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In
geography Geography (from Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and phenomena of the Earth and planets. The first person to use the word γεωγρα ...

geography
, latitude is a
geographic coordinate A geographic coordinate system (GCS) is a coordinate system associated with position (geometry), positions on Earth (geographic position). A GCS can give positions: *as Geodetic coordinates, spherical coordinate system using latitude, long ...
that specifies the
north North is one of the four compass points or cardinal directions. It is the opposite of south and is perpendicular to East and West. ''North'' is a noun, adjective, or adverb indicating Direction (geometry), direction or geography. Etymology The ...

north
south South is one of the cardinal directions or compass points. South is the opposite of north and is perpendicular to the east and west. Etymology The word ''south'' comes from Old English ''sūþ'', from earlier Proto-Germanic language, Proto-Germa ...

south
position of a point on the Earth's surface. Latitude is an angle (defined below) which ranges from 0° at the
Equator The Earth's equator is an imaginary planetary line that is about long in circumference. The equator divides the planet into the Northern Hemisphere and Southern Hemisphere and is located at 0 degrees latitude In geography, latitude is ...

Equator
to 90° (North or South) at the poles. Lines of constant latitude, or ''parallels'', run east–west as circles parallel to the equator. Latitude is used together with
longitude Longitude (, ) is a geographic coordinate A geographic coordinate system (GCS) is a coordinate system associated with position (geometry), positions on Earth (geographic position). A GCS can give positions: *as Geodetic coordinates, ...

longitude
to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the ''geodetic latitude'' as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular (or '' normal'') to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six ''auxiliary latitudes'' that are used in special applications.


Background

Two levels of abstraction are employed in the definitions of latitude and longitude. In the first step the physical surface is modeled by the
geoid The geoid () is the shape that the ocean The ocean (also the sea The sea, connected as the world ocean or simply the ocean The ocean (also the sea or the world ocean) is the body of salt water which covers approximately ...

geoid
, a surface which approximates the
mean sea level There are several kinds of mean in mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

mean sea level
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
sphere A sphere (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 m ...

sphere
, but the geoid is more accurately modeled by an ellipsoid. 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 graticule 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 normal to the reference surface, which passes through the point on the physical surface. Latitude and longitude together with some specification of
height 200px, A cuboid demonstrating the dimensions length, width">length.html" ;"title="cuboid demonstrating the dimensions length">cuboid demonstrating the dimensions length, width, and height. Height is measure of vertical distance, either vertical ...

height
constitute a
geographic coordinate system A geographic coordinate system (GCS) is a coordinate system associated with positions on Earth (geographic position). A GCS can give positions: *as spherical coordinate system using latitude In geography, latitude is a geographic c ...
as defined in the specification of the ISO 19111 standard. Since there are many different
reference ellipsoid Image:OblateSpheroid.PNG, 250px, Flattened sphere In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the geoid, which is the truer, imperfect figure of the Earth, or other planetary body, as opposed to a perfect, ...
s, 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
Global Positioning System The Global Positioning System (GPS), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government The federal government of the United States (U.S. federal government) is the national ...
(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
phi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialects of the Greek ...
( or ). It is measured in
degrees Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
, minutes and seconds or
decimal degrees Decimal degrees (DD) is a notation for expressing latitude In geography Geography (from Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and phenom ...
, north or south of the equator. For navigational purposes positions are given in degrees and decimal minutes. For instance,
The Needles The Needles is a row of three stacks of chalk Chalk is a soft, white, porous, sedimentary carbonate rock, a form of limestone Limestone is a common type of carbonate rock, carbonate sedimentary rock. It is composed mostly of the ...

The Needles
lighthouse is at 50°39.734′ N 001°35.500′ W. This article relates to coordinate systems for the Earth: it may be adapted to cover the Moon, planets and other celestial objects ( planetographic latitude). For a brief history see History of latitude.


Determination

In
celestial navigation Celestial navigation, also known as astronavigation, is the ancient and modern practice of position fixing that enables a navigator to transition through a space without having to rely on estimated calculations, or dead reckoning, to know their p ...

celestial navigation
, latitude is determined with the meridian altitude method. More precise measurement of latitude requires an understanding of the gravitational field of the Earth, either to set up
theodolite A theodolite is a precision optical instrument for measuring angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), ...

theodolite
s or to determine GPS satellite orbits. The study of the
figure of the Earth Figure of the Earth is a term of art Jargon is the specialized terminology Terminology is a general word for the group of specialized words or meanings relating to a particular field, and also the study of such terms and their use. This is ...
together with its gravitational field is the science of
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's geometric shape, orientation in space, and gravitational field. The field also incorporates studies of how these properties change over time and equivalent measu ...
.


Latitude on the sphere


The graticule on the sphere

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
poles The Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a nation A nation is a community of people formed on the basis of a combination of shared features such as language, history, ethnicity, culture ...
where the axis of rotation of the Earth intersects the reference surface. Planes which contain the rotation axis intersect the surface at the meridians; and the angle between any one meridian plane and that through Greenwich (the
Prime Meridian#REDIRECT Prime meridian A prime meridian is the meridian (geography), meridian (a line of longitude) in a geographic coordinate system at which longitude is defined to be 0°. Together, a prime meridian and its anti-meridian (the 180th meridian ...

Prime Meridian
) 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
Equator The Earth's equator is an imaginary planetary line that is about long in circumference. The equator divides the planet into the Northern Hemisphere and Southern Hemisphere and is located at 0 degrees latitude In geography, latitude is ...

Equator
. 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
North Pole Sea ice in 2006 as observed from the National Oceanic and Atmospheric Administration North Pole Web Cam, part of the North Pole Environmental Observatory The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is ...
has a latitude of 90° North (written 90° N or +90°), and the
South Pole The South Pole, also known as the Geographic South Pole, Terrestrial South Pole or 90th Parallel South, is one of the where intersects its surface. It is the southernmost point on Earth and lies on the of Earth from the . Situated on the ...
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 radius 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.


Named latitudes on the Earth

Besides the equator, four other parallels are of significance: : The plane of the Earth's orbit about the Sun is called the
ecliptic The ecliptic is the plane (geometry), plane of Earth's orbit around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic against the ...

ecliptic
, 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 . The latitude of the tropical circles is equal to 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
epoch In chronology 222px, Joseph Scaliger's ''De emendatione temporum'' (1583) began the modern science of chronology Chronology (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-E ...
. The time variation is discussed more fully in the article on
axial tilt In , axial tilt, also known as obliquity, is the angle between an object's and its al axis, or, equivalently, the angle between its ial plane and . It differs from . At an obliquity of 0 degrees, the two axes point in the same direction; i.e., ...
. The figure shows the geometry of a cross-section of the plane perpendicular to the ecliptic and through the centres of the Earth and the Sun at the December
solstice A solstice is an event that occurs when the Sun appears to reach its most northerly or southerly excursion relative to the celestial equator on the celestial sphere In astronomy and navigation, the celestial sphere is an abstraction, abstr ...

solstice
when the Sun is overhead at some point of the
Tropic of Capricorn The Tropic of Capricorn (or the Southern Tropic) is the that contains the at the December (or southern) . It is thus the southernmost latitude where the Sun can be seen directly overhead. It also reaches 90 degrees below the horizon at solar ...
. The south polar latitudes below the
Antarctic Circle 350px, ''Map of the Antarctic with the Antarctic Circle in blue.'' The Antarctic Circle is the most southerly of the five major circle of latitude, circles of latitude that mark maps of the Earth. The region south of this circle is known as the An ...

Antarctic Circle
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
tropics The tropics are the region of Earth Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% i ...

tropics
is it possible for the Sun to be directly overhead (at the
zenith The zenith is an imaginary point directly "above" a particular location, on the imaginary celestial sphere. "Above" means in the vertical direction (plumb line) opposite to the gravity direction at that location (nadir). The zenith is the "high ...

zenith
). On
map projections In cartography Cartography (; from Greek χάρτης ''chartēs'', "papyrus, sheet of paper, map"; and γράφειν ''graphein'', "write") is the study and practice of making and using maps. Combining science Science (from the Lati ...
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
Mercator projection The Mercator projection () is a cylindrical map projection presented by Flemish Flemish (''Vlaams'') is a Low Franconian dialect cluster of the Dutch language. It is sometimes referred to as Flemish Dutch (), Belgian Dutch ( ), or Souther ...

Mercator projection
and the
Transverse Mercator projection The transverse Mercator map projection In cartography, a map projection is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the Surf ...
. 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.


Latitude on the ellipsoid


Ellipsoids

In 1687
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

Isaac Newton
published the ''
Philosophiæ Naturalis Principia Mathematica (from Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it bec ...
'', in which he proved that a rotating self-gravitating fluid body in equilibrium takes the form of an
oblate In Christianity Christianity is an , based on the and of . It is the , with about 2.5 billion followers. Its adherents, known as , make up a majority of the population in , and believe that is the , whose coming as the was in the (ca ...
ellipsoid. (This article uses the term ''ellipsoid'' in preference to the older term ''spheroid''.) Newton's result was confirmed by geodetic measurements in the 18th century. (See
Meridian arc In geodesy Geodesy () is the Earth science Earth science or geoscience includes all fields of natural science Natural science is a branch of science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic ...
.) An oblate ellipsoid is the three-dimensional surface generated by the rotation of an ellipse about its shorter axis (minor axis). "Oblate ellipsoid of revolution" is abbreviated to 'ellipsoid' in the remainder of this article. (Ellipsoids which do not have an axis of symmetry are termed triaxial.) Many different reference ellipsoids have been used in the history of
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's geometric shape, orientation in space, and gravitational field. The field also incorporates studies of how these properties change over time and equivalent measu ...
. In pre-satellite days they were devised to give a good fit to the
geoid The geoid () is the shape that the ocean The ocean (also the sea The sea, connected as the world ocean or simply the ocean The ocean (also the sea or the world ocean) is the body of salt water which covers approximately ...

geoid
over the limited area of a survey but, with the advent of
GPS The Global Positioning System (GPS), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government The federal government of the United States (U.S. federal government) is the national ...

GPS
, it has become natural to use reference ellipsoids (such as
WGS84 The World Geodetic System (WGS) is a standard for use in cartography Cartography (; from Greek χάρτης ''chartēs'', "papyrus, sheet of paper, map"; and γράφειν ''graphein'', "write") is the study and practice of making and us ...
) 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 of the geoid. Since latitude is defined with respect to an ellipsoid, the position of a given point is different on each ellipsoid: one cannot exactly specify the latitude and longitude of a geographical feature without specifying the ellipsoid used. Many maps maintained by national agencies are based on older ellipsoids, so one must know how the latitude and longitude values are transformed from one ellipsoid to another. GPS handsets include software to carry out datum transformations which link WGS84 to the local reference ellipsoid with its associated grid.


The geometry of the ellipsoid

The shape of an ellipsoid of revolution is determined by the shape of the
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

ellipse
which is rotated about its minor (shorter) axis. Two parameters are required. One is invariably the equatorial radius, which is the
semi-major axis In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

semi-major axis
, . The other parameter is usually (1) the polar radius or
semi-minor axis In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

semi-minor axis
, ; or (2) the (first)
flattening Flattening is a measure of the compression of a circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre; equiv ...
, ; or (3) the
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off- center, in geometry * Eccentricity (graph theory) of a ...

eccentricity
, . These parameters are not independent: they are related by :f=\frac, \qquad e^2=2f-f^2,\qquad b=a(1-f)=a\sqrt\,. Many other parameters (see
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

ellipse
,
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a Surface (mathemat ...

ellipsoid
) appear in the study of geodesy, geophysics and map projections but they can all be expressed in terms of one or two members of the set , , and . Both and are small and often appear in series expansions in calculations; they are of the order and 0.0818 respectively. Values for a number of ellipsoids are given in
Figure of the Earth Figure of the Earth is a Jargon, term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model. The Spherical Earth, sphere is an app ...
. Reference ellipsoids are usually defined by the semi-major axis and the ''inverse'' flattening, . For example, the defining values for the
WGS84 The World Geodetic System (WGS) is a standard for use in cartography Cartography (; from Greek χάρτης ''chartēs'', "papyrus, sheet of paper, map"; and γράφειν ''graphein'', "write") is the study and practice of making and us ...
ellipsoid, used by all GPS devices, are * (equatorial radius): exactly * (inverse flattening): exactly from which are derived * (polar radius): * (eccentricity squared): The difference between the semi-major and semi-minor axes is about and as fraction of the semi-major axis it equals the flattening; on a computer monitor the ellipsoid could be sized as 300 by 299 pixels. This would barely be distinguishable from a 300-by-300-pixel sphere, so illustrations usually exaggerate the flattening.


Geodetic and geocentric latitudes

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: *''
Geodetic latitude In geography, latitude is a Geographic coordinate system, geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle (defined below) which ranges from 0° at the Equator to 90° (North ...

Geodetic latitude
'': the angle between the normal and the equatorial plane. The standard notation in English publications is . This is the definition assumed when the word latitude is used without qualification. The definition must be accompanied with a specification of the ellipsoid. *'' Geocentric latitude'' (also known as ''spherical latitude'', after the 3D polar angle): the angle between the radius (from centre to the point on the surface) and the equatorial plane. (Figure
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
). There is no standard notation: examples from various texts include , , , , , . This article uses . Geographic latitude must be used with care, as some authors use it as a synonym for geodetic latitude whilst others use it as an alternative to the
astronomical latitude In geography, latitude is a Geographic coordinate system, geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle (defined below) which ranges from 0° at the Equator to 90° (North ...
. "Latitude" (unqualified) should normally refer to the geodetic latitude. The importance of specifying the reference datum may be illustrated by a simple example. On the reference ellipsoid for WGS84, the centre of the
Eiffel Tower The Eiffel Tower ( ; french: links=yes, tour Eiffel ) is a wrought-iron Wrought iron is an iron Iron () is a chemical element with Symbol (chemistry), symbol Fe (from la, Wikt:ferrum, ferrum) and atomic number 26. It is a metal th ...

Eiffel Tower
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
ED50 {{Geodesy ED50 ("European Datum 1950") is a datum (geodesy), geodetic datum which was defined after World War II for the international connection of geodetic networks. Background Some of the important battles of World War II were fought on the b ...
define a point on the ground which is distant from the tower. A web search may produce several different values for the latitude of the tower; the reference ellipsoid is rarely specified.


Meridian distance

The length of a degree of latitude depends on the
figure of the Earth Figure of the Earth is a term of art Jargon is the specialized terminology Terminology is a general word for the group of specialized words or meanings relating to a particular field, and also the study of such terms and their use. This is ...
assumed.


Meridian distance on the sphere

On the sphere the normal passes through the centre and the latitude () is therefore equal to the angle subtended at the centre by the meridian arc from the equator to the point concerned. If the meridian distance is denoted by then : m(\phi)=\fracR\phi_\mathrm = R\phi_\mathrm where denotes the mean radius of the Earth. is equal to . No higher accuracy is appropriate for since higher-precision results necessitate an ellipsoid model. With this value for the meridian length of 1 degree of latitude on the sphere is (60.0 nautical miles). The length of 1 minute of latitude is (1.00 nautical miles), while the length of 1 second of latitude is (see
nautical mile A nautical mile is a unit of length A unit of length refers to any arbitrarily chosen and accepted reference standard for measurement of length. The most common units in modern use are the metric system, metric units, used in every country gl ...
).


Meridian distance on the ellipsoid

In
Meridian arc In geodesy Geodesy () is the Earth science Earth science or geoscience includes all fields of natural science Natural science is a branch of science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic ...
and standard texts it is shown that the distance along a meridian from latitude to the equator is given by ( in radians) :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)^\, d\phi' where is the meridional
radius of curvature In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differentia ...
. The '' quarter meridian'' distance from the equator to the pole is :m_\mathrm = m\left(\frac\right)\, For
WGS84 The World Geodetic System (WGS) is a standard for use in cartography Cartography (; from Greek χάρτης ''chartēs'', "papyrus, sheet of paper, map"; and γράφειν ''graphein'', "write") is the study and practice of making and us ...
this distance is . The evaluation of the meridian distance integral is central to many studies in geodesy and map projection. It can be evaluated by expanding the integral by the binomial series and integrating term by term: see
Meridian arc In geodesy Geodesy () is the Earth science Earth science or geoscience includes all fields of natural science Natural science is a branch of science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic ...
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 for LaTeX code and figures. :\delta m(\phi) = M(\phi)\, \delta\phi = a\left(1 - e^2\right) \left(1 - e^2 \sin^2\phi\right)^\, \delta\phi When the latitude difference is 1 degree, corresponding to radians, the arc distance is about :\Delta^1_\text = \frac The distance in metres (correct to 0.01 metre) between latitudes \phi − 0.5 degrees and \phi + 0.5 degrees on the WGS84 spheroid is :\Delta^1_\text = 111\,132.954 - 559.822\cos 2\phi + 1.175\cos 4\phi The variation of this distance with latitude (on
WGS84 The World Geodetic System (WGS) is a standard for use in cartography Cartography (; from Greek χάρτης ''chartēs'', "papyrus, sheet of paper, map"; and γράφειν ''graphein'', "write") is the study and practice of making and us ...
) is shown in the table along with the length of a degree of longitude (east–west distance): :\Delta^1_\text = \frac\, A calculator for any latitude is provided by the U.S. Government's
National Geospatial-Intelligence Agency The National Geospatial-Intelligence Agency (NGA) is a within the whose primary mission is collecting, analyzing, and distributing (GEOINT) in support of . Initially known as the National Imagery and Mapping Agency (NIMA) from 1996 to 2003, it ...

National Geospatial-Intelligence Agency
(NGA). The following graph illustrates the variation of both a degree of latitude and a degree of longitude with latitude.


Auxiliary latitudes

There are six auxiliary latitudes that have applications to special problems in geodesy, geophysics and the theory of map projections: * Geocentric latitude * Parametric (or reduced) latitude * Rectifying latitude * Authalic latitude * Conformal latitude * Isometric latitude 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
geographic coordinate system A geographic coordinate system (GCS) is a coordinate system associated with positions on Earth (geographic position). A GCS can give positions: *as spherical coordinate system using latitude In geography, latitude is a geographic c ...
as discussed
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
. 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, , and the eccentricity, . (For inverses see
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
.) 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. Derivations of these expressions may be found in Adams (''Note'': Adams uses the nomenclature isometric latitude for the conformal latitude of this article (and throughout the modern literature).) and online publications by Osborne and Rapp.


Geocentric latitude

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 () and the geodetic latitude () is: :\theta(\phi) = \tan^\left(\left(1 - e^2\right)\tan\phi\right) = \tan^\left((1 - f)^2\tan\phi\right)\,. For points not on the surface of the ellipsoid, the relationship involves additionally the
ellipsoidal height 250px, Flattened sphere In geodesy Geodesy () is the Earth science Earth science or geoscience includes all fields of natural science Natural science is a branch of science Science (from the Latin word ''scientia'', meaning "kno ...
''h'': : \theta(\phi,h) = \tan^\left( \frac\tan\phi \right) 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 \phi\theta may be shown to be about 11.5 minutes of arc at a geodetic latitude of approximately 45° 6′.


Parametric latitude (or reduced latitude)

The parametric latitude or reduced latitude, , is defined by the radius drawn from the centre of the ellipsoid to that point on the surrounding sphere (of radius ) which is the projection parallel to the Earth's axis of a point on the ellipsoid at latitude . It was introduced by Legendre and Bessel
Translation:
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, , is also used in the current literature. The parametric latitude is related to the geodetic latitude by: :\beta(\phi) = \tan^\left(\sqrt\tan\phi\right) = \tan^\left((1 - f)\tan\phi\right) The alternative name arises from the parameterization of the equation of the ellipse describing a meridian section. In terms of Cartesian coordinates , the distance from the minor axis, and , the distance above the equatorial plane, the equation of the
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

ellipse
is: : \frac + \frac =1\, . The Cartesian coordinates of the point are parameterized by : p = a\cos\beta\,, \qquad z = b\sin\beta\,; Cayley suggested the term ''parametric latitude'' because of the form of these equations. The parametric latitude is not used in the theory of map projections. Its most important application is in the theory of ellipsoid geodesics, ( Vincenty, Karney).


Rectifying latitude

The rectifying latitude, , is the meridian distance scaled so that its value at the poles is equal to 90 degrees or radians: :\mu(\phi) = \frac\frac where the meridian distance from the equator to a latitude is (see
Meridian arc In geodesy Geodesy () is the Earth science Earth science or geoscience includes all fields of natural science Natural science is a branch of science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic ...
) :m(\phi) = a\left(1 - e^2\right)\int_0^\phi \left(1 - e^2 \sin^2 \phi'\right)^\, d\phi'\,, and the length of the meridian quadrant from the equator to the pole (the polar distance) is :m_\mathrm = m\left(\frac\right)\,. Using the rectifying latitude to define a latitude on a sphere of radius :R = \frac 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 equirectangular projection 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 equidistant conic projection. (Snyder, Section 16). The rectifying latitude is also of great importance in the construction of the
Transverse Mercator projection The transverse Mercator map projection In cartography, a map projection is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the Surf ...
.


Authalic latitude

The authalic latitude (after the Greek for " same area"), , gives an area-preserving transformation to a sphere. :\xi(\phi) = \sin^\left(\frac\right) where :\begin q(\phi) &= \frac - \frac\ln \left(\frac\right) \\ &= \frac + \frac\tanh^(e\sin\phi) \end and :\begin q_\mathrm = q\left(\frac\right) &= 1 - \frac \ln\left(\frac\right) \\ &= 1 + \frac\tanh^e \end and the radius of the sphere is taken as :R_q = a\sqrt\,. An example of the use of the authalic latitude is the
Albers equal-area conic projection Image:Usgs map albers equal area conic.PNG, frame, An Albers projection shows areas accurately, but distorts shapes. The Albers equal-area conic projection, or Albers projection (named after Heinrich C. Albers), is a Map projection#Conic, conic, E ...
.


Conformal latitude

The conformal latitude, , gives an angle-preserving () transformation to the sphere. :\begin \chi(\phi) &= 2\tan^\left \left(\frac\right) \left(\frac\right)^e\right \frac - \frac \\ &= 2\tan^\left \tan\left(\frac + \frac\right) \left(\frac\right)^\frac \right- \frac \\ &= \tan^\left sinh\left(\sinh^(\tan\phi) - e\tanh^(e\sin\phi)\right)\right\\ &= \operatorname\left operatorname^(\phi) - e\tanh^(e\sin\phi)\right\end where is the
Gudermannian function of the Gudermannian function The Gudermannian function, named after Christoph Gudermann (1798–1852), relates the circular functions and hyperbolic function Hyperbolic is an adjective describing something that resembles or pertains to a hyperbo ...
. (See also
Mercator projection The Mercator projection () is a cylindrical map projection presented by Flemish Flemish (''Vlaams'') is a Low Franconian dialect cluster of the Dutch language. It is sometimes referred to as Flemish Dutch (), Belgian Dutch ( ), or Souther ...
.) 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
Transverse Mercator projection The transverse Mercator map projection In cartography, a map projection is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the Surf ...
on the ellipsoid is not a double projection. (It does, however, involve a generalisation of the conformal latitude to the complex plane).


Isometric latitude

The isometric latitude, , is used in the development of the ellipsoidal versions of the normal
Mercator projection The Mercator projection () is a cylindrical map projection presented by Flemish Flemish (''Vlaams'') is a Low Franconian dialect cluster of the Dutch language. It is sometimes referred to as Flemish Dutch (), Belgian Dutch ( ), or Souther ...

Mercator projection
and the
Transverse Mercator projection The transverse Mercator map projection In cartography, a map projection is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the Surf ...
. The name "isometric" arises from the fact that at any point on the ellipsoid equal increments of and longitude give rise to equal distance displacements along the meridians and parallels respectively. The graticule defined by the lines of constant and constant , 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): :\begin \psi(\phi) &= \ln\left tan\left(\frac + \frac\right)\right+ \frac\ln\left frac\right\\ &= \sinh^(\tan\phi) -e\tanh^(e\sin\phi) \\ &= \operatorname^(\phi)-e\tanh^(e\sin\phi). \end 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 (units of length or pixels) then the distance, , of a parallel of latitude from the equator is :y(\phi) = \frac\psi(\phi)\,. The isometric latitude is closely related to the conformal latitude : :\psi(\phi) = \operatorname^ \chi(\phi)\,.


Inverse formulae and series

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. * The first is a numerical inversion of the defining equation for each and every particular value of the auxiliary latitude. The methods available are
fixed-point iteration In numerical analysis, fixed-point iteration is a method of computing fixed point (mathematics), fixed points of a function. More specifically, given a function f defined on the real numbers with real values and given a point x_0 in the domain of ...
and
Newton–Raphson In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better Numerical analysis, approximations to the root of a fun ...

Newton–Raphson
root finding. ** When converting from isometric or conformal to geodetic, two iterations of Newton-Raphson gives
double precision Double-precision floating-point format (sometimes called FP64 or float64) is a computer number format A computer number format is the internal representation of numeric values in digital device hardware and software, such as in programmable comput ...
accuracy. * 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 Lagrange reversion. Such series are presented by Adams who uses Taylor series expansions and gives coefficients in terms of the eccentricity. Osborne derives series to arbitrary order by using the computer algebra package Maxima and expresses the coefficients in terms of both eccentricity and flattening. The series method is not applicable to the isometric latitude and one must find the conformal latitude in an intermediate step.


Numerical comparison of auxiliary latitudes

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. To first order in the flattening ''f'', the auxiliary latitudes can be expressed as ''ζ'' = ''ϕ'' − ''Cf'' sin 2''ϕ'' where the constant ''C'' takes on the values , , 1, 1for ''ζ'' = 'β'', ''ξ'', ''μ'', ''χ'', ''θ''


Latitude and coordinate systems

The geodetic latitude, or any of the auxiliary latitudes defined on the reference ellipsoid, constitutes with longitude a two-dimensional coordinate system 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.


Geodetic coordinates

At an arbitrary point consider the line which is normal to the reference ellipsoid. The geodetic coordinates are the latitude and longitude of the point on the ellipsoid and the distance . 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 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.


Spherical polar coordinates

The geocentric latitude is the complement of the ''polar angle'' or ''colatitude'' in conventional spherical polar coordinates in which the coordinates of a point are where is the distance of from the centre , is the angle between the radius vector and the polar axis and is longitude. Since the normal at a general point on the ellipsoid does not pass through the centre it is clear that points 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.


Ellipsoidal-harmonic coordinates

The parametric latitude can also be extended to a three-dimensional coordinate system. For a point not on the reference ellipsoid (semi-axes and ) construct an auxiliary ellipsoid which is confocal (same foci , ) with the reference ellipsoid: the necessary condition is that the product of semi-major axis and eccentricity is the same for both ellipsoids. Let be the semi-minor axis () of the auxiliary ellipsoid. Further let be the parametric latitude of on the auxiliary ellipsoid. The set define the ellipsoidal-harmonic coordinatesHolfmann-Wellenfor & Moritz (2006) ''Physical Geodesy'', p.240, eq. (6-6) to (6-10). or simply ''ellipsoidal coordinates'' (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 oblate spheroidal coordinates); for a generalization, see triaxial ellipsoidal coordinates.


Coordinate conversions

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 Geographic coordinate conversion. The relation of Cartesian and spherical polars is given in Spherical coordinate system. The relation of Cartesian and ellipsoidal coordinates is discussed in Torge.


Astronomical latitude

Astronomical latitude () is the angle between the equatorial plane and the true vertical direction at a point on the surface. The true vertical, the direction of a plumb line, is also the gravity direction (the resultant of the gravitational acceleration (mass-based) and the centrifugal acceleration) at that latitude. Astronomic latitude is calculated from angles measured between the
zenith The zenith is an imaginary point directly "above" a particular location, on the imaginary celestial sphere. "Above" means in the vertical direction (plumb line) opposite to the gravity direction at that location (nadir). The zenith is the "high ...

zenith
and stars whose declination 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 angle between the astronomic and geodetic normals is called ''vertical deflection'' and is usually a few seconds of arc but it is important in geodesy. The reason why it differs from the normal to the geoid is, because the geoid is an idealized, theoretical shape "at mean sea level". Points on the real surface of the earth are usually above or below this idealized geoid surface and here the true vertical can vary slightly. Also, the true vertical at a point at a specific time is influenced by tidal forces, which the theoretical geoid averages out. Astronomical latitude is not to be confused with declination, the coordinate astronomers use in a similar way to specify the angular position of stars north/south of the celestial equator (see equatorial coordinates), nor with ecliptic latitude, the coordinate that astronomers use to specify the angular position of stars north/south of the
ecliptic The ecliptic is the plane (geometry), plane of Earth's orbit around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic against the ...

ecliptic
(see ecliptic coordinates).


See also

*Altitude (sea level, mean sea level) *Bowditch's American Practical Navigator *Cardinal direction *Circle of latitude *Colatitude *Declination on celestial sphere *Degree Confluence Project *Geodesy *Geodetic datum *Geographic coordinate system *Geographical distance *Geomagnetic latitude *Geotagging *Great-circle distance * History of latitude *Horse latitudes *International Latitude Service *List of countries by latitude *Longitude *Natural Area Code *Navigation *Orders of magnitude (length) *World Geodetic System


References


Footnotes


Citations


External links


GEONets Names Server
access to the
National Geospatial-Intelligence Agency The National Geospatial-Intelligence Agency (NGA) is a within the whose primary mission is collecting, analyzing, and distributing (GEOINT) in support of . Initially known as the National Imagery and Mapping Agency (NIMA) from 1996 to 2003, it ...

National Geospatial-Intelligence Agency
's (NGA) database of foreign geographic feature names.
Resources for determining your latitude and longitude
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