ (66.57°) S </td></tr></tbody></table></dd></dl> The plane of the Earth's orbit about the Sun is called the <a href="/wiki/Ecliptic" title="Ecliptic">ecliptic</a>, and the plane perpendicular to the rotation axis of the Earth is the equatorial plane. The angle between the ecliptic and the equatorial plane is called variously the axial tilt, the obliquity, or the inclination of the ecliptic, and it is conventionally denoted by i. The latitude of the tropical circles is equal to i and the latitude of the polar circles is its complement (90° - i). The axis of rotation varies slowly over time and the values given here are those for the current <a href="/wiki/Epoch_(astronomy)" title="Epoch (astronomy)">epoch</a>. The time variation is discussed more fully in the article on <a href="/wiki/Axial_tilt" title="Axial tilt">axial tilt</a>.<a href="#cite_note-3">[a]</a> The figure shows the geometry of a <a href="/wiki/Cross_section_(geometry)" title="Cross section (geometry)">cross-section</a> of the plane perpendicular to the ecliptic and through the centres of the Earth and the Sun at the December <a href="/wiki/Solstice" title="Solstice">solstice</a> when the Sun is overhead at some point of the <a href="/wiki/Tropic_of_Capricorn" title="Tropic of Capricorn">Tropic of Capricorn</a>. The south polar latitudes below the <a href="/wiki/Antarctic_Circle" title="Antarctic Circle">Antarctic Circle</a> are in daylight, whilst the north polar latitudes above the Arctic Circle are in night. The situation is reversed at the June solstice, when the Sun is overhead at the Tropic of Cancer. Only at latitudes in between the two <a href="/wiki/Tropics" title="Tropics">tropics</a> is it possible for the Sun to be directly overhead (at the <a href="/wiki/Zenith" title="Zenith">zenith</a>). On <a href="/wiki/Map_projections" class="mw-redirect" title="Map projections">map projections</a> there is no universal rule as to how meridians and parallels should appear. The examples below show the named parallels (as red lines) on the commonly used <a href="/wiki/Mercator_projection" title="Mercator projection">Mercator projection</a> and the <a href="/wiki/Transverse_Mercator_projection" title="Transverse Mercator projection">Transverse Mercator projection</a>. On the former the parallels are horizontal and the meridians are vertical, whereas on the latter there is no exact relationship of parallels and meridians with horizontal and vertical: both are complicated curves. <table style="margin: 1em auto 1em auto"> <tbody><tr valign="top"> <th width="1%"> </th> <th width="36%">Normal Mercator </th> <th width="3%"> </th> <th width="36%">Transverse Mercator </th> <th width="1%"> </th></tr> <tr valign="top"> <td> </td> <td align="center" width="200px"><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:MercNormSph_enhanced.png" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/MercNormSph_enhanced.png/200px-MercNormSph_enhanced.png" decoding="async" width="200" height="199" class="mw-file-element" data-file-width="800" data-file-height="797"></noscript><span class="lazy-image-placeholder" style="width: 200px;height: 199px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/MercNormSph_enhanced.png/200px-MercNormSph_enhanced.png" data-width="200" data-height="199" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/MercNormSph_enhanced.png/300px-MercNormSph_enhanced.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/MercNormSph_enhanced.png/400px-MercNormSph_enhanced.png 2x" data-class="mw-file-element"> </a><figcaption></figcaption></figure> </td> <td> \ </td> <td align="center" width="200px"><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:MercTranSph_enhanced.png" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/MercTranSph_enhanced.png/200px-MercTranSph_enhanced.png" decoding="async" width="200" height="201" class="mw-file-element" data-file-width="1068" data-file-height="1073"></noscript><span class="lazy-image-placeholder" style="width: 200px;height: 201px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/MercTranSph_enhanced.png/200px-MercTranSph_enhanced.png" data-width="200" data-height="201" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/MercTranSph_enhanced.png/300px-MercTranSph_enhanced.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/MercTranSph_enhanced.png/400px-MercTranSph_enhanced.png 2x" data-class="mw-file-element"> </a><figcaption></figcaption></figure> </td></tr></tbody></table> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><h2 id="Latitude_on_the_ellipsoid">Latitude on the ellipsoid</h2> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=6" title="Edit section: Latitude on the ellipsoid" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <div class="mw-heading mw-heading3"><h3 id="Ellipsoids">Ellipsoids</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=7" title="Edit section: Ellipsoids" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Ellipsoid_of_revolution" class="mw-redirect" title="Ellipsoid of revolution">Ellipsoid of revolution</a></div> In 1687 <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> published the <a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Philosophiæ Naturalis Principia Mathematica</a>, in which he proved that a rotating self-gravitating fluid body in equilibrium takes the form of an <a href="/wiki/Oblate_spheroid" class="mw-redirect" title="Oblate spheroid">oblate</a> ellipsoid.<a href="#cite_note-newton-4">[3]</a> (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 <a href="/wiki/Meridian_arc" title="Meridian arc">Meridian arc</a>.) 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 <a href="/wiki/Triaxial_ellipsoid" class="mw-redirect" title="Triaxial ellipsoid">triaxial</a>.) Many different <a href="/wiki/Figure_of_the_Earth" title="Figure of the Earth">reference ellipsoids</a> have been used in the history of <a href="/wiki/Geodesy" title="Geodesy">geodesy</a>. In pre-satellite days they were devised to give a good fit to the <a href="/wiki/Geoid" title="Geoid">geoid</a> over the limited area of a survey but, with the advent of <a href="/wiki/GPS" class="mw-redirect" title="GPS">GPS</a>, it has become natural to use reference ellipsoids (such as <a href="/wiki/WGS84" class="mw-redirect" title="WGS84">WGS84</a>) 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 100 m (330 ft) 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 <a href="/wiki/Datum_(geodesy)" class="mw-redirect" title="Datum (geodesy)">datum transformations</a> which link WGS84 to the local reference ellipsoid with its associated grid. <div class="mw-heading mw-heading3"><h3 id="The_geometry_of_the_ellipsoid">The geometry of the ellipsoid</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=8" title="Edit section: The geometry of the ellipsoid" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Ellipsoid_parametric_euler_mono.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Ellipsoid_parametric_euler_mono.svg/200px-Ellipsoid_parametric_euler_mono.svg.png" decoding="async" width="200" height="200" class="mw-file-element" data-file-width="480" data-file-height="480"></noscript><span class="lazy-image-placeholder" style="width: 200px;height: 200px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Ellipsoid_parametric_euler_mono.svg/200px-Ellipsoid_parametric_euler_mono.svg.png" data-width="200" data-height="200" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Ellipsoid_parametric_euler_mono.svg/300px-Ellipsoid_parametric_euler_mono.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Ellipsoid_parametric_euler_mono.svg/400px-Ellipsoid_parametric_euler_mono.svg.png 2x" data-class="mw-file-element"> </a><figcaption>A sphere of radius a compressed along the z axis to form an oblate ellipsoid of revolution.</figcaption></figure> The shape of an ellipsoid of revolution is determined by the shape of the <a href="/wiki/Ellipse" title="Ellipse">ellipse</a> which is rotated about its minor (shorter) axis. Two parameters are required. One is invariably the equatorial radius, which is the <a href="/wiki/Ellipse" title="Ellipse">semi-major axis</a>, a. The other parameter is usually (1) the polar radius or <a href="/wiki/Ellipse" title="Ellipse">semi-minor axis</a>, b; or (2) the (first) <a href="/wiki/Flattening" title="Flattening">flattening</a>, f; or (3) the <a href="/wiki/Ellipse" title="Ellipse">eccentricity</a>, e. These parameters are not independent: they are related by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={\frac {a-b}{a}},\qquad e^{2}=2f-f^{2},\qquad b=a(1-f)=a{\sqrt {1-e^{2}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="2em"></mspace> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mi>f</mi> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="2em"></mspace> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f={\frac {a-b}{a}},\qquad e^{2}=2f-f^{2},\qquad b=a(1-f)=a{\sqrt {1-e^{2}}}\,.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d176b5a394f9d591c6242a6e8754f0c3a6e76e90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:60.775ex; height:5.343ex;" alt="{\displaystyle f={\frac {a-b}{a}},\qquad e^{2}=2f-f^{2},\qquad b=a(1-f)=a{\sqrt {1-e^{2}}}\,.}"></noscript><span class="lazy-image-placeholder" style="width: 60.775ex;height: 5.343ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d176b5a394f9d591c6242a6e8754f0c3a6e76e90" data-alt="{\displaystyle f={\frac {a-b}{a}},\qquad e^{2}=2f-f^{2},\qquad b=a(1-f)=a{\sqrt {1-e^{2}}}\,.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> Many other parameters (see <a href="/wiki/Ellipse" title="Ellipse">ellipse</a>, <a href="/wiki/Ellipsoid" title="Ellipsoid">ellipsoid</a>) 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 a, b, f and e. Both f and e are small and often appear in series expansions in calculations; they are of the order <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠1/298⁠ and 0.0818 respectively. Values for a number of ellipsoids are given in <a href="/wiki/Figure_of_the_Earth" title="Figure of the Earth">Figure of the Earth</a>. Reference ellipsoids are usually defined by the semi-major axis and the inverse flattening, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/f⁠. For example, the defining values for the <a href="/wiki/WGS84" class="mw-redirect" title="WGS84">WGS84</a> ellipsoid, used by all GPS devices, are<a href="#cite_note-5">[4]</a> <ul><li>a (equatorial radius): 6378137.0 m exactly</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/f⁠ (inverse flattening): 298.257223563 exactly</li></ul> from which are derived <ul><li>b (polar radius): 6356752.31425 m</li> <li>e2 (eccentricity squared): 0.00669437999014</li></ul> The difference between the semi-major and semi-minor axes is about 21 km (13 miles) 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. <div class="mw-heading mw-heading3"><h3 id="Geodetic_and_geocentric_latitudes">Geodetic and geocentric latitudes</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=9" title="Edit section: Geodetic and geocentric latitudes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Geodetic_coordinates#Geodetic_vs._geocentric_coordinates" title="Geodetic coordinates">Geodetic coordinates § Geodetic vs. geocentric coordinates</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Latitude_and_longitude_graticule_on_an_ellipsoid.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Latitude_and_longitude_graticule_on_an_ellipsoid.svg/200px-Latitude_and_longitude_graticule_on_an_ellipsoid.svg.png" decoding="async" width="200" height="182" class="mw-file-element" data-file-width="825" data-file-height="750"></noscript><span class="lazy-image-placeholder" style="width: 200px;height: 182px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Latitude_and_longitude_graticule_on_an_ellipsoid.svg/200px-Latitude_and_longitude_graticule_on_an_ellipsoid.svg.png" data-width="200" data-height="182" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Latitude_and_longitude_graticule_on_an_ellipsoid.svg/300px-Latitude_and_longitude_graticule_on_an_ellipsoid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Latitude_and_longitude_graticule_on_an_ellipsoid.svg/400px-Latitude_and_longitude_graticule_on_an_ellipsoid.svg.png 2x" data-class="mw-file-element"> </a><figcaption>The definition of geodetic latitude (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></noscript><span class="lazy-image-placeholder" style="width: 1.385ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" data-alt="{\displaystyle \phi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) and longitude (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></noscript><span class="lazy-image-placeholder" style="width: 1.355ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" data-alt="{\displaystyle \lambda }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) on an ellipsoid. The normal to the surface does not pass through the centre, except at the equator and at the poles.</figcaption></figure> 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: <ul><li><a href="/wiki/Geodetic_latitude" class="mw-redirect" title="Geodetic latitude">Geodetic latitude</a>: the angle between the normal and the equatorial plane. The standard notation in English publications is 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.</li> <li><a href="/wiki/Geocentric_latitude" class="mw-redirect" title="Geocentric latitude">Geocentric latitude</a> (also known as spherical latitude, after the <a href="/wiki/3D_polar_angle" class="mw-redirect" title="3D polar angle">3D polar angle</a>): the angle between the radius (from centre to the point on the surface) and the equatorial plane. (Figure <a href="#Geocentric_latitude">below</a>). There is no standard notation: examples from various texts include θ, ψ, q, ϕ′, ϕc, ϕg. This article uses θ.</li></ul> 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 <a href="#Astronomical_latitude">astronomical latitude</a>. "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 <a href="/wiki/Eiffel_Tower" title="Eiffel Tower">Eiffel Tower</a> has a geodetic latitude of 48° 51′ 29″ N, or 48.8583° N and longitude of 2° 17′ 40″ E or 2.2944°E. The same coordinates on the datum <a href="/wiki/ED50" title="ED50">ED50</a> define a point on the ground which is 140 metres (460 feet) distant from the tower.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] A web search may produce several different values for the latitude of the tower; the reference ellipsoid is rarely specified. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><h2 id="Meridian_distance">Meridian distance</h2> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=10" title="Edit section: Meridian distance" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Meridian_arc" title="Meridian arc">Meridian arc</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Length_of_a_degree_of_longitude" class="mw-redirect" title="Length of a degree of longitude">Length of a degree of longitude</a></div> The length of a degree of latitude depends on the <a href="/wiki/Figure_of_the_Earth" title="Figure of the Earth">figure of the Earth</a> assumed. <div class="mw-heading mw-heading3"><h3 id="Meridian_distance_on_the_sphere">Meridian distance on the sphere</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=11" title="Edit section: Meridian distance on the sphere" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> On the sphere the normal passes through the centre and the latitude (ϕ) is therefore equal to the angle subtended at the centre by the meridian arc from the equator to the point concerned. If the <a href="/wiki/Meridian_arc" title="Meridian arc">meridian distance</a> is denoted by m(ϕ) then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m(\phi )={\frac {\pi }{180^{\circ }}}R\phi _{\mathrm {degrees} }=R\phi _{\mathrm {radians} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mfrac> </mrow> <mi>R</mi> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>R</mi> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(\phi )={\frac {\pi }{180^{\circ }}}R\phi _{\mathrm {degrees} }=R\phi _{\mathrm {radians} }}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/508adf4208d93637637efa0bca3d57f2a6613bc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.013ex; height:4.843ex;" alt="{\displaystyle m(\phi )={\frac {\pi }{180^{\circ }}}R\phi _{\mathrm {degrees} }=R\phi _{\mathrm {radians} }}"></noscript><span class="lazy-image-placeholder" style="width: 34.013ex;height: 4.843ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/508adf4208d93637637efa0bca3d57f2a6613bc0" data-alt="{\displaystyle m(\phi )={\frac {\pi }{180^{\circ }}}R\phi _{\mathrm {degrees} }=R\phi _{\mathrm {radians} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> where R denotes the <a href="/wiki/Earth_radius#Mean_radii" title="Earth radius">mean radius</a> of the Earth. R is equal to 6,371 km or 3,959 miles. No higher accuracy is appropriate for R since higher-precision results necessitate an ellipsoid model. With this value for R the meridian length of 1 degree of latitude on the sphere is 111.2 km (69.1 statute miles) (60.0 nautical miles). The length of one minute of latitude is 1.853 km (1.151 statute miles) (1.00 nautical miles), while the length of 1 second of latitude is 30.8 m or 101 feet (see <a href="/wiki/Nautical_mile" title="Nautical mile">nautical mile</a>). <div class="mw-heading mw-heading3"><h3 id="Meridian_distance_on_the_ellipsoid">Meridian distance on the ellipsoid</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=12" title="Edit section: Meridian distance on the ellipsoid" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> In <a href="/wiki/Meridian_arc" title="Meridian arc">Meridian arc</a> and standard texts<a href="#cite_note-torge-6">[5]</a><a href="#cite_note-osborne-7">[6]</a><a href="#cite_note-rapp-8">[7]</a> it is shown that the distance along a meridian from latitude ϕ to the equator is given by (ϕ in radians) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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 '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msubsup> <mi>M</mi> <mo stretchy="false">(</mo> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 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 '}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/105b6ed018f6f75a9118e3b7cb02309733fd4e0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:61.539ex; height:6.343ex;" alt="{\displaystyle 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 '}"></noscript><span class="lazy-image-placeholder" style="width: 61.539ex;height: 6.343ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/105b6ed018f6f75a9118e3b7cb02309733fd4e0d" data-alt="{\displaystyle 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 '}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> where M(ϕ) is the meridional <a href="/wiki/Radius_of_curvature_(applications)" class="mw-redirect" title="Radius of curvature (applications)">radius of curvature</a>. The <a href="/wiki/Quarter_meridian" class="mw-redirect" title="Quarter meridian">quarter meridian</a> distance from the equator to the pole is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\mathrm {p} }=m\left({\frac {\pi }{2}}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\mathrm {p} }=m\left({\frac {\pi }{2}}\right)\,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f09cc186defc619dca86a937548c4a0de7d00279" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.043ex; height:4.843ex;" alt="{\displaystyle m_{\mathrm {p} }=m\left({\frac {\pi }{2}}\right)\,}"></noscript><span class="lazy-image-placeholder" style="width: 14.043ex;height: 4.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f09cc186defc619dca86a937548c4a0de7d00279" data-alt="{\displaystyle m_{\mathrm {p} }=m\left({\frac {\pi }{2}}\right)\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> For <a href="/wiki/WGS84" class="mw-redirect" title="WGS84">WGS84</a> this distance is 10001.965729 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 <a href="/wiki/Meridian_arc" title="Meridian arc">Meridian arc</a> 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<a href="#cite_note-osborne-7">[6]</a><a href="#cite_note-rapp-8">[7]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>δ</mi> <mi>ϕ</mi> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>δ</mi> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \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 }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9722821ccc0c337366b9ab1d994c3bf2fd72827" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:51.389ex; height:4.509ex;" alt="{\displaystyle \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 }"></noscript><span class="lazy-image-placeholder" style="width: 51.389ex;height: 4.509ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9722821ccc0c337366b9ab1d994c3bf2fd72827" data-alt="{\displaystyle \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 }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> <table class="wikitable" style="margin:1em auto 1em auto;float:right;clear:right;"> <tbody><tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></noscript><span class="lazy-image-placeholder" style="width: 1.385ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" data-alt="{\displaystyle \phi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th> <th>Δ1 lat</th> <th>Δ1 long </th></tr> <tr style="text-align:right;"> <td>0°</td> <td>110.574 km</td> <td>111.320 km </td></tr> <tr style="text-align:right;"> <td>15°</td> <td>110.649 km</td> <td>107.550 km </td></tr> <tr style="text-align:right;"> <td>30°</td> <td>110.852 km</td> <td>96.486 km </td></tr> <tr style="text-align:right;"> <td>45°</td> <td>111.132 km</td> <td>78.847 km </td></tr> <tr style="text-align:right;"> <td>60°</td> <td>111.412 km</td> <td>55.800 km </td></tr> <tr style="text-align:right;"> <td>75°</td> <td>111.618 km</td> <td>28.902 km </td></tr> <tr style="text-align:right;"> <td>90°</td> <td>111.694 km</td> <td>0.000 km </td></tr></tbody></table> When the latitude difference is 1 degree, corresponding to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/180⁠ radians, the arc distance is about <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{\text{lat}}^{1}={\frac {\pi a\left(1-e^{2}\right)}{180^{\circ }\left(1-e^{2}\sin ^{2}\phi \right)^{\frac {3}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>lat</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{\text{lat}}^{1}={\frac {\pi a\left(1-e^{2}\right)}{180^{\circ }\left(1-e^{2}\sin ^{2}\phi \right)^{\frac {3}{2}}}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23040206d0511e78a15b4d5b540bab2063e2c393" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:28.639ex; height:8.509ex;" alt="{\displaystyle \Delta _{\text{lat}}^{1}={\frac {\pi a\left(1-e^{2}\right)}{180^{\circ }\left(1-e^{2}\sin ^{2}\phi \right)^{\frac {3}{2}}}}}"></noscript><span class="lazy-image-placeholder" style="width: 28.639ex;height: 8.509ex;vertical-align: -4.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23040206d0511e78a15b4d5b540bab2063e2c393" data-alt="{\displaystyle \Delta _{\text{lat}}^{1}={\frac {\pi a\left(1-e^{2}\right)}{180^{\circ }\left(1-e^{2}\sin ^{2}\phi \right)^{\frac {3}{2}}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> The distance in metres (correct to 0.01 metre) between latitudes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></noscript><span class="lazy-image-placeholder" style="width: 1.385ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" data-alt="{\displaystyle \phi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  − 0.5 degrees and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></noscript><span class="lazy-image-placeholder" style="width: 1.385ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" data-alt="{\displaystyle \phi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  + 0.5 degrees on the WGS84 spheroid is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{\text{lat}}^{1}=111\,132.954-559.822\cos 2\phi +1.175\cos 4\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>lat</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mn>111</mn> <mspace width="thinmathspace"></mspace> <mn>132.954</mn> <mo>−</mo> <mn>559.822</mn> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>ϕ</mi> <mo>+</mo> <mn>1.175</mn> <mi>cos</mi> <mo>⁡</mo> <mn>4</mn> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{\text{lat}}^{1}=111\,132.954-559.822\cos 2\phi +1.175\cos 4\phi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c1269b3f1e5227960b5981f7f89b63bdd51a95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:50.147ex; height:3.176ex;" alt="{\displaystyle \Delta _{\text{lat}}^{1}=111\,132.954-559.822\cos 2\phi +1.175\cos 4\phi }"></noscript><span class="lazy-image-placeholder" style="width: 50.147ex;height: 3.176ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c1269b3f1e5227960b5981f7f89b63bdd51a95" data-alt="{\displaystyle \Delta _{\text{lat}}^{1}=111\,132.954-559.822\cos 2\phi +1.175\cos 4\phi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> The variation of this distance with latitude (on <a href="/wiki/WGS84" class="mw-redirect" title="WGS84">WGS84</a>) is shown in the table along with the <a href="/wiki/Length_of_a_degree_of_longitude" class="mw-redirect" title="Length of a degree of longitude">length of a degree of longitude</a> (east–west distance): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{\text{long}}^{1}={\frac {\pi a\cos \phi }{180^{\circ }{\sqrt {1-e^{2}\sin ^{2}\phi }}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>long</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mrow> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{\text{long}}^{1}={\frac {\pi a\cos \phi }{180^{\circ }{\sqrt {1-e^{2}\sin ^{2}\phi }}}}\,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc4eef2001bf30bead7f2674bec10a21f89710d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:28.581ex; height:8.176ex;" alt="{\displaystyle \Delta _{\text{long}}^{1}={\frac {\pi a\cos \phi }{180^{\circ }{\sqrt {1-e^{2}\sin ^{2}\phi }}}}\,}"></noscript><span class="lazy-image-placeholder" style="width: 28.581ex;height: 8.176ex;vertical-align: -4.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc4eef2001bf30bead7f2674bec10a21f89710d2" data-alt="{\displaystyle \Delta _{\text{long}}^{1}={\frac {\pi a\cos \phi }{180^{\circ }{\sqrt {1-e^{2}\sin ^{2}\phi }}}}\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> A calculator for any latitude is provided by the U.S. Government's <a href="/wiki/National_Geospatial-Intelligence_Agency" title="National Geospatial-Intelligence Agency">National Geospatial-Intelligence Agency</a> (NGA).<a href="#cite_note-9">[8]</a> The following graph illustrates the variation of both a degree of latitude and a degree of longitude with latitude. <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:WGS84_angle_to_distance_conversion.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/WGS84_angle_to_distance_conversion.svg/410px-WGS84_angle_to_distance_conversion.svg.png" decoding="async" width="410" height="328" class="mw-file-element" data-file-width="512" data-file-height="410"></noscript><span class="lazy-image-placeholder" style="width: 410px;height: 328px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/WGS84_angle_to_distance_conversion.svg/410px-WGS84_angle_to_distance_conversion.svg.png" data-width="410" data-height="328" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/WGS84_angle_to_distance_conversion.svg/615px-WGS84_angle_to_distance_conversion.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/WGS84_angle_to_distance_conversion.svg/820px-WGS84_angle_to_distance_conversion.svg.png 2x" data-class="mw-file-element"> </a><figcaption>The definition of geodetic latitude (ϕ) and geocentric latitude (θ).</figcaption></figure> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><h2 id="Auxiliary_latitudes">Auxiliary latitudes</h2> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=13" title="Edit section: Auxiliary latitudes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> There are six auxiliary latitudes that have applications to special problems in geodesy, geophysics and the theory of map projections: <ul><li><a href="/wiki/Geocentric_latitude" class="mw-redirect" title="Geocentric latitude">Geocentric latitude</a></li> <li>Parametric (or reduced) latitude</li> <li>Rectifying latitude</li> <li><a href="/wiki/Authalic_latitude" class="mw-redirect" title="Authalic latitude">Authalic latitude</a></li> <li>Conformal latitude</li> <li>Isometric latitude</li></ul> The definitions given in this section all relate to locations on the reference ellipsoid but the first two auxiliary latitudes, like the geodetic latitude, can be extended to define a three-dimensional <a href="/wiki/Geographic_coordinate_system" title="Geographic coordinate system">geographic coordinate system</a> as discussed <a href="#Latitude_and_coordinate_systems">below</a>. The remaining latitudes are not used in this way; they are used only as intermediate constructs in map projections of the reference ellipsoid to the plane or in calculations of geodesics on the ellipsoid. Their numerical values are not of interest. For example, no one would need to calculate the authalic latitude of the Eiffel Tower. The expressions below give the auxiliary latitudes in terms of the geodetic latitude, the semi-major axis, a, and the eccentricity, e. (For inverses see <a href="#Inverse_formulae_and_series">below</a>.) The forms given are, apart from notational variants, those in the standard reference for map projections, namely "Map projections: a working manual" by J. P. Snyder.<a href="#cite_note-snyder-10">[9]</a> Derivations of these expressions may be found in Adams<a href="#cite_note-adams1921-11">[10]</a> and online publications by Osborne<a href="#cite_note-osborne-7">[6]</a> and Rapp.<a href="#cite_note-rapp-8">[7]</a> <div class="mw-heading mw-heading3"><h3 id="Geocentric_latitude">Geocentric latitude</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=14" title="Edit section: Geocentric latitude" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="#Geodetic_and_geocentric_latitudes">§ Geodetic and geocentric latitudes</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Geocentric_coords_03.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Geocentric_coords_03.svg/250px-Geocentric_coords_03.svg.png" decoding="async" width="250" height="207" class="mw-file-element" data-file-width="316" data-file-height="262"></noscript><span class="lazy-image-placeholder" style="width: 250px;height: 207px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Geocentric_coords_03.svg/250px-Geocentric_coords_03.svg.png" data-width="250" data-height="207" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Geocentric_coords_03.svg/375px-Geocentric_coords_03.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Geocentric_coords_03.svg/500px-Geocentric_coords_03.svg.png 2x" data-class="mw-file-element"> </a><figcaption>The definition of geodetic latitude (ϕ) and geocentric latitude (θ)</figcaption></figure> The geocentric latitude is the angle between the equatorial plane and the radius from the centre to a point of interest. When the point is on the surface of the ellipsoid, the relation between the geocentric latitude (θ) and the geodetic latitude (ϕ) is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta (\phi )=\tan ^{-1}\left(\left(1-e^{2}\right)\tan \phi \right)=\tan ^{-1}\left((1-f)^{2}\tan \phi \right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>tan</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>tan</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta (\phi )=\tan ^{-1}\left(\left(1-e^{2}\right)\tan \phi \right)=\tan ^{-1}\left((1-f)^{2}\tan \phi \right)\,.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be12bdcfb52b31afe5eb175478a51743a38ce0f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:54.614ex; height:3.343ex;" alt="{\displaystyle \theta (\phi )=\tan ^{-1}\left(\left(1-e^{2}\right)\tan \phi \right)=\tan ^{-1}\left((1-f)^{2}\tan \phi \right)\,.}"></noscript><span class="lazy-image-placeholder" style="width: 54.614ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be12bdcfb52b31afe5eb175478a51743a38ce0f5" data-alt="{\displaystyle \theta (\phi )=\tan ^{-1}\left(\left(1-e^{2}\right)\tan \phi \right)=\tan ^{-1}\left((1-f)^{2}\tan \phi \right)\,.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> For points not on the surface of the ellipsoid, the relationship involves additionally the <a href="/wiki/Ellipsoidal_height" class="mw-redirect" title="Ellipsoidal height">ellipsoidal height</a> h: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta (\phi ,h)=\tan ^{-1}\left({\frac {N(1-f)^{2}+h}{N+h}}\tan \phi \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>h</mi> </mrow> <mrow> <mi>N</mi> <mo>+</mo> <mi>h</mi> </mrow> </mfrac> </mrow> <mi>tan</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta (\phi ,h)=\tan ^{-1}\left({\frac {N(1-f)^{2}+h}{N+h}}\tan \phi \right)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c91758246aec3c0613c68412150d7d776ded5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.874ex; height:7.509ex;" alt="{\displaystyle \theta (\phi ,h)=\tan ^{-1}\left({\frac {N(1-f)^{2}+h}{N+h}}\tan \phi \right)}"></noscript><span class="lazy-image-placeholder" style="width: 39.874ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c91758246aec3c0613c68412150d7d776ded5d" data-alt="{\displaystyle \theta (\phi ,h)=\tan ^{-1}\left({\frac {N(1-f)^{2}+h}{N+h}}\tan \phi \right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> where N is the prime vertical radius of curvature. The geodetic and geocentric latitudes are equal at the equator and at the poles but at other latitudes they differ by a few minutes of arc. Taking the value of the squared eccentricity as 0.0067 (it depends on the choice of ellipsoid) the maximum difference of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi {-}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi {-}\theta }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fad050a806bb61857c84d009cfc72a3fec9ec8b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.284ex; height:2.509ex;" alt="{\displaystyle \phi {-}\theta }"></noscript><span class="lazy-image-placeholder" style="width: 4.284ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fad050a806bb61857c84d009cfc72a3fec9ec8b4" data-alt="{\displaystyle \phi {-}\theta }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  may be shown to be about 11.5 minutes of arc at a geodetic latitude of approximately 45° 6′.<a href="#cite_note-12">[b]</a> <div class="mw-heading mw-heading3"><h3 id="Parametric_latitude_(or_reduced_latitude)">Parametric latitude (or reduced latitude)</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=15" title="Edit section: Parametric latitude (or reduced latitude)" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Ellipsoid_reduced_angle_definition.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Ellipsoid_reduced_angle_definition.svg/200px-Ellipsoid_reduced_angle_definition.svg.png" decoding="async" width="200" height="195" class="mw-file-element" data-file-width="330" data-file-height="321"></noscript><span class="lazy-image-placeholder" style="width: 200px;height: 195px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Ellipsoid_reduced_angle_definition.svg/200px-Ellipsoid_reduced_angle_definition.svg.png" data-width="200" data-height="195" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Ellipsoid_reduced_angle_definition.svg/300px-Ellipsoid_reduced_angle_definition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Ellipsoid_reduced_angle_definition.svg/400px-Ellipsoid_reduced_angle_definition.svg.png 2x" data-class="mw-file-element"> </a><figcaption>Definition of the parametric latitude (β) on the ellipsoid</figcaption></figure> The parametric latitude or reduced latitude, β, is defined by the radius drawn from the centre of the ellipsoid to that point Q on the surrounding sphere (of radius a) which is the projection parallel to the Earth's axis of a point P on the ellipsoid at latitude ϕ. It was introduced by Legendre<a href="#cite_note-legendre-13">[11]</a> and Bessel<a href="#cite_note-bessel-14">[12]</a> who solved problems for geodesics on the ellipsoid by transforming them to an equivalent problem for spherical geodesics by using this smaller latitude. Bessel's notation, u(ϕ), is also used in the current literature. The parametric latitude is related to the geodetic latitude by:<a href="#cite_note-osborne-7">[6]</a><a href="#cite_note-rapp-8">[7]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta (\phi )=\tan ^{-1}\left({\sqrt {1-e^{2}}}\tan \phi \right)=\tan ^{-1}\left((1-f)\tan \phi \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mi>tan</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mi>tan</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta (\phi )=\tan ^{-1}\left({\sqrt {1-e^{2}}}\tan \phi \right)=\tan ^{-1}\left((1-f)\tan \phi \right)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73a42a66b39844ce024d63ac0ffeb3c4ac3b08d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:53.287ex; height:4.843ex;" alt="{\displaystyle \beta (\phi )=\tan ^{-1}\left({\sqrt {1-e^{2}}}\tan \phi \right)=\tan ^{-1}\left((1-f)\tan \phi \right)}"></noscript><span class="lazy-image-placeholder" style="width: 53.287ex;height: 4.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73a42a66b39844ce024d63ac0ffeb3c4ac3b08d4" data-alt="{\displaystyle \beta (\phi )=\tan ^{-1}\left({\sqrt {1-e^{2}}}\tan \phi \right)=\tan ^{-1}\left((1-f)\tan \phi \right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> The alternative name arises from the parameterization of the equation of the ellipse describing a meridian section. In terms of Cartesian coordinates p, the distance from the minor axis, and z, the distance above the equatorial plane, the equation of the <a href="/wiki/Ellipse" title="Ellipse">ellipse</a> is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {p^{2}}{a^{2}}}+{\frac {z^{2}}{b^{2}}}=1\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p^{2}}{a^{2}}}+{\frac {z^{2}}{b^{2}}}=1\,.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b55d30cf33f6eccc1558881edcfd2dfe423c5dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.236ex; height:6.009ex;" alt="{\displaystyle {\frac {p^{2}}{a^{2}}}+{\frac {z^{2}}{b^{2}}}=1\,.}"></noscript><span class="lazy-image-placeholder" style="width: 14.236ex;height: 6.009ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b55d30cf33f6eccc1558881edcfd2dfe423c5dd" data-alt="{\displaystyle {\frac {p^{2}}{a^{2}}}+{\frac {z^{2}}{b^{2}}}=1\,.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> The Cartesian coordinates of the point are parameterized by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=a\cos \beta \,,\qquad z=b\sin \beta \,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mspace width="thinmathspace"></mspace> <mo>,</mo> <mspace width="2em"></mspace> <mi>z</mi> <mo>=</mo> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> <mspace width="thinmathspace"></mspace> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=a\cos \beta \,,\qquad z=b\sin \beta \,;}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab1ea1af54648e83df7ac4fba6091bbc9bc65b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:28.051ex; height:2.509ex;" alt="{\displaystyle p=a\cos \beta \,,\qquad z=b\sin \beta \,;}"></noscript><span class="lazy-image-placeholder" style="width: 28.051ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab1ea1af54648e83df7ac4fba6091bbc9bc65b34" data-alt="{\displaystyle p=a\cos \beta \,,\qquad z=b\sin \beta \,;}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> Cayley suggested the term parametric latitude because of the form of these equations.<a href="#cite_note-cayley-15">[13]</a> The parametric latitude is not used in the theory of map projections. Its most important application is in the theory of ellipsoid geodesics, (<a href="/wiki/Vincenty%27s_formulae" title="Vincenty's formulae">Vincenty</a>, Karney<a href="#cite_note-Karney-16">[14]</a>). <div class="mw-heading mw-heading3"><h3 id="Rectifying_latitude">Rectifying latitude</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=16" title="Edit section: Rectifying latitude" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Rectifying_radius" class="mw-redirect" title="Rectifying radius">Rectifying radius</a></div> The rectifying latitude, μ, is the meridian distance scaled so that its value at the poles is equal to 90 degrees or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/2⁠ radians: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (\phi )={\frac {\pi }{2}}{\frac {m(\phi )}{m_{\mathrm {p} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (\phi )={\frac {\pi }{2}}{\frac {m(\phi )}{m_{\mathrm {p} }}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/349ea19e122303e5b69ddfd8be93244b25788c3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.934ex; height:6.343ex;" alt="{\displaystyle \mu (\phi )={\frac {\pi }{2}}{\frac {m(\phi )}{m_{\mathrm {p} }}}}"></noscript><span class="lazy-image-placeholder" style="width: 15.934ex;height: 6.343ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/349ea19e122303e5b69ddfd8be93244b25788c3e" data-alt="{\displaystyle \mu (\phi )={\frac {\pi }{2}}{\frac {m(\phi )}{m_{\mathrm {p} }}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> where the meridian distance from the equator to a latitude ϕ is (see <a href="/wiki/Meridian_arc" title="Meridian arc">Meridian arc</a>) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m(\phi )=a\left(1-e^{2}\right)\int _{0}^{\phi }\left(1-e^{2}\sin ^{2}\phi '\right)^{-{\frac {3}{2}}}\,d\phi '\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> <mspace width="thinmathspace"></mspace> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(\phi )=a\left(1-e^{2}\right)\int _{0}^{\phi }\left(1-e^{2}\sin ^{2}\phi '\right)^{-{\frac {3}{2}}}\,d\phi '\,,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d68ac2a2a87186fe170c3413ad1d6b214ad387c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.417ex; height:6.343ex;" alt="{\displaystyle m(\phi )=a\left(1-e^{2}\right)\int _{0}^{\phi }\left(1-e^{2}\sin ^{2}\phi '\right)^{-{\frac {3}{2}}}\,d\phi '\,,}"></noscript><span class="lazy-image-placeholder" style="width: 45.417ex;height: 6.343ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d68ac2a2a87186fe170c3413ad1d6b214ad387c" data-alt="{\displaystyle m(\phi )=a\left(1-e^{2}\right)\int _{0}^{\phi }\left(1-e^{2}\sin ^{2}\phi '\right)^{-{\frac {3}{2}}}\,d\phi '\,,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> and the length of the meridian quadrant from the equator to the pole (the <a href="/wiki/Meridian_arc#Polar_distance" title="Meridian arc">polar distance</a>) is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\mathrm {p} }=m\left({\frac {\pi }{2}}\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\mathrm {p} }=m\left({\frac {\pi }{2}}\right)\,.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feaed75d16edd488d2ae9b3fcf3d164198d1d4fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.077ex; height:4.843ex;" alt="{\displaystyle m_{\mathrm {p} }=m\left({\frac {\pi }{2}}\right)\,.}"></noscript><span class="lazy-image-placeholder" style="width: 15.077ex;height: 4.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feaed75d16edd488d2ae9b3fcf3d164198d1d4fc" data-alt="{\displaystyle m_{\mathrm {p} }=m\left({\frac {\pi }{2}}\right)\,.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> Using the rectifying latitude to define a latitude on a sphere of radius <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R={\frac {2m_{\mathrm {p} }}{\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> <mi>π</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\frac {2m_{\mathrm {p} }}{\pi }}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3c5625835b98da55e6ea02beb168219a8abef06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.048ex; height:5.509ex;" alt="{\displaystyle R={\frac {2m_{\mathrm {p} }}{\pi }}}"></noscript><span class="lazy-image-placeholder" style="width: 10.048ex;height: 5.509ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3c5625835b98da55e6ea02beb168219a8abef06" data-alt="{\displaystyle R={\frac {2m_{\mathrm {p} }}{\pi }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> defines a projection from the ellipsoid to the sphere such that all meridians have true length and uniform scale. The sphere may then be projected to the plane with an <a href="/wiki/Equirectangular_projection" title="Equirectangular projection">equirectangular projection</a> to give a double projection from the ellipsoid to the plane such that all meridians have true length and uniform meridian scale. An example of the use of the rectifying latitude is the <a href="/wiki/Equidistant_conic_projection" title="Equidistant conic projection">equidistant conic projection</a>. (Snyder, Section 16).<a href="#cite_note-snyder-10">[9]</a> The rectifying latitude is also of great importance in the construction of the <a href="/wiki/Transverse_Mercator_projection" title="Transverse Mercator projection">Transverse Mercator projection</a>. <div class="mw-heading mw-heading3"><h3 id="Authalic_latitude">Authalic latitude</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=17" title="Edit section: Authalic latitude" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Authalic_radius" class="mw-redirect" title="Authalic radius">Authalic radius</a></div> The authalic latitude (after the Greek for "<a href="https://en.wiktionary.org/wiki/authalic" class="extiw" title="wiktionary:authalic">same area</a>"), ξ, gives an <a href="/wiki/Equal-area_projection" title="Equal-area projection">equal-area projection</a> to a sphere. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi (\phi )=\sin ^{-1}\left({\frac {q(\phi )}{q_{\mathrm {p} }}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi (\phi )=\sin ^{-1}\left({\frac {q(\phi )}{q_{\mathrm {p} }}}\right)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc893a2c538edce543f6f2c8e4f8eed929ea83d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.033ex; height:6.343ex;" alt="{\displaystyle \xi (\phi )=\sin ^{-1}\left({\frac {q(\phi )}{q_{\mathrm {p} }}}\right)}"></noscript><span class="lazy-image-placeholder" style="width: 21.033ex;height: 6.343ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc893a2c538edce543f6f2c8e4f8eed929ea83d5" data-alt="{\displaystyle \xi (\phi )=\sin ^{-1}\left({\frac {q(\phi )}{q_{\mathrm {p} }}}\right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> where <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}q(\phi )&={\frac {\left(1-e^{2}\right)\sin \phi }{1-e^{2}\sin ^{2}\phi }}-{\frac {1-e^{2}}{2e}}\ln \left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)\\[2pt]&={\frac {\left(1-e^{2}\right)\sin \phi }{1-e^{2}\sin ^{2}\phi }}+{\frac {1-e^{2}}{e}}\tanh ^{-1}(e\sin \phi )\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>q</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>e</mi> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>e</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>e</mi> </mfrac> </mrow> <msup> <mi>tanh</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>e</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}q(\phi )&={\frac {\left(1-e^{2}\right)\sin \phi }{1-e^{2}\sin ^{2}\phi }}-{\frac {1-e^{2}}{2e}}\ln \left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)\\[2pt]&={\frac {\left(1-e^{2}\right)\sin \phi }{1-e^{2}\sin ^{2}\phi }}+{\frac {1-e^{2}}{e}}\tanh ^{-1}(e\sin \phi )\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b256b2b2f95ca49231dfb329dd724cdbd2de7d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:48.738ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}q(\phi )&={\frac {\left(1-e^{2}\right)\sin \phi }{1-e^{2}\sin ^{2}\phi }}-{\frac {1-e^{2}}{2e}}\ln \left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)\\[2pt]&={\frac {\left(1-e^{2}\right)\sin \phi }{1-e^{2}\sin ^{2}\phi }}+{\frac {1-e^{2}}{e}}\tanh ^{-1}(e\sin \phi )\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 48.738ex;height: 14.843ex;vertical-align: -6.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b256b2b2f95ca49231dfb329dd724cdbd2de7d1" data-alt="{\displaystyle {\begin{aligned}q(\phi )&={\frac {\left(1-e^{2}\right)\sin \phi }{1-e^{2}\sin ^{2}\phi }}-{\frac {1-e^{2}}{2e}}\ln \left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)\\[2pt]&={\frac {\left(1-e^{2}\right)\sin \phi }{1-e^{2}\sin ^{2}\phi }}+{\frac {1-e^{2}}{e}}\tanh ^{-1}(e\sin \phi )\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}q_{\mathrm {p} }=q\left({\frac {\pi }{2}}\right)&=1-{\frac {1-e^{2}}{2e}}\ln \left({\frac {1-e}{1+e}}\right)\\&=1+{\frac {1-e^{2}}{e}}\tanh ^{-1}e\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>q</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>e</mi> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>e</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>e</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>e</mi> </mfrac> </mrow> <msup> <mi>tanh</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>e</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}q_{\mathrm {p} }=q\left({\frac {\pi }{2}}\right)&=1-{\frac {1-e^{2}}{2e}}\ln \left({\frac {1-e}{1+e}}\right)\\&=1+{\frac {1-e^{2}}{e}}\tanh ^{-1}e\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7447963fa3c0dd43284a844f87ac2ae8732810b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:38.182ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}q_{\mathrm {p} }=q\left({\frac {\pi }{2}}\right)&=1-{\frac {1-e^{2}}{2e}}\ln \left({\frac {1-e}{1+e}}\right)\\&=1+{\frac {1-e^{2}}{e}}\tanh ^{-1}e\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 38.182ex;height: 12.176ex;vertical-align: -5.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7447963fa3c0dd43284a844f87ac2ae8732810b2" data-alt="{\displaystyle {\begin{aligned}q_{\mathrm {p} }=q\left({\frac {\pi }{2}}\right)&=1-{\frac {1-e^{2}}{2e}}\ln \left({\frac {1-e}{1+e}}\right)\\&=1+{\frac {1-e^{2}}{e}}\tanh ^{-1}e\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> and the radius of the sphere is taken as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{q}=a{\sqrt {\frac {q_{\mathrm {p} }}{2}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mn>2</mn> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{q}=a{\sqrt {\frac {q_{\mathrm {p} }}{2}}}\,.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/410e3824d4063dc2ceef3089f832b3f559284a28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.458ex; height:6.343ex;" alt="{\displaystyle R_{q}=a{\sqrt {\frac {q_{\mathrm {p} }}{2}}}\,.}"></noscript><span class="lazy-image-placeholder" style="width: 13.458ex;height: 6.343ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/410e3824d4063dc2ceef3089f832b3f559284a28" data-alt="{\displaystyle R_{q}=a{\sqrt {\frac {q_{\mathrm {p} }}{2}}}\,.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> An example of the use of the authalic latitude is the <a href="/wiki/Albers_projection" title="Albers projection">Albers equal-area conic projection</a>.<a href="#cite_note-snyder-10">[9]</a>: §14 <div class="mw-heading mw-heading3"><h3 id="Conformal_latitude">Conformal latitude</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=18" title="Edit section: Conformal latitude" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The conformal latitude, χ, gives an angle-preserving (<a href="/wiki/Conformal_map" title="Conformal map">conformal</a>) transformation to the sphere. <a href="#cite_note-17">[15]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\chi (\phi )&=2\tan ^{-1}\left[\left({\frac {1+\sin \phi }{1-\sin \phi }}\right)\left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)^{e}\right]^{\frac {1}{2}}-{\frac {\pi }{2}}\\[2pt]&=2\tan ^{-1}\left[\tan \left({\frac {\phi }{2}}+{\frac {\pi }{4}}\right)\left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)^{\frac {e}{2}}\right]-{\frac {\pi }{2}}\\[2pt]&=\tan ^{-1}\left[\sinh \left(\sinh ^{-1}(\tan \phi )-e\tanh ^{-1}(e\sin \phi )\right)\right]\\&=\operatorname {gd} \left[\operatorname {gd} ^{-1}(\phi )-e\tanh ^{-1}(e\sin \phi )\right]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.5em 0.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <msup> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>e</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>e</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>sinh</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>tan</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>e</mi> <msup> <mi>tanh</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>e</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>gd</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>e</mi> <msup> <mi>tanh</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>e</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\chi (\phi )&=2\tan ^{-1}\left[\left({\frac {1+\sin \phi }{1-\sin \phi }}\right)\left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)^{e}\right]^{\frac {1}{2}}-{\frac {\pi }{2}}\\[2pt]&=2\tan ^{-1}\left[\tan \left({\frac {\phi }{2}}+{\frac {\pi }{4}}\right)\left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)^{\frac {e}{2}}\right]-{\frac {\pi }{2}}\\[2pt]&=\tan ^{-1}\left[\sinh \left(\sinh ^{-1}(\tan \phi )-e\tanh ^{-1}(e\sin \phi )\right)\right]\\&=\operatorname {gd} \left[\operatorname {gd} ^{-1}(\phi )-e\tanh ^{-1}(e\sin \phi )\right]\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea5beec16e6a78772ab9d8ffd9123b9ae6e89e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.005ex; width:55.036ex; height:23.176ex;" alt="{\displaystyle {\begin{aligned}\chi (\phi )&=2\tan ^{-1}\left[\left({\frac {1+\sin \phi }{1-\sin \phi }}\right)\left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)^{e}\right]^{\frac {1}{2}}-{\frac {\pi }{2}}\\[2pt]&=2\tan ^{-1}\left[\tan \left({\frac {\phi }{2}}+{\frac {\pi }{4}}\right)\left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)^{\frac {e}{2}}\right]-{\frac {\pi }{2}}\\[2pt]&=\tan ^{-1}\left[\sinh \left(\sinh ^{-1}(\tan \phi )-e\tanh ^{-1}(e\sin \phi )\right)\right]\\&=\operatorname {gd} \left[\operatorname {gd} ^{-1}(\phi )-e\tanh ^{-1}(e\sin \phi )\right]\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 55.036ex;height: 23.176ex;vertical-align: -11.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea5beec16e6a78772ab9d8ffd9123b9ae6e89e0" data-alt="{\displaystyle {\begin{aligned}\chi (\phi )&=2\tan ^{-1}\left[\left({\frac {1+\sin \phi }{1-\sin \phi }}\right)\left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)^{e}\right]^{\frac {1}{2}}-{\frac {\pi }{2}}\\[2pt]&=2\tan ^{-1}\left[\tan \left({\frac {\phi }{2}}+{\frac {\pi }{4}}\right)\left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)^{\frac {e}{2}}\right]-{\frac {\pi }{2}}\\[2pt]&=\tan ^{-1}\left[\sinh \left(\sinh ^{-1}(\tan \phi )-e\tanh ^{-1}(e\sin \phi )\right)\right]\\&=\operatorname {gd} \left[\operatorname {gd} ^{-1}(\phi )-e\tanh ^{-1}(e\sin \phi )\right]\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> where gd(x) is the <a href="/wiki/Gudermannian_function" title="Gudermannian function">Gudermannian function</a>. (See also <a href="/wiki/Mercator_projection#Alternative_expressions" title="Mercator projection">Mercator projection</a>.) The conformal latitude defines a transformation from the ellipsoid to a sphere of arbitrary radius such that the angle of intersection between any two lines on the ellipsoid is the same as the corresponding angle on the sphere (so that the shape of small elements is well preserved). A further conformal transformation from the sphere to the plane gives a conformal double projection from the ellipsoid to the plane. This is not the only way of generating such a conformal projection. For example, the 'exact' version of the <a href="/wiki/Transverse_Mercator_projection" title="Transverse Mercator projection">Transverse Mercator projection</a> on the ellipsoid is not a double projection. (It does, however, involve a generalisation of the conformal latitude to the complex plane). <div class="mw-heading mw-heading3"><h3 id="Isometric_latitude">Isometric latitude</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=19" title="Edit section: Isometric latitude" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The isometric latitude, ψ, is used in the development of the ellipsoidal versions of the normal <a href="/wiki/Mercator_projection" title="Mercator projection">Mercator projection</a> and the <a href="/wiki/Transverse_Mercator_projection" title="Transverse Mercator projection">Transverse Mercator projection</a>. The name "isometric" arises from the fact that at any point on the ellipsoid equal increments of ψ and longitude λ give rise to equal distance displacements along the meridians and parallels respectively. The <a href="/wiki/Geographic_coordinate_system" title="Geographic coordinate system">graticule</a> defined by the lines of constant ψ and constant λ, divides the surface of the ellipsoid into a mesh of squares (of varying size). The isometric latitude is zero at the equator but rapidly diverges from the geodetic latitude, tending to infinity at the poles. The conventional notation is given in Snyder (page 15):<a href="#cite_note-snyder-10">[9]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\psi (\phi )&=\ln \left[\tan \left({\frac {\pi }{4}}+{\frac {\phi }{2}}\right)\right]+{\frac {e}{2}}\ln \left[{\frac {1-e\sin \phi }{1+e\sin \phi }}\right]\\&=\sinh ^{-1}(\tan \phi )-e\tanh ^{-1}(e\sin \phi )\\&=\operatorname {gd} ^{-1}(\phi )-e\tanh ^{-1}(e\sin \phi ).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>e</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>sinh</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>tan</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>e</mi> <msup> <mi>tanh</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>e</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>e</mi> <msup> <mi>tanh</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>e</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\psi (\phi )&=\ln \left[\tan \left({\frac {\pi }{4}}+{\frac {\phi }{2}}\right)\right]+{\frac {e}{2}}\ln \left[{\frac {1-e\sin \phi }{1+e\sin \phi }}\right]\\&=\sinh ^{-1}(\tan \phi )-e\tanh ^{-1}(e\sin \phi )\\&=\operatorname {gd} ^{-1}(\phi )-e\tanh ^{-1}(e\sin \phi ).\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2999bd8461a831ee601ecad69062c5b30adec5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:47.521ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\psi (\phi )&=\ln \left[\tan \left({\frac {\pi }{4}}+{\frac {\phi }{2}}\right)\right]+{\frac {e}{2}}\ln \left[{\frac {1-e\sin \phi }{1+e\sin \phi }}\right]\\&=\sinh ^{-1}(\tan \phi )-e\tanh ^{-1}(e\sin \phi )\\&=\operatorname {gd} ^{-1}(\phi )-e\tanh ^{-1}(e\sin \phi ).\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 47.521ex;height: 12.843ex;vertical-align: -5.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2999bd8461a831ee601ecad69062c5b30adec5f" data-alt="{\displaystyle {\begin{aligned}\psi (\phi )&=\ln \left[\tan \left({\frac {\pi }{4}}+{\frac {\phi }{2}}\right)\right]+{\frac {e}{2}}\ln \left[{\frac {1-e\sin \phi }{1+e\sin \phi }}\right]\\&=\sinh ^{-1}(\tan \phi )-e\tanh ^{-1}(e\sin \phi )\\&=\operatorname {gd} ^{-1}(\phi )-e\tanh ^{-1}(e\sin \phi ).\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> For the normal Mercator projection (on the ellipsoid) this function defines the spacing of the parallels: if the length of the equator on the projection is E (units of length or pixels) then the distance, y, of a parallel of latitude ϕ from the equator is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(\phi )={\frac {E}{2\pi }}\psi (\phi )\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(\phi )={\frac {E}{2\pi }}\psi (\phi )\,.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c3c8fdb803b473efe5685cc97cdcfe6622aa472" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.521ex; height:5.176ex;" alt="{\displaystyle y(\phi )={\frac {E}{2\pi }}\psi (\phi )\,.}"></noscript><span class="lazy-image-placeholder" style="width: 16.521ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c3c8fdb803b473efe5685cc97cdcfe6622aa472" data-alt="{\displaystyle y(\phi )={\frac {E}{2\pi }}\psi (\phi )\,.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> The isometric latitude ψ is closely related to the conformal latitude χ: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\phi )=\operatorname {gd} ^{-1}\chi (\phi )\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>gd</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\phi )=\operatorname {gd} ^{-1}\chi (\phi )\,.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eaf60be567d5882f6b8a5b07f11fc3aafa2c157" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.665ex; height:3.176ex;" alt="{\displaystyle \psi (\phi )=\operatorname {gd} ^{-1}\chi (\phi )\,.}"></noscript><span class="lazy-image-placeholder" style="width: 18.665ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eaf60be567d5882f6b8a5b07f11fc3aafa2c157" data-alt="{\displaystyle \psi (\phi )=\operatorname {gd} ^{-1}\chi (\phi )\,.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> <div class="mw-heading mw-heading3"><h3 id="Inverse_formulae_and_series">Inverse formulae and series</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=20" title="Edit section: Inverse formulae and series" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The formulae in the previous sections give the auxiliary latitude in terms of the geodetic latitude. The expressions for the geocentric and parametric latitudes may be inverted directly but this is impossible in the four remaining cases: the rectifying, authalic, conformal, and isometric latitudes. There are two methods of proceeding. <ul><li>The first is a numerical inversion of the defining equation for each and every particular value of the auxiliary latitude. The methods available are <a href="/wiki/Fixed-point_iteration" title="Fixed-point iteration">fixed-point iteration</a> and <a href="/wiki/Newton%27s_method" title="Newton's method">Newton–Raphson</a> root finding. <ul><li>When converting from isometric or conformal to geodetic, two iterations of Newton-Raphson gives <a href="/wiki/Double_precision" class="mw-redirect" title="Double precision">double precision</a> accuracy.<a href="#cite_note-18">[16]</a></li></ul></li> <li>The other, more useful, approach is to express the auxiliary latitude as a series in terms of the geodetic latitude and then invert the series by the method of <a href="/wiki/Lagrange_reversion" class="mw-redirect" title="Lagrange reversion">Lagrange reversion</a>. Such series are presented by Adams who uses Taylor series expansions and gives coefficients in terms of the eccentricity.<a href="#cite_note-adams1921-11">[10]</a> Orihuela<a href="#cite_note-19">[17]</a> gives series for the conversions between all pairs of auxiliary latitudes in terms of the third flattening, n = (a - b)/(a + b). Karney<a href="#cite_note-20">[18]</a> establishes that the truncation errors for such series are consistently smaller that the equivalent series in terms of the eccentricity. The series method is not applicable to the isometric latitude and one must find the conformal latitude in an intermediate step.<a href="#cite_note-osborne-7">[6]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Numerical_comparison_of_auxiliary_latitudes">Numerical comparison of auxiliary latitudes</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=21" title="Edit section: Numerical comparison of auxiliary latitudes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/File:Auxiliary_Latitudes_Difference.svg" class="mw-file-description" title="inline"><noscript><img alt="inline" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Auxiliary_Latitudes_Difference.svg/540px-Auxiliary_Latitudes_Difference.svg.png" decoding="async" width="540" height="360" class="mw-file-element" data-file-width="540" data-file-height="360"></noscript><span class="lazy-image-placeholder" style="width: 540px;height: 360px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Auxiliary_Latitudes_Difference.svg/540px-Auxiliary_Latitudes_Difference.svg.png" data-alt="inline" data-width="540" data-height="360" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Auxiliary_Latitudes_Difference.svg/810px-Auxiliary_Latitudes_Difference.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Auxiliary_Latitudes_Difference.svg/1080px-Auxiliary_Latitudes_Difference.svg.png 2x" data-class="mw-file-element"> </a><figcaption>inline</figcaption></figure> The plot to the right shows the difference between the geodetic latitude and the auxiliary latitudes other than the isometric latitude (which diverges to infinity at the poles) for the case of the WGS84 ellipsoid. The differences shown on the plot are in arc minutes. In the Northern hemisphere (positive latitudes), θ ≤ χ ≤ μ ≤ ξ ≤ β ≤ ϕ; in the Southern hemisphere (negative latitudes), the inequalities are reversed, with equality at the equator and the poles. Although the graph appears symmetric about 45°, the minima of the curves actually lie between 45° 2′ and 45° 6′. Some representative data points are given in the table below. The conformal and geocentric latitudes are nearly indistinguishable, a fact that was exploited in the days of hand calculators to expedite the construction of map projections.<a href="#cite_note-snyder-10">[9]</a>: 108 To first order in the flattening f, the auxiliary latitudes can be expressed as ζ = ϕ − Cf sin 2ϕ where the constant C takes on the values [<style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>1⁄2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">2⁄3, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">3⁄4, 1, 1] for ζ = [β, ξ, μ, χ, θ]. <table class="wikitable" style="margin: 1em auto 1em auto; text-align:right;"> <caption>Approximate difference from geodetic latitude (ϕ) </caption> <tbody><tr> <th>ϕ </th> <th>Parametric β − ϕ </th> <th>Authalic ξ − ϕ </th> <th>Rectifying μ − ϕ </th> <th>Conformal χ − ϕ </th> <th>Geocentric θ − ϕ </th></tr> <tr> <td>0°</td> <td>0.00′</td> <td>0.00′</td> <td>0.00′</td> <td>0.00′</td> <td>0.00′ </td></tr> <tr> <td>15°</td> <td>−2.88′</td> <td>−3.84′</td> <td>−4.32′</td> <td>−5.76′</td> <td>−5.76′ </td></tr> <tr> <td>30°</td> <td>−5.00′</td> <td>−6.66′</td> <td>−7.49′</td> <td>−9.98′</td> <td>−9.98′ </td></tr> <tr> <td>45°</td> <td>−5.77′</td> <td>−7.70′</td> <td>−8.66′</td> <td>−11.54′</td> <td>−11.55′ </td></tr> <tr> <td>60°</td> <td>−5.00′</td> <td>−6.67′</td> <td>−7.51′</td> <td>−10.01′</td> <td>−10.02′ </td></tr> <tr> <td>75°</td> <td>−2.89′</td> <td>−3.86′</td> <td>−4.34′</td> <td>−5.78′</td> <td>−5.79′ </td></tr> <tr> <td>90°</td> <td>0.00′</td> <td>0.00′</td> <td>0.00′</td> <td>0.00′</td> <td>0.00′ </td></tr></tbody></table> <div style="clear:both;" class=""></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><h2 id="Latitude_and_coordinate_systems">Latitude and coordinate systems</h2> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=22" title="Edit section: Latitude and coordinate systems" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> The geodetic latitude, or any of the auxiliary latitudes defined on the reference ellipsoid, constitutes with longitude a two-dimensional <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a> on that ellipsoid. To define the position of an arbitrary point it is necessary to extend such a coordinate system into three dimensions. Three latitudes are used in this way: the geodetic, geocentric and parametric latitudes are used in geodetic coordinates, spherical polar coordinates and ellipsoidal coordinates respectively. <div class="mw-heading mw-heading3"><h3 id="Geodetic_coordinates">Geodetic coordinates</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=23" title="Edit section: Geodetic coordinates" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Geodetic_coordinates" title="Geodetic coordinates">Geodetic coordinates</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Geodetic_coordinates.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Geodetic_coordinates.svg/200px-Geodetic_coordinates.svg.png" decoding="async" width="200" height="179" class="mw-file-element" data-file-width="300" data-file-height="268"></noscript><span class="lazy-image-placeholder" style="width: 200px;height: 179px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Geodetic_coordinates.svg/200px-Geodetic_coordinates.svg.png" data-width="200" data-height="179" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Geodetic_coordinates.svg/300px-Geodetic_coordinates.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Geodetic_coordinates.svg/400px-Geodetic_coordinates.svg.png 2x" data-class="mw-file-element"> </a><figcaption> Geodetic coordinates P(ɸ,λ,h)</figcaption></figure> At an arbitrary point P consider the line PN which is normal to the reference ellipsoid. The geodetic coordinates P(ɸ,λ,h) are the latitude and longitude of the point N on the ellipsoid and the distance PN. This height differs from the height above the geoid or a reference height such as that above mean sea level at a specified location. The direction of PN will also differ from the direction of a vertical plumb line. The relation of these different heights requires knowledge of the shape of the geoid and also the gravity field of the Earth. <div class="mw-heading mw-heading3"><h3 id="Spherical_polar_coordinates">Spherical polar coordinates</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=24" title="Edit section: Spherical polar coordinates" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Geocentric_coords_02.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Geocentric_coords_02.svg/200px-Geocentric_coords_02.svg.png" decoding="async" width="200" height="195" class="mw-file-element" data-file-width="330" data-file-height="321"></noscript><span class="lazy-image-placeholder" style="width: 200px;height: 195px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Geocentric_coords_02.svg/200px-Geocentric_coords_02.svg.png" data-width="200" data-height="195" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Geocentric_coords_02.svg/300px-Geocentric_coords_02.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/Geocentric_coords_02.svg/400px-Geocentric_coords_02.svg.png 2x" data-class="mw-file-element"> </a><figcaption> Geocentric coordinate related to spherical polar coordinates P(r,θ′,λ)</figcaption></figure> The geocentric latitude θ is the complement of the polar angle or <a href="/wiki/Colatitude" title="Colatitude">colatitude</a> θ′ in conventional <a href="/wiki/Spherical_polar_coordinates" class="mw-redirect" title="Spherical polar coordinates">spherical polar coordinates</a> in which the coordinates of a point are P(r,θ′,λ) where r is the distance of P from the centre O, θ′ is the angle between the radius vector and the polar axis and λ is longitude. Since the normal at a general point on the ellipsoid does not pass through the centre it is clear that points P' on the normal, which all have the same geodetic latitude, will have differing geocentric latitudes. Spherical polar coordinate systems are used in the analysis of the gravity field. <div class="mw-heading mw-heading3"><h3 id="Ellipsoidal-harmonic_coordinates">Ellipsoidal-harmonic coordinates</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=25" title="Edit section: Ellipsoidal-harmonic coordinates" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Ellipsoidal_coordinates.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Ellipsoidal_coordinates.svg/200px-Ellipsoidal_coordinates.svg.png" decoding="async" width="200" height="193" class="mw-file-element" data-file-width="299" data-file-height="288"></noscript><span class="lazy-image-placeholder" style="width: 200px;height: 193px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Ellipsoidal_coordinates.svg/200px-Ellipsoidal_coordinates.svg.png" data-width="200" data-height="193" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Ellipsoidal_coordinates.svg/300px-Ellipsoidal_coordinates.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Ellipsoidal_coordinates.svg/400px-Ellipsoidal_coordinates.svg.png 2x" data-class="mw-file-element"> </a><figcaption>Ellipsoidal coordinates P(u,β,λ)</figcaption></figure> The parametric latitude can also be extended to a three-dimensional coordinate system. For a point P not on the reference ellipsoid (semi-axes OA and OB) construct an auxiliary ellipsoid which is confocal (same foci F, F′) with the reference ellipsoid: the necessary condition is that the product ae of semi-major axis and eccentricity is the same for both ellipsoids. Let u be the semi-minor axis (OD) of the auxiliary ellipsoid. Further let β be the parametric latitude of P on the auxiliary ellipsoid. The set (u,β,λ) define the ellipsoidal-harmonic coordinates<a href="#cite_note-21">[19]</a> or simply ellipsoidal coordinates<a href="#cite_note-torge-6">[5]</a>: §4.2.2 (although that term is also used to refer to geodetic coordinate). These coordinates are the natural choice in models of the gravity field for a rotating ellipsoidal body. The above applies to a biaxial ellipsoid (a spheroid, as in <a href="/wiki/Oblate_spheroidal_coordinates" title="Oblate spheroidal coordinates">oblate spheroidal coordinates</a>); for a generalization, see <a href="/wiki/Triaxial_ellipsoidal_coordinates" class="mw-redirect" title="Triaxial ellipsoidal coordinates">triaxial ellipsoidal coordinates</a>. <div class="mw-heading mw-heading3"><h3 id="Coordinate_conversions">Coordinate conversions</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=26" title="Edit section: Coordinate conversions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The relations between the above coordinate systems, and also Cartesian coordinates are not presented here. The transformation between geodetic and Cartesian coordinates may be found in <a href="/wiki/Geographic_coordinate_conversion" title="Geographic coordinate conversion">geographic coordinate conversion</a>. The relation of Cartesian and spherical polars is given in <a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">spherical coordinate system</a>. The relation of Cartesian and ellipsoidal coordinates is discussed in Torge.<a href="#cite_note-torge-6">[5]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><h2 id="Astronomical_latitude">Astronomical latitude</h2> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=27" title="Edit section: Astronomical latitude" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-7 collapsible-block" id="mf-section-7"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Geoida.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Geoida.svg/220px-Geoida.svg.png" decoding="async" width="220" height="102" class="mw-file-element" data-file-width="730" data-file-height="338"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 102px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Geoida.svg/220px-Geoida.svg.png" data-width="220" data-height="102" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Geoida.svg/330px-Geoida.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Geoida.svg/440px-Geoida.svg.png 2x" data-class="mw-file-element"> </a><figcaption><div><ol><li>Ocean</li><li>Ellipsoid</li><li>Local plumb line</li><li>Continent</li><li>Geoid</li></ol></div></figcaption></figure> Astronomical latitude (Φ) is the angle between the equatorial plane and the true <a href="/wiki/Vertical_direction" class="mw-redirect" title="Vertical direction">vertical direction</a> at a point on the surface. The true vertical, the direction of a <a href="/wiki/Plumb_line" class="mw-redirect" title="Plumb line">plumb line</a>, is also the <a href="/wiki/Gravity_direction" class="mw-redirect" title="Gravity direction">gravity direction</a> (the resultant of the <a href="/wiki/Gravitational_acceleration" title="Gravitational acceleration">gravitational acceleration</a> (mass-based) and the <a href="/wiki/Centrifugal_acceleration" class="mw-redirect" title="Centrifugal acceleration">centrifugal acceleration</a>) at that latitude.<a href="#cite_note-torge-6">[5]</a> Astronomic latitude is calculated from angles measured between the <a href="/wiki/Zenith" title="Zenith">zenith</a> and stars whose <a href="/wiki/Declination" title="Declination">declination</a> is accurately known. In general the true vertical at a point on the surface does not exactly coincide with either the normal to the reference ellipsoid or the normal to the geoid. The geoid is an idealized, theoretical shape "at mean sea level". Points on land do not lie precisely on the geoid, and the vertical at a point at a specific time is influenced by tidal forces which the theoretical geoid averages out. The angle between the astronomic and geodetic normals is called <a href="/wiki/Vertical_deflection" title="Vertical deflection">vertical deflection</a> and is usually a few seconds of arc but it is important in geodesy.<a href="#cite_note-torge-6">[5]</a><a href="#cite_note-wellenhofmoritz-22">[20]</a> Astronomical latitude is not to be confused with <a href="/wiki/Declination" title="Declination">declination</a>, the coordinate <a href="/wiki/Astronomer" title="Astronomer">astronomers</a> use in a similar way to specify the angular position of stars north–south of the <a href="/wiki/Celestial_equator" title="Celestial equator">celestial equator</a> (see <a href="/wiki/Equatorial_coordinates" class="mw-redirect" title="Equatorial coordinates">equatorial coordinates</a>), nor with <a href="/wiki/Ecliptic_latitude" class="mw-redirect" title="Ecliptic latitude">ecliptic latitude</a>, the coordinate that astronomers use to specify the angular position of stars north–south of the <a href="/wiki/Ecliptic" title="Ecliptic">ecliptic</a> (see <a href="/wiki/Ecliptic_coordinates" class="mw-redirect" title="Ecliptic coordinates">ecliptic coordinates</a>). </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><h2 id="See_also">See also</h2> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=28" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-8 collapsible-block" id="mf-section-8"> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Altitude" title="Altitude">Altitude</a> (<a href="/wiki/Sea_level" title="Sea level">mean sea level</a>)</li> <li><a href="/wiki/Bowditch%27s_American_Practical_Navigator" title="Bowditch's American Practical Navigator">Bowditch's American Practical Navigator</a></li> <li><a href="/wiki/Cardinal_direction" title="Cardinal direction">Cardinal direction</a></li> <li><a href="/wiki/Circle_of_latitude" title="Circle of latitude">Circle of latitude</a></li> <li><a href="/wiki/Colatitude" title="Colatitude">Colatitude</a></li> <li><a href="/wiki/Declination" title="Declination">Declination</a> on <a href="/wiki/Celestial_sphere" title="Celestial sphere">celestial sphere</a></li> <li><a href="/wiki/Degree_Confluence_Project" title="Degree Confluence Project">Degree Confluence Project</a></li> <li><a href="/wiki/Geodesy" title="Geodesy">Geodesy</a></li> <li><a href="/wiki/Geodetic_datum" title="Geodetic datum">Geodetic datum</a></li> <li><a href="/wiki/Geographic_coordinate_system" title="Geographic coordinate system">Geographic coordinate system</a></li> <li><a href="/wiki/Geographical_distance" title="Geographical distance">Geographical distance</a></li> <li><a href="/wiki/Geomagnetic_latitude" title="Geomagnetic latitude">Geomagnetic latitude</a></li> <li><a href="/wiki/Geotagging" title="Geotagging">Geotagging</a></li> <li><a href="/wiki/Great-circle_distance" title="Great-circle distance">Great-circle distance</a></li> <li><a href="/wiki/History_of_latitude" title="History of latitude">History of latitude</a></li> <li><a href="/wiki/Horse_latitudes" title="Horse latitudes">Horse latitudes</a></li> <li><a href="/wiki/International_Latitude_Service" title="International Latitude Service">International Latitude Service</a></li> <li><a href="/wiki/List_of_countries_by_latitude" title="List of countries by latitude">List of countries by latitude</a></li> <li><a href="/wiki/Longitude" title="Longitude">Longitude</a></li> <li><a href="/wiki/Natural_Area_Code" title="Natural Area Code">Natural Area Code</a></li> <li><a href="/wiki/Navigation" title="Navigation">Navigation</a></li> <li><a href="/wiki/Orders_of_magnitude_(length)" title="Orders of magnitude (length)">Orders of magnitude (length)</a></li> <li><a href="/wiki/World_Geodetic_System" title="World Geodetic System">World Geodetic System</a></li></ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"><h2 id="References">References</h2> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=29" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-9 collapsible-block" id="mf-section-9"> <div class="mw-heading mw-heading3"><h3 id="Footnotes">Footnotes</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=30" title="Edit section: Footnotes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-3"><a href="#cite_ref-3">^</a> The value of this angle today is 23°26′09.8″ (or 23.43605°). This figure is provided by <a href="/wiki/Template:Circle_of_latitude" title="Template:Circle of latitude">Template:Circle of latitude</a>. </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> An elementary calculation involves differentiation to find the maximum difference of the geodetic and geocentric latitudes. </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=31" title="Edit section: Citations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://committee.iso.org/sites/tc211/home/projects/projects---complete-list/iso-19111.html">"ISO 19111 Geographic information — Referencing by coordinates"</a>. ISO. 2021-06-01. Retrieved 2022-01-16.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ISO&rft.atitle=ISO+19111+Geographic+information+%E2%80%94+Referencing+by+coordinates&rft.date=2021-06-01&rft_id=https%3A%2F%2Fcommittee.iso.org%2Fsites%2Ftc211%2Fhome%2Fprojects%2Fprojects---complete-list%2Fiso-19111.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALatitude" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThe_Corporation_of_Trinity_House2020" class="citation web cs1">The Corporation of Trinity House (10 January 2020). <a rel="nofollow" class="external text" href="https://www.trinityhouse.co.uk/notice-to-mariners/1/2020-needles-lighthouse">"1/2020 Needles Lighthouse"</a>. Notices to Mariners. 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Translated by Motte, Andrew. p. <a rel="nofollow" class="external text" href="https://archive.org/details/100878576/page/407">407</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Book+III+Proposition+XIX+Problem+III&rft.btitle=Philosophi%C3%A6+Naturalis+Principia+Mathematica&rft.pages=407&rft.aulast=Newton&rft.aufirst=Isaac&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2F100878576&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALatitude" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNational_Imagery_and_Mapping_Agency2004" class="citation web cs1">National Imagery and Mapping Agency (23 June 2004). <a rel="nofollow" class="external text" href="https://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf">"Department of Defense World Geodetic System 1984"</a> (PDF). National Imagery and Mapping Agency. p. 3-1. TR8350.2. Retrieved 25 April 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Department+of+Defense+World+Geodetic+System+1984&rft.pages=3%26%2345%3B1&rft.pub=National+Imagery+and+Mapping+Agency&rft.date=2004-06-23&rft.au=National+Imagery+and+Mapping+Agency&rft_id=https%3A%2F%2Fearth-info.nga.mil%2FGandG%2Fpublications%2Ftr8350.2%2Fwgs84fin.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALatitude" class="Z3988"> </li> <li id="cite_note-torge-6">^ <a href="#cite_ref-torge_6-0">a</a> <a href="#cite_ref-torge_6-1">b</a> <a href="#cite_ref-torge_6-2">c</a> <a href="#cite_ref-torge_6-3">d</a> <a href="#cite_ref-torge_6-4">e</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTorge2001" class="citation book cs1">Torge, W. (2001). Geodesy (3rd ed.). De Gruyter. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-11-017072-8" title="Special:BookSources/3-11-017072-8"><bdi>3-11-017072-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geodesy&rft.edition=3rd&rft.pub=De+Gruyter&rft.date=2001&rft.isbn=3-11-017072-8&rft.aulast=Torge&rft.aufirst=W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALatitude" class="Z3988"> </li> <li id="cite_note-osborne-7">^ <a href="#cite_ref-osborne_7-0">a</a> <a href="#cite_ref-osborne_7-1">b</a> <a href="#cite_ref-osborne_7-2">c</a> <a href="#cite_ref-osborne_7-3">d</a> <a href="#cite_ref-osborne_7-4">e</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOsborne2013" class="citation book cs1">Osborne, Peter (2013). "Chapters 5,6". The Mercator Projections. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5281%2Fzenodo.35392">10.5281/zenodo.35392</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapters+5%2C6&rft.btitle=The+Mercator+Projections&rft.date=2013&rft_id=info%3Adoi%2F10.5281%2Fzenodo.35392&rft.aulast=Osborne&rft.aufirst=Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALatitude" class="Z3988"> for LaTeX code and figures. </li> <li id="cite_note-rapp-8">^ <a href="#cite_ref-rapp_8-0">a</a> <a href="#cite_ref-rapp_8-1">b</a> <a href="#cite_ref-rapp_8-2">c</a> <a href="#cite_ref-rapp_8-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRapp1991" class="citation book cs1">Rapp, Richard H. (1991). "Chapter 3". Geometric Geodesy, Part I. 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National Geospatial-Intelligence Agency. Archived from <a rel="nofollow" class="external text" href="http://msi.nga.mil/MSISiteContent/StaticFiles/Calculators/degree.html">the original</a> on 2012-12-11. Retrieved 2011-02-08.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Length+of+degree+calculator&rft.pub=National+Geospatial-Intelligence+Agency&rft_id=http%3A%2F%2Fmsi.nga.mil%2FMSISiteContent%2FStaticFiles%2FCalculators%2Fdegree.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALatitude" class="Z3988"> </li> <li id="cite_note-snyder-10">^ <a href="#cite_ref-snyder_10-0">a</a> <a href="#cite_ref-snyder_10-1">b</a> <a href="#cite_ref-snyder_10-2">c</a> <a href="#cite_ref-snyder_10-3">d</a> <a href="#cite_ref-snyder_10-4">e</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSnyder1987" class="citation book cs1">Snyder, John P. 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Retrieved 2017-09-02.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Map+Projections%3A+A+Working+Manual&rft.place=Washington%2C+DC&rft.series=U.S.+Geological+Survey+Professional+Paper+1395&rft.pub=United+States+Government+Printing+Office&rft.date=1987&rft.aulast=Snyder&rft.aufirst=John+P.&rft_id=https%3A%2F%2Fpubs.er.usgs.gov%2Fpubs%2Fpp%2Fpp1395&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALatitude" class="Z3988"> </li> <li id="cite_note-adams1921-11">^ <a href="#cite_ref-adams1921_11-0">a</a> <a href="#cite_ref-adams1921_11-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdams1921" class="citation book cs1"><a href="/wiki/Oscar_S._Adams" title="Oscar S. Adams">Adams, Oscar S.</a> (1921). <a rel="nofollow" class="external text" href="https://geodesy.noaa.gov/library/pdfs/Special_Publication_No_67.pdf">Latitude Developments Connected With Geodesy and Cartography (with tables, including a table for Lambert equal area meridional projection</a> (PDF). Special Publication No. 67. US Coast and Geodetic Survey.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Latitude+Developments+Connected+With+Geodesy+and+Cartography+%28with+tables%2C+including+a+table+for+Lambert+equal+area+meridional+projection&rft.series=Special+Publication+No.+67&rft.pub=US+Coast+and+Geodetic+Survey&rft.date=1921&rft.aulast=Adams&rft.aufirst=Oscar+S.&rft_id=https%3A%2F%2Fgeodesy.noaa.gov%2Flibrary%2Fpdfs%2FSpecial_Publication_No_67.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALatitude" class="Z3988"> (Note: Adams uses the nomenclature isometric latitude for the conformal latitude of this article (and throughout the modern literature).) </li> <li id="cite_note-legendre-13"><a href="#cite_ref-legendre_13-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLegendre1806" class="citation journal cs1 cs1-prop-long-vol">Legendre, A. M. (1806). 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Nachr. 4 (86): 241–254. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0908.1824">0908.1824</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010AN....331..852K">2010AN....331..852K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fasna.201011352">10.1002/asna.201011352</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118760590">118760590</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Astron.+Nachr.&rft.atitle=%C3%9Cber+die+Berechnung+der+geographischen+Langen+und+Breiten+aus+geodatischen+Vermessungen&rft.volume=4&rft.issue=86&rft.pages=241-254&rft.date=1825&rft_id=info%3Aarxiv%2F0908.1824&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118760590%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1002%2Fasna.201011352&rft_id=info%3Abibcode%2F2010AN....331..852K&rft.aulast=Bessel&rft.aufirst=F.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALatitude" class="Z3988"> Translation: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarneyDeakin2010" class="citation journal cs1">Karney, C. 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Nachr. 331 (8): 852–861. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0908.1824">0908.1824</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1825AN......4..241B">1825AN......4..241B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fasna.18260041601">10.1002/asna.18260041601</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118630614">118630614</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Astron.+Nachr.&rft.atitle=The+calculation+of+longitude+and+latitude+from+geodesic+measurements&rft.volume=331&rft.issue=8&rft.pages=852-861&rft.date=2010&rft_id=info%3Aarxiv%2F0908.1824&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118630614%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1002%2Fasna.18260041601&rft_id=info%3Abibcode%2F1825AN......4..241B&rft.aulast=Karney&rft.aufirst=C.+F.+F.&rft.au=Deakin%2C+R.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALatitude" class="Z3988"> </li> <li id="cite_note-cayley-15"><a href="#cite_ref-cayley_15-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCayley1870" class="citation journal cs1">Cayley, A. 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Journal of Geodesy. 87 (1): 43–55. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1109.4448">1109.4448</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013JGeod..87...43K">2013JGeod..87...43K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00190-012-0578-z">10.1007/s00190-012-0578-z</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119310141">119310141</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Geodesy&rft.atitle=Algorithms+for+geodesics&rft.volume=87&rft.issue=1&rft.pages=43-55&rft.date=2013&rft_id=info%3Aarxiv%2F1109.4448&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119310141%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00190-012-0578-z&rft_id=info%3Abibcode%2F2013JGeod..87...43K&rft.aulast=Karney&rft.aufirst=C.+F.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALatitude" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLagrange1779" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange, Joseph-Louis</a> (1779). <a rel="nofollow" class="external text" href="https://archive.org/details/oeuvresdelagrang04lagr/page/663">"Sur la Construction des Cartes Géographiques"</a>. Oevres (in French). Vol. IV. p. 667.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Sur+la+Construction+des+Cartes+G%C3%A9ographiques&rft.btitle=Oevres&rft.pages=667&rft.date=1779&rft.aulast=Lagrange&rft.aufirst=Joseph-Louis&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Foeuvresdelagrang04lagr%2Fpage%2F663&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALatitude" class="Z3988"> </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarney2011" class="citation journal cs1">Karney, Charles F. F. (August 2011). "Transverse Mercator with an accuracy of a few nanometers". Journal of Geodesy. 85 (8): 475–485. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1002.1417">1002.1417</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011JGeod..85..475K">2011JGeod..85..475K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00190-011-0445-3">10.1007/s00190-011-0445-3</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118619524">118619524</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Geodesy&rft.atitle=Transverse+Mercator+with+an+accuracy+of+a+few+nanometers&rft.volume=85&rft.issue=8&rft.pages=475-485&rft.date=2011-08&rft_id=info%3Aarxiv%2F1002.1417&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118619524%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00190-011-0445-3&rft_id=info%3Abibcode%2F2011JGeod..85..475K&rft.aulast=Karney&rft.aufirst=Charles+F.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALatitude" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOrihuela2013" class="citation web cs1">Orihuela, Sebastián (2013). <a rel="nofollow" class="external text" href="https://www.academia.edu/7580468">"Funciones de Latitud"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Funciones+de+Latitud&rft.date=2013&rft.aulast=Orihuela&rft.aufirst=Sebasti%C3%A1n&rft_id=https%3A%2F%2Fwww.academia.edu%2F7580468&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALatitude" class="Z3988"> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarney2023" class="citation journal cs1">Karney, Charles F. F. (2023). "On auxiliary latitudes". Survey Review. 56 (395): 165–180. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2212.05818">2212.05818</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00396265.2023.2217604">10.1080/00396265.2023.2217604</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Survey+Review&rft.atitle=On+auxiliary+latitudes&rft.volume=56&rft.issue=395&rft.pages=165-180&rft.date=2023&rft_id=info%3Aarxiv%2F2212.05818&rft_id=info%3Adoi%2F10.1080%2F00396265.2023.2217604&rft.aulast=Karney&rft.aufirst=Charles+F.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALatitude" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> Holfmann-Wellenfor & Moritz (2006) Physical Geodesy, p.240, eq. (6-6) to (6-10). </li> <li id="cite_note-wellenhofmoritz-22"><a href="#cite_ref-wellenhofmoritz_22-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHofmann-WellenhofMoritz2006" class="citation book cs1">Hofmann-Wellenhof, B.; Moritz, H. (2006). Physical Geodesy (2nd ed.). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-211-33544-7" title="Special:BookSources/3-211-33544-7"><bdi>3-211-33544-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+Geodesy&rft.edition=2nd&rft.date=2006&rft.isbn=3-211-33544-7&rft.aulast=Hofmann-Wellenhof&rft.aufirst=B.&rft.au=Moritz%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALatitude" class="Z3988"> </li> </ol></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"><h2 id="External_links">External links</h2> <a role="button" href="/w/index.php?title=Latitude&action=edit&section=32" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-10 collapsible-block" id="mf-section-10"> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1250146164">.mw-parser-output .sister-box .side-box-abovebelow{padding:0.75em 0;text-align:center}.mw-parser-output .sister-box .side-box-abovebelow>b{display:block}.mw-parser-output .sister-box .side-box-text>ul{border-top:1px solid #aaa;padding:0.75em 0;width:217px;margin:0 auto}.mw-parser-output .sister-box .side-box-text>ul>li{min-height:31px}.mw-parser-output .sister-logo{display:inline-block;width:31px;line-height:31px;vertical-align:middle;text-align:center}.mw-parser-output .sister-link{display:inline-block;margin-left:4px;width:182px;vertical-align:middle}@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-v2.svg"]{background-color:white}}</style><div role="navigation" aria-labelledby="sister-projects" class="side-box metadata side-box-right sister-box sistersitebox plainlinks"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-abovebelow"> Latitude at Wikipedia's <a href="/wiki/Wikipedia:Wikimedia_sister_projects" title="Wikipedia:Wikimedia sister projects">sister projects</a></div> <div class="side-box-flex"> <div class="side-box-text plainlist"><ul><li><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/27px-Wiktionary-logo-v2.svg.png" decoding="async" width="27" height="27" class="mw-file-element" data-file-width="391" data-file-height="391"></noscript><span class="lazy-image-placeholder" style="width: 27px;height: 27px;" 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data-width="20" data-height="27" data-srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 2x" data-class="mw-file-element"> <a href="https://commons.wikimedia.org/wiki/Category:Latitudes" class="extiw" title="c:Category:Latitudes">Media</a> from Commons</li><li><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/27px-Wikinews-logo.svg.png" decoding="async" width="27" height="15" class="mw-file-element" data-file-width="759" data-file-height="415"></noscript><span class="lazy-image-placeholder" style="width: 27px;height: 15px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/27px-Wikinews-logo.svg.png" data-alt="" data-width="27" data-height="15" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/41px-Wikinews-logo.svg.png 1.5x, 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href="https://en.wikiquote.org/wiki/Special:Search/Latitude" class="extiw" title="q:Special:Search/Latitude">Quotations</a> from Wikiquote</li><li><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/26px-Wikisource-logo.svg.png" decoding="async" width="26" height="27" class="mw-file-element" data-file-width="410" data-file-height="430"></noscript><span class="lazy-image-placeholder" style="width: 26px;height: 27px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/26px-Wikisource-logo.svg.png" data-alt="" data-width="26" data-height="27" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/39px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/51px-Wikisource-logo.svg.png 2x" data-class="mw-file-element"> <a href="https://en.wikisource.org/wiki/Special:Search/Latitude" class="extiw" title="s:Special:Search/Latitude">Texts</a> from Wikisource</li><li><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/27px-Wikibooks-logo.svg.png" decoding="async" width="27" height="27" class="mw-file-element" data-file-width="300" data-file-height="300"></noscript><span class="lazy-image-placeholder" style="width: 27px;height: 27px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/27px-Wikibooks-logo.svg.png" data-alt="" data-width="27" data-height="27" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/41px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/54px-Wikibooks-logo.svg.png 2x" data-class="mw-file-element"> <a href="https://en.wikibooks.org/wiki/Special:Search/Latitude" class="extiw" title="b:Special:Search/Latitude">Textbooks</a> from Wikibooks</li><li><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/27px-Wikiversity_logo_2017.svg.png" decoding="async" width="27" height="22" class="mw-file-element" data-file-width="626" data-file-height="512"></noscript><span class="lazy-image-placeholder" style="width: 27px;height: 22px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/27px-Wikiversity_logo_2017.svg.png" data-alt="" data-width="27" data-height="22" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/41px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/54px-Wikiversity_logo_2017.svg.png 2x" data-class="mw-file-element"> <a href="https://en.wikiversity.org/wiki/Special:Search/Latitude" class="extiw" title="v:Special:Search/Latitude">Resources</a> from Wikiversity</li></ul></div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://earth-info.nga.mil/gns/html/">GEONets Names Server</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080309045124/http://earth-info.nga.mil/gns/html/">Archived</a> 2008-03-09 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. access to the <a href="/wiki/National_Geospatial-Intelligence_Agency" title="National Geospatial-Intelligence Agency">National Geospatial-Intelligence Agency</a>'s (NGA) database of foreign geographic feature names.</li> <li><a rel="nofollow" class="external text" href="http://jan.ucc.nau.edu/~cvm/latlon_find_location.html">Resources for determining your latitude and longitude</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080519082322/http://jan.ucc.nau.edu/~cvm/latlon_find_location.html">Archived</a> 2008-05-19 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://geography.about.com/library/howto/htdegrees.htm">Convert decimal degrees into degrees, minutes, seconds</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121107190244/http://geography.about.com/library/howto/htdegrees.htm">Archived</a> 2012-11-07 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> – info about decimal to <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> conversion.</li> <li><a rel="nofollow" class="external text" href="https://www.fcc.gov/mb/audio/bickel/DDDMMSS-decimal.html">Convert decimal degrees into degrees, minutes, seconds</a></li> <li><a rel="nofollow" class="external text" href="http://www.marinewaypoints.com/learn/greatcircle.shtml">Distance calculation based on latitude and longitude</a> – JavaScript version</li> <li><a rel="nofollow" class="external text" href="https://www.academia.edu/12297694/16th_Century_Latitude_Survey">16th Century Latitude Survey</a></li> <li><a rel="nofollow" class="external text" href="http://www.longcamp.com/nav.html">Determination of Latitude by Francis Drake on the Coast of California in 1579</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": 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href="https://en.wikipedia.org/w/index.php?title=Latitude&oldid=1257340328">https://en.wikipedia.org/w/index.php?title=Latitude&oldid=1257340328</a>"</div></div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"> <a class="last-modified-bar" href="/w/index.php?title=Latitude&action=history"> <div class="post-content last-modified-bar__content"> Last edited on 14 November 2024, at 12:15 </div> </a> <div class="post-content footer-content"> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Breedtegraad" title="Breedtegraad – Afrikaans" lang="af" hreflang="af" data-title="Breedtegraad" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Geografische_Breite" title="Geografische Breite – Alemannic" lang="gsw" hreflang="gsw" data-title="Geografische Breite" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%B1%D8%B6_(%D8%AC%D8%BA%D8%B1%D8%A7%D9%81%D9%8A%D8%A7)" title="عرض (جغرافيا) – Arabic" lang="ar" hreflang="ar" data-title="عرض (جغرافيا)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Latitut" title="Latitut – Aragonese" lang="an" hreflang="an" data-title="Latitut" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-frp mw-list-item"><a href="https://frp.wikipedia.org/wiki/Latituda" title="Latituda – Arpitan" lang="frp" hreflang="frp" data-title="Latituda" data-language-autonym="Arpetan" data-language-local-name="Arpitan" class="interlanguage-link-target">Arpetan</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Llatit%C3%BA" title="Llatitú – Asturian" lang="ast" hreflang="ast" data-title="Llatitú" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-awa mw-list-item"><a href="https://awa.wikipedia.org/wiki/%E0%A4%85%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%BE%E0%A4%82%E0%A4%B6" title="अक्षांश – Awadhi" lang="awa" hreflang="awa" data-title="अक्षांश" data-language-autonym="अवधी" data-language-local-name="Awadhi" class="interlanguage-link-target">अवधी</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Co%C4%9Frafi_enlik" title="Coğrafi enlik – Azerbaijani" lang="az" hreflang="az" data-title="Coğrafi enlik" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A7%D8%A6%D9%86%D9%84%D9%85" title="ائنلم – South Azerbaijani" lang="azb" hreflang="azb" data-title="ائنلم" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A6%BE%E0%A6%82%E0%A6%B6" title="অক্ষাংশ – Bangla" lang="bn" hreflang="bn" data-title="অক্ষাংশ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/H%C5%ABi-t%C5%8D%CD%98" title="Hūi-tō͘ – Minnan" lang="nan" hreflang="nan" data-title="Hūi-tō͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B3%D1%80%D0%B0%D1%84%D0%B8%D0%BA_%D0%BA%D0%B8%D2%A3%D0%BB%D0%B5%D0%BA" title="Географик киңлек – Bashkir" lang="ba" hreflang="ba" data-title="Географик киңлек" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A8%D1%8B%D1%80%D0%B0%D1%82%D0%B0" title="Шырата – Belarusian" lang="be" hreflang="be" data-title="Шырата" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A8%D1%8B%D1%80%D0%B0%D1%82%D0%B0" title="Шырата – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Шырата" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%85%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%BE%E0%A4%82%E0%A4%B6" title="अक्षांश – Bhojpuri" lang="bh" hreflang="bh" data-title="अक्षांश" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target">भोजपुरी</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B3%D1%80%D0%B0%D1%84%D1%81%D0%BA%D0%B0_%D1%88%D0%B8%D1%80%D0%B8%D0%BD%D0%B0" title="Географска ширина – Bulgarian" lang="bg" hreflang="bg" data-title="Географска ширина" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%A0%E0%BD%95%E0%BE%B2%E0%BD%BA%E0%BD%91%E0%BC%8B%E0%BD%90%E0%BD%B2%E0%BD%82" title="འཕྲེད་ཐིག – Tibetan" lang="bo" hreflang="bo" data-title="འཕྲེད་ཐིག" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target">བོད་ཡིག</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Geografska_%C5%A1irina" title="Geografska širina – Bosnian" lang="bs" hreflang="bs" data-title="Geografska širina" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Ledred" title="Ledred – Breton" lang="br" hreflang="br" data-title="Ledred" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Latitud" title="Latitud – Catalan" lang="ca" hreflang="ca" data-title="Latitud" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%90%D0%BD_(%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82)" title="Ан (координат) – Chuvash" lang="cv" hreflang="cv" data-title="Ан (координат)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-ceb mw-list-item"><a href="https://ceb.wikipedia.org/wiki/Latitud" title="Latitud – Cebuano" lang="ceb" hreflang="ceb" data-title="Latitud" data-language-autonym="Cebuano" data-language-local-name="Cebuano" class="interlanguage-link-target">Cebuano</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Zem%C4%9Bpisn%C3%A1_%C5%A1%C3%AD%C5%99ka" title="Zeměpisná šířka – Czech" lang="cs" hreflang="cs" data-title="Zeměpisná šířka" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Mbariro_dzeTaranyika" title="Mbariro dzeTaranyika – Shona" lang="sn" hreflang="sn" data-title="Mbariro dzeTaranyika" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target">ChiShona</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Lledred" title="Lledred – Welsh" lang="cy" hreflang="cy" data-title="Lledred" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Geographische_Breite" title="Geographische Breite – German" lang="de" hreflang="de" data-title="Geographische Breite" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-dsb mw-list-item"><a href="https://dsb.wikipedia.org/wiki/Geografiska_%C5%A1yrina" title="Geografiska šyrina – Lower Sorbian" lang="dsb" hreflang="dsb" data-title="Geografiska šyrina" data-language-autonym="Dolnoserbski" data-language-local-name="Lower Sorbian" class="interlanguage-link-target">Dolnoserbski</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Geograafiline_laius" title="Geograafiline laius – Estonian" lang="et" hreflang="et" data-title="Geograafiline laius" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CE%B5%CF%89%CE%B3%CF%81%CE%B1%CF%86%CE%B9%CE%BA%CF%8C_%CF%80%CE%BB%CE%AC%CF%84%CE%BF%CF%82" title="Γεωγραφικό πλάτος – Greek" lang="el" hreflang="el" data-title="Γεωγραφικό πλάτος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Latitud" title="Latitud – Spanish" lang="es" hreflang="es" data-title="Latitud" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Latitudo" title="Latitudo – Esperanto" lang="eo" hreflang="eo" data-title="Latitudo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Latitude" title="Latitude – Basque" lang="eu" hreflang="eu" data-title="Latitude" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%B1%D8%B6_%D8%AC%D8%BA%D8%B1%D8%A7%D9%81%DB%8C%D8%A7%DB%8C%DB%8C" title="عرض جغرافیایی – Persian" lang="fa" hreflang="fa" data-title="عرض جغرافیایی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Latitude" title="Latitude – French" lang="fr" hreflang="fr" data-title="Latitude" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Breedtegraad" title="Breedtegraad – Western Frisian" lang="fy" hreflang="fy" data-title="Breedtegraad" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target">Frysk</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Domhanleithead" title="Domhanleithead – Irish" lang="ga" hreflang="ga" data-title="Domhanleithead" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Latitude" title="Latitude – Galician" lang="gl" hreflang="gl" data-title="Latitude" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9C%84%EB%8F%84" title="위도 – Korean" lang="ko" hreflang="ko" data-title="위도" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%B7%D5%AD%D5%A1%D6%80%D5%B0%D5%A1%D5%A3%D6%80%D5%A1%D5%AF%D5%A1%D5%B6_%D5%AC%D5%A1%D5%B5%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Աշխարհագրական լայնություն – Armenian" lang="hy" hreflang="hy" data-title="Աշխարհագրական լայնություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%BE%E0%A4%82%E0%A4%B6_%E0%A4%B0%E0%A5%87%E0%A4%96%E0%A4%BE%E0%A4%8F%E0%A4%81" title="अक्षांश रेखाएँ – Hindi" lang="hi" hreflang="hi" data-title="अक्षांश रेखाएँ" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/Geografiska_%C5%A1%C4%9Brokos%C4%87" title="Geografiska šěrokosć – Upper Sorbian" lang="hsb" hreflang="hsb" data-title="Geografiska šěrokosć" data-language-autonym="Hornjoserbsce" data-language-local-name="Upper Sorbian" class="interlanguage-link-target">Hornjoserbsce</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Zemljopisna_%C5%A1irina" title="Zemljopisna širina – Croatian" lang="hr" hreflang="hr" data-title="Zemljopisna širina" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Latitudo" title="Latitudo – Ido" lang="io" hreflang="io" data-title="Latitudo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-ig mw-list-item"><a href="https://ig.wikipedia.org/wiki/Latitude" title="Latitude – Igbo" lang="ig" hreflang="ig" data-title="Latitude" data-language-autonym="Igbo" data-language-local-name="Igbo" class="interlanguage-link-target">Igbo</a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Latitud" title="Latitud – Iloko" lang="ilo" hreflang="ilo" data-title="Latitud" data-language-autonym="Ilokano" data-language-local-name="Iloko" class="interlanguage-link-target">Ilokano</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Garis_lintang" title="Garis lintang – Indonesian" lang="id" hreflang="id" data-title="Garis lintang" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%A3%C3%A6%D1%80%D1%85%D0%B0%D0%B4" title="Уæрхад – Ossetic" lang="os" hreflang="os" data-title="Уæрхад" data-language-autonym="Ирон" data-language-local-name="Ossetic" class="interlanguage-link-target">Ирон</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Breiddargr%C3%A1%C3%B0a" title="Breiddargráða – Icelandic" lang="is" hreflang="is" data-title="Breiddargráða" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Latitudine" title="Latitudine – Italian" lang="it" hreflang="it" data-title="Latitudine" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%95_%D7%A8%D7%95%D7%97%D7%91" title="קו רוחב – Hebrew" lang="he" hreflang="he" data-title="קו רוחב" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%85%E0%B2%95%E0%B3%8D%E0%B2%B7%E0%B2%BE%E0%B2%82%E0%B2%B6" title="ಅಕ್ಷಾಂಶ – Kannada" lang="kn" hreflang="kn" data-title="ಅಕ್ಷಾಂಶ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%92%E1%83%90%E1%83%9C%E1%83%94%E1%83%93%E1%83%98" title="განედი – Georgian" lang="ka" hreflang="ka" data-title="განედი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%95%D0%BD%D0%B4%D1%96%D0%BA" title="Ендік – Kazakh" lang="kk" hreflang="kk" data-title="Ендік" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Latitid" title="Latitid – Haitian Creole" lang="ht" hreflang="ht" data-title="Latitid" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/H%C3%AAl%C3%AEpan" title="Hêlîpan – Kurdish" lang="ku" hreflang="ku" data-title="Hêlîpan" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%B5%D2%A3%D0%B4%D0%B8%D0%BA" title="Кеңдик – Kyrgyz" lang="ky" hreflang="ky" data-title="Кеңдик" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BB%80%E0%BA%AA%E0%BA%B1%E0%BB%89%E0%BA%99%E0%BA%82%E0%BA%B0%E0%BB%9C%E0%BA%B2%E0%BA%99" title="ເສັ້ນຂະໜານ – Lao" lang="lo" hreflang="lo" data-title="ເສັ້ນຂະໜານ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target">ລາວ</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Latitudo_geographica" title="Latitudo geographica – Latin" lang="la" hreflang="la" data-title="Latitudo geographica" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/%C4%A2eogr%C4%81fiskais_platums" title="Ģeogrāfiskais platums – Latvian" lang="lv" hreflang="lv" data-title="Ģeogrāfiskais platums" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Geographesch_Breet" title="Geographesch Breet – Luxembourgish" lang="lb" hreflang="lb" data-title="Geographesch Breet" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Platuma" title="Platuma – Lithuanian" lang="lt" hreflang="lt" data-title="Platuma" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-ln mw-list-item"><a href="https://ln.wikipedia.org/wiki/Monk%C9%94%CC%81l%C9%94%CC%81t%C9%94%CC%81_mw%C3%A2_libale" title="Monkɔ́lɔ́tɔ́ mwâ libale – Lingala" lang="ln" hreflang="ln" data-title="Monkɔ́lɔ́tɔ́ mwâ libale" data-language-autonym="Lingála" data-language-local-name="Lingala" class="interlanguage-link-target">Lingála</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Latitudin" title="Latitudin – Lombard" lang="lmo" hreflang="lmo" data-title="Latitudin" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-mai mw-list-item"><a href="https://mai.wikipedia.org/wiki/%E0%A4%85%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%BE%E0%A4%82%E0%A4%B6" title="अक्षांश – Maithili" lang="mai" hreflang="mai" data-title="अक्षांश" data-language-autonym="मैथिली" data-language-local-name="Maithili" class="interlanguage-link-target">मैथिली</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B3%D1%80%D0%B0%D1%84%D1%81%D0%BA%D0%B0_%D1%88%D0%B8%D1%80%D0%B8%D0%BD%D0%B0" title="Географска ширина – Macedonian" lang="mk" hreflang="mk" data-title="Географска ширина" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Laharam-pehintany" title="Laharam-pehintany – Malagasy" lang="mg" hreflang="mg" data-title="Laharam-pehintany" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%95%E0%B5%8D%E0%B4%B7%E0%B4%BE%E0%B4%82%E0%B4%B6%E0%B4%82" title="അക്ഷാംശം – Malayalam" lang="ml" hreflang="ml" data-title="അക്ഷാംശം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%85%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%BE%E0%A4%82%E0%A4%B6" title="अक्षांश – Marathi" lang="mr" hreflang="mr" data-title="अक्षांश" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Latitud" title="Latitud – Malay" lang="ms" hreflang="ms" data-title="Latitud" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/%C5%AAi-d%C3%B4" title="Ūi-dô – Mindong" lang="cdo" hreflang="cdo" data-title="Ūi-dô" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target">閩東語 / Mìng-dĕ̤ng-ngṳ̄</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9C%E1%80%90%E1%80%B9%E1%80%90%E1%80%AE%E1%80%80%E1%80%BB%E1%80%B0" title="လတ္တီကျူ – Burmese" lang="my" hreflang="my" data-title="လတ္တီကျူ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Breedtegraad" title="Breedtegraad – Dutch" lang="nl" hreflang="nl" data-title="Breedtegraad" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-nds-nl mw-list-item"><a href="https://nds-nl.wikipedia.org/wiki/Breedtegraod" title="Breedtegraod – Low Saxon" lang="nds-NL" hreflang="nds-NL" data-title="Breedtegraod" data-language-autonym="Nedersaksies" data-language-local-name="Low Saxon" class="interlanguage-link-target">Nedersaksies</a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%85%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%BE%E0%A4%82%E0%A4%B6" title="अक्षांश – Nepali" lang="ne" hreflang="ne" data-title="अक्षांश" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B7%AF%E5%BA%A6" title="緯度 – Japanese" lang="ja" hreflang="ja" data-title="緯度" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Geograafisk_breetje" title="Geograafisk breetje – Northern Frisian" lang="frr" hreflang="frr" data-title="Geograafisk breetje" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Breddegrad" title="Breddegrad – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Breddegrad" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Breiddegrad" title="Breiddegrad – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Breiddegrad" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Latitud" title="Latitud – Occitan" lang="oc" hreflang="oc" data-title="Latitud" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%9B%D0%BE%D0%BF%D1%82%D1%8B%D0%BA" title="Лоптык – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Лоптык" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target">Олык марий</a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%85%E0%AC%95%E0%AD%8D%E0%AC%B7%E0%AC%BE%E0%AC%82%E0%AC%B6" title="ଅକ୍ଷାଂଶ – Odia" lang="or" hreflang="or" data-title="ଅକ୍ଷାଂଶ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target">ଓଡ଼ିଆ</a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Sararran_Dagalee" title="Sararran Dagalee – Oromo" lang="om" hreflang="om" data-title="Sararran Dagalee" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target">Oromoo</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Kenglik" title="Kenglik – Uzbek" lang="uz" hreflang="uz" data-title="Kenglik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%85%E0%A8%95%E0%A8%B8%E0%A8%BC%E0%A8%BE%E0%A8%82%E0%A8%B8%E0%A8%BC_%E0%A8%B0%E0%A9%87%E0%A8%96%E0%A8%BE" title="ਅਕਸ਼ਾਂਸ਼ ਰੇਖਾ – Punjabi" lang="pa" hreflang="pa" data-title="ਅਕਸ਼ਾਂਸ਼ ਰੇਖਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B9%D8%B1%D8%B6_%D8%A8%D9%84%D8%AF" title="عرض بلد – Western Punjabi" lang="pnb" hreflang="pnb" data-title="عرض بلد" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-blk mw-list-item"><a href="https://blk.wikipedia.org/wiki/%E1%80%9C%E1%80%90%E1%80%B9%E1%80%90%E1%80%AE%E1%80%85%E1%80%BB%E1%80%AF" title="လတ္တီစျု – Pa'O" lang="blk" hreflang="blk" data-title="လတ္တီစျု" data-language-autonym="ပအိုဝ်ႏဘာႏသာႏ" data-language-local-name="Pa'O" class="interlanguage-link-target">ပအိုဝ်ႏဘာႏသာႏ</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Szeroko%C5%9B%C4%87_geograficzna" title="Szerokość geograficzna – Polish" lang="pl" hreflang="pl" data-title="Szerokość geograficzna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Latitude" title="Latitude – Portuguese" lang="pt" hreflang="pt" data-title="Latitude" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Latitudine" title="Latitudine – Romanian" lang="ro" hreflang="ro" data-title="Latitudine" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A8%D0%B8%D1%80%D0%BE%D1%82%D0%B0" title="Широта – Russian" lang="ru" hreflang="ru" data-title="Широта" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9A%D1%8D%D1%82%D0%B8%D1%80%D1%8D%D1%8D%D2%BB%D0%B8%D0%BD" title="Кэтирээһин – Yakut" lang="sah" hreflang="sah" data-title="Кэтирээһин" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-sat mw-list-item"><a href="https://sat.wikipedia.org/wiki/%E1%B1%9A%E1%B1%A0%E1%B1%AA%E1%B1%B7%E1%B1%9F%E1%B1%B8%E1%B1%A5_%E1%B1%9C%E1%B1%9F%E1%B1%A8" title="ᱚᱠᱪᱷᱟᱸᱥ ᱜᱟᱨ – Santali" lang="sat" hreflang="sat" data-title="ᱚᱠᱪᱷᱟᱸᱥ ᱜᱟᱨ" data-language-autonym="ᱥᱟᱱᱛᱟᱲᱤ" data-language-local-name="Santali" class="interlanguage-link-target">ᱥᱟᱱᱛᱟᱲᱤ</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Gjer%C3%ABsia_gjeografike" title="Gjerësia gjeografike – Albanian" lang="sq" hreflang="sq" data-title="Gjerësia gjeografike" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Latit%C3%B9dini" title="Latitùdini – Sicilian" lang="scn" hreflang="scn" data-title="Latitùdini" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%85%E0%B6%9A%E0%B7%8A%E0%B7%82%E0%B7%8F%E0%B6%82%E0%B7%81" title="අක්ෂාංශ – Sinhala" lang="si" hreflang="si" data-title="අක්ෂාංශ" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Latitude" title="Latitude – Simple English" lang="en-simple" hreflang="en-simple" data-title="Latitude" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Zemepisn%C3%A1_%C5%A1%C3%ADrka" title="Zemepisná šírka – Slovak" lang="sk" hreflang="sk" data-title="Zemepisná šírka" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Zemljepisna_%C5%A1irina" title="Zemljepisna širina – Slovenian" lang="sl" hreflang="sl" data-title="Zemljepisna širina" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Szyroko%C5%9B%C4%87_geograficzno" title="Szyrokość geograficzno – Silesian" lang="szl" hreflang="szl" data-title="Szyrokość geograficzno" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target">Ślůnski</a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Lool" title="Lool – Somali" lang="so" hreflang="so" data-title="Lool" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target">Soomaaliga</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%BE%DB%8E%DA%B5%DB%8C_%D9%BE%D8%A7%D9%86%DB%8C" title="ھێڵی پانی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ھێڵی پانی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B3%D1%80%D0%B0%D1%84%D1%81%D0%BA%D0%B0_%D1%88%D0%B8%D1%80%D0%B8%D0%BD%D0%B0" title="Географска ширина – Serbian" lang="sr" hreflang="sr" data-title="Географска ширина" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Geografska_%C5%A1irina" title="Geografska širina – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Geografska širina" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Garis_datar" title="Garis datar – Sundanese" lang="su" hreflang="su" data-title="Garis datar" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Leveysaste" title="Leveysaste – Finnish" lang="fi" hreflang="fi" data-title="Leveysaste" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Latitud" title="Latitud – Swedish" lang="sv" hreflang="sv" data-title="Latitud" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl badge-Q70893996 mw-list-item" title=""><a href="https://tl.wikipedia.org/wiki/Latitud" title="Latitud – Tagalog" lang="tl" hreflang="tl" data-title="Latitud" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AE%BF%E0%AE%B2%E0%AE%A8%E0%AF%87%E0%AE%B0%E0%AF%8D%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%81" title="நிலநேர்க்கோடு – Tamil" lang="ta" hreflang="ta" data-title="நிலநேர்க்கோடு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tehri" title="Tehri – Kabyle" lang="kab" hreflang="kab" data-title="Tehri" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target">Taqbaylit</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%85%E0%B0%95%E0%B1%8D%E0%B0%B7%E0%B0%BE%E0%B0%82%E0%B0%B6%E0%B0%82" title="అక్షాంశం – Telugu" lang="te" hreflang="te" data-title="అక్షాంశం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A5%E0%B8%B0%E0%B8%95%E0%B8%B4%E0%B8%88%E0%B8%B9%E0%B8%94" title="ละติจูด – Thai" lang="th" hreflang="th" data-title="ละติจูด" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D1%80%D0%B7%D0%B8_%D2%B7%D1%83%D2%93%D1%80%D0%BE%D1%84%D0%B8%D1%91%D3%A3" title="Арзи ҷуғрофиёӣ – Tajik" lang="tg" hreflang="tg" data-title="Арзи ҷуғрофиёӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Enlem" title="Enlem – Turkish" lang="tr" hreflang="tr" data-title="Enlem" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A8%D0%B8%D1%80%D0%BE%D1%82%D0%B0" title="Широта – Ukrainian" lang="uk" hreflang="uk" data-title="Широта" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B9%D8%B1%D8%B6_%D8%A8%D9%84%D8%AF" title="عرض بلد – Urdu" lang="ur" hreflang="ur" data-title="عرض بلد" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-ug mw-list-item"><a href="https://ug.wikipedia.org/wiki/%D9%83%DB%95%DA%AD%D9%84%D9%89%D9%83" title="كەڭلىك – Uyghur" lang="ug" hreflang="ug" data-title="كەڭلىك" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="Uyghur" class="interlanguage-link-target">ئۇيغۇرچە / Uyghurche</a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Latitudine" title="Latitudine – Venetian" lang="vec" hreflang="vec" data-title="Latitudine" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target">Vèneto</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/V%C4%A9_%C4%91%E1%BB%99" title="Vĩ độ – Vietnamese" lang="vi" hreflang="vi" data-title="Vĩ độ" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Br%C3%AAedtegroad" title="Brêedtegroad – West Flemish" lang="vls" hreflang="vls" data-title="Brêedtegroad" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target">West-Vlams</a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Latitud" title="Latitud – Waray" lang="war" hreflang="war" data-title="Latitud" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target">Winaray</a></li><li class="interlanguage-link interwiki-wo mw-list-item"><a href="https://wo.wikipedia.org/wiki/Tus-wu-gaar" title="Tus-wu-gaar – Wolof" lang="wo" hreflang="wo" data-title="Tus-wu-gaar" data-language-autonym="Wolof" data-language-local-name="Wolof" class="interlanguage-link-target">Wolof</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%B7%AF%E5%BA%A6" title="緯度 – Wu" lang="wuu" hreflang="wuu" data-title="緯度" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%92%D7%90%D7%A8%D7%98%D7%9C_%D7%9C%D7%99%D7%A0%D7%99%D7%A2_(%D7%92%D7%A2%D7%90%D7%92%D7%A8%D7%90%D7%A4%D7%99%D7%A2)" title="גארטל ליניע (געאגראפיע) – Yiddish" lang="yi" hreflang="yi" data-title="גארטל ליניע (געאגראפיע)" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target">ייִדיש</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%B7%AF" title="緯 – Cantonese" lang="yue" hreflang="yue" data-title="緯" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Ver%C4%B1niye" title="Verıniye – Zazaki" lang="diq" hreflang="diq" data-title="Verıniye" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target">Zazaki</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%BA%AC%E5%BA%A6" title="纬度 – Chinese" lang="zh" hreflang="zh" data-title="纬度" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li></ul> </section> </div> <div class="minerva-footer-logo"><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod"> This page was last edited on 14 November 2024, at 12:15 (UTC).</li> <li id="footer-info-copyright">Content is available under <a class="external" rel="nofollow" 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