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In
geography Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, an ...
, latitude is a
coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is si ...
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 or geography. Etymology The word ''nor ...
south South is one of the cardinal directions or compass points. The direction is the opposite of north and is perpendicular to both east and west. Etymology The word ''south'' comes from Old English ''sūþ'', from earlier Proto-Germanic ''*sun ...
position of a point on the surface of
the Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surface ...
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
Equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term ca ...
. Lines of constant latitude, or ''parallels'', run east–west as circles parallel to the equator. Latitude and ''
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek l ...
'' are used together as a coordinate pair to specify a location on the surface of the Earth. 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 '' normal'') to the ellipsoidal surface from the point, and the plane of the equator.


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 surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is exte ...
, a surface which approximates the
mean sea level There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' 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
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
, 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 Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is ab ...
constitute a
geographic coordinate system The geographic coordinate system (GCS) is a spherical or ellipsoidal coordinate system for measuring and communicating positions directly on the Earth as latitude and longitude. It is the simplest, oldest and most widely used of the variou ...
as defined in the specification of the ISO 19111 standard. Since there are many different
reference ellipsoid An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximatio ...
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 and operated by the United States Space Force. It is one of the global navigation satellite ...
(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: ''fi'' ) is the 21st letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
( or ). It is measured in degrees, minutes and seconds or
decimal degrees Decimal degrees (DD) is a notation for expressing latitude and longitude geographic coordinates as decimal fractions of a degree. DD are used in many geographic information systems (GIS), web mapping applications such as OpenStreetMap, and GPS ...
, 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 that rise about out of the sea off the western extremity of the Isle of Wight in the English Channel, United Kingdom, close to Alum Bay and Scratchell's Bay, and part of Totland, the wester ...
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 A planetary coordinate system is a generalization of the geographic coordinate system and the geocentric coordinate system for planets other than Earth. Similar coordinate systems are defined for other solid celestial bodies, such as in the ''sele ...
). For a brief history see
History of latitude The Greeks studied the results of the measurements of latitude by the explorer Pytheas who voyaged to Britain and beyond, as far as the Arctic Circle (observing the midnight sun), in 325 BC. They used several methods to measure latitude, includ ...
.


Determination

In
celestial navigation Celestial navigation, also known as astronavigation, is the practice of position fixing using stars and other celestial bodies that enables a navigator to accurately determine their actual current physical position in space (or on the surface o ...
, latitude is determined with the
meridian altitude Meridian altitude is a method of celestial navigation to calculate an observer's latitude. It notes the altitude angle of an astronomical object above the horizon at culmination. Principle Meridian altitude is the simplest calculation of celestia ...
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 angles between designated visible points in the horizontal and vertical planes. The traditional use has been for land surveying, but it is also used extensively for building and ...
s or to determine GPS satellite orbits. The study of the
figure of the Earth Figure of the Earth is a 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. A sphere is a well-known historical approxima ...
together with its gravitational field is the science of
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure ( geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equiv ...
.


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 Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Cen ...
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 A prime meridian is an arbitrary 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 in a 360°-system) form a great ...
) 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 equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term ca ...
. 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 The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface. It is called the True North Pole to distinguish from the Ma ...
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 two points where Earth's axis of rotation intersects its surface. It is the southernmost point on Earth and lies antipod ...
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.


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 or ecliptic plane is the orbital plane of the Earth 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 agains ...
, 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 and periodization, an epoch or reference epoch is an instant in time chosen as the origin of a particular calendar era. The "epoch" serves as a reference point from which time is measured. The moment of epoch is usually decided ...
. The time variation is discussed more fully in the article on
axial tilt In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and orbi ...
. 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. Two solstices occur annually, around June 21 and December 21. In many coun ...
when the Sun is overhead at some point of the
Tropic of Capricorn The Tropic of Capricorn (or the Southern Tropic) is the circle of latitude that contains the subsolar point at the December (or southern) solstice. It is thus the southernmost latitude where the Sun can be seen directly overhead. It also reac ...
. The south polar latitudes below the
Antarctic Circle The Antarctic Circle is the most southerly of the five major circles of latitude that mark maps of Earth. The region south of this circle is known as the Antarctic, and the zone immediately to the north is called the Southern Temperate Zone. S ...
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 regions of Earth surrounding the Equator. They are defined in latitude by the Tropic of Cancer in the Northern Hemisphere at N and the Tropic of Capricorn in the Southern Hemisphere at S. The tropics are also referred ...
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 celestial sphere. "Above" means in the vertical direction ( plumb line) opposite to the gravity direction at that location ( nadir). The zenith is the "highes ...
). On
map projections In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitu ...
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 geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because it is unique in representing north as up and ...
and the
Transverse Mercator projection The transverse Mercator map projection (TM, TMP) is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the Universal Transverse Merca ...
. 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, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the ...
published the ''
Philosophiæ Naturalis Principia Mathematica ( English: ''Mathematical Principles of Natural Philosophy'') often referred to as simply the (), is a book by Isaac Newton that expounds Newton's laws of motion and his law of universal gravitation. The ''Principia'' is written in Latin ...
'', in which he proved that a rotating self-gravitating fluid body in equilibrium takes the form of an
oblate In Christianity (especially in the Roman Catholic, Orthodox, Anglican and Methodist traditions), an oblate is a person who is specifically dedicated to God or to God's service. Oblates are individuals, either laypersons or clergy, normally li ...
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 and navigation, a meridian arc is the curve between two points on the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length. The purpose of measuring meridian arcs is to d ...
.) 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 figure ( geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equiv ...
. In pre-satellite days they were devised to give a good fit to the
geoid The geoid () is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is exte ...
over the limited area of a survey but, with the advent of GPS, it has become natural to use reference ellipsoids (such as
WGS84 The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also descr ...
) 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 mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of 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, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the long ...
, . The other parameter is usually (1) the polar radius or
semi-minor axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the long ...
, ; or (2) the (first)
flattening Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surf ...
, ; 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 ...
, . 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 mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse ...
,
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as t ...
) 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 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. A sphere is a well-known historical approxima ...
. 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 used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also descr ...
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 Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a ''reference ellipsoid''. They include geodetic latitude (north/south) , ''longitude'' (east/west) , and ellipsoidal height (also known as ge ...
'': 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 In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
'' (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). 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 coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north po ...
. "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 lattice tower on the Champ de Mars in Paris, France. It is named after the engineer Gustave Eiffel, whose company designed and built the tower. Locally nicknamed ...
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 ED50 ("European Datum 1950", EPSG:4230) is a 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 borders of Ger ...
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 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. A sphere is a well-known historical approxima ...
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 used in air, marine, and space navigation, and for the definition of territorial waters. Historically, it was defined as the meridian arc length corresponding to one minute ( of a degree) of latitude. To ...
).


Meridian distance on the ellipsoid

In
Meridian arc In geodesy and navigation, a meridian arc is the curve between two points on the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length. The purpose of measuring meridian arcs is to d ...
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. 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 used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also descr ...
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 and navigation, a meridian arc is the curve between two points on the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length. The purpose of measuring meridian arcs is to d ...
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 used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also descr ...
) 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 combat support agency within the United States Department of Defense whose primary mission is collecting, analyzing, and distributing geospatial intelligence (GEOINT) in support of nation ...
(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 In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
* 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 The geographic coordinate system (GCS) is a spherical or ellipsoidal coordinate system for measuring and communicating positions directly on the Earth as latitude and longitude. It is the simplest, oldest and most widely used of the variou ...
as discussed below. 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.) 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 Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a '' reference ellipsoid''. They include geodetic latitude (north/south) , ''longitude'' (east/west) , and ellipsoidal height (also known as g ...
''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 mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of 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 and navigation, a meridian arc is the curve between two points on the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length. The purpose of measuring meridian arcs is to d ...
) :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 The equirectangular projection (also called the equidistant cylindrical projection or la carte parallélogrammatique projection), and which includes the special case of the plate carrée projection (also called the geographic projection, lat/lon ...
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 (TM, TMP) is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the Universal Transverse Merca ...
.


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) \\ pt &= \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.


Conformal latitude

The conformal latitude, , gives an angle-preserving ( conformal) transformation to the sphere. :\begin \chi(\phi) &= 2\tan^\left \left(\frac\right) \left(\frac\right)^e\right \frac - \frac \\ pt &= 2\tan^\left \tan\left(\frac + \frac\right) \left(\frac\right)^\frac \right- \frac \\ pt &= \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. (See also
Mercator projection The Mercator projection () is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because it is unique in representing north as up and ...
.) 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 (TM, TMP) is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the Universal Transverse Merca ...
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 geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because it is unique in representing north as up and ...
and the
Transverse Mercator projection The transverse Mercator map projection (TM, TMP) is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the Universal Transverse Merca ...
. 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 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 approximations to the roots (or zeroes) of a real ...
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 floating-point number format, usually occupying 64 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. Fl ...
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 In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
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 a spherical coordinate system, a colatitude is the complementary angle of a given latitude, i.e. the difference between a right angle and the latitude. Here Southern latitudes are defined to be negative, and as a result the colatitude is a non ...
'' 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 In geodesy, conversion among different geographic coordinate systems is made necessary by the different geographic coordinate systems in use across the world and over time. Coordinate conversion is composed of a number of different types of convers ...
. The relation of Cartesian and spherical polars is given in
spherical coordinate system In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' mea ...
. 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 In astronomy, geography, and related sciences and contexts, a '' direction'' or '' plane'' passing by a given point is said to be vertical if it contains the local gravity direction at that point. Conversely, a direction or plane is said to be h ...
at a point on the surface. The true vertical, the direction of a
plumb line A plumb bob, plumb bob level, or plummet, is a weight, usually with a pointed tip on the bottom, suspended from a string and used as a vertical reference line, or plumb-line. It is a precursor to the spirit level and used to establish a verti ...
, is also the gravity direction (the resultant of the
gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodi ...
(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 celestial sphere. "Above" means in the vertical direction ( plumb line) opposite to the gravity direction at that location ( nadir). The zenith is the "highes ...
and stars whose
declination In astronomy, declination (abbreviated dec; symbol ''δ'') is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of th ...
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 The vertical deflection (VD) or deflection of the vertical (DoV), also known as deflection of the plumb line and astro-geodetic deflection, is a measure of how far the gravity direction at a given point of interest is rotated by local mass anom ...
'' 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 In astronomy, declination (abbreviated dec; symbol ''δ'') is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of th ...
, the coordinate
astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either ...
s use in a similar way to specify the angular position of stars north/south of the
celestial equator The celestial equator is the great circle of the imaginary celestial sphere on the same plane as the equator of Earth. This plane of reference bases the equatorial coordinate system. In other words, the celestial equator is an abstract proje ...
(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 or ecliptic plane is the orbital plane of the Earth 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 agains ...
(see
ecliptic coordinates The ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions, orbits, and pole orientations of Solar System objects. Because most planets (except Mercury) and many small Solar System bod ...
).


See also

*
Altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
(
mean sea level There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' a ...
) *
Bowditch's American Practical Navigator ''The American Practical Navigator'' (colloquially often referred to as ''Bowditch''), originally written by Nathaniel Bowditch, is an encyclopedia of navigation. It serves as a valuable handbook on oceanography and meteorology, and contains use ...
*
Cardinal direction The four cardinal directions, or cardinal points, are the four main compass directions: north, east, south, and west, commonly denoted by their initials N, E, S, and W respectively. Relative to north, the directions east, south, and west ar ...
*
Circle of latitude A circle of latitude or line of latitude on Earth is an abstract east– west small circle connecting all locations around Earth (ignoring elevation) at a given latitude coordinate line. Circles of latitude are often called parallels be ...
*
Colatitude In a spherical coordinate system, a colatitude is the complementary angle of a given latitude, i.e. the difference between a right angle and the latitude. Here Southern latitudes are defined to be negative, and as a result the colatitude is a non ...
*
Declination In astronomy, declination (abbreviated dec; symbol ''δ'') is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of th ...
on
celestial sphere In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial s ...
* Degree Confluence Project *
Geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure ( geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equiv ...
*
Geodetic datum A geodetic datum or geodetic system (also: geodetic reference datum, geodetic reference system, or geodetic reference frame) is a global datum reference or reference frame for precisely representing the position of locations on Earth or other p ...
*
Geographic coordinate system The geographic coordinate system (GCS) is a spherical or ellipsoidal coordinate system for measuring and communicating positions directly on the Earth as latitude and longitude. It is the simplest, oldest and most widely used of the variou ...
*
Geographical distance Geographical distance or geodetic distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitud ...
* Geomagnetic latitude *
Geotagging Geotagging, or GeoTagging, is the process of adding geographical identification metadata to various media such as a geotagged photograph or video, websites, SMS messages, QR Codes or RSS feeds and is a form of geospatial metadata. This dat ...
*
Great-circle distance The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle. It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a ...
*
History of latitude The Greeks studied the results of the measurements of latitude by the explorer Pytheas who voyaged to Britain and beyond, as far as the Arctic Circle (observing the midnight sun), in 325 BC. They used several methods to measure latitude, includ ...
*
Horse latitudes The horse latitudes are the latitudes about 30 degrees north and south of the Equator. They are characterized by sunny skies, calm winds, and very little precipitation. They are also known as subtropical ridges, or highs. It is a high-pressure ...
*
International Latitude Service __NOTOC__ The International Latitude Service was created by the International Geodetic Association in 1899 to study variations in latitude caused by polar motion, precession, or "wobble" of the Earth's axis. The original ''International Lati ...
* List of countries by latitude *
Longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek l ...
* Natural Area Code *
Navigation Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
*
Orders of magnitude (length) The following are examples of orders of magnitude for different lengths. __TOC__ Overview Detailed list To help compare different orders of magnitude, the following list describes various lengths between 1.6 \times 10^ metres and 10 ...
*
World Geodetic System The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also descr ...


References


Footnotes


Citations


External links


GEONets Names Server
, access to the
National Geospatial-Intelligence Agency The National Geospatial-Intelligence Agency (NGA) is a combat support agency within the United States Department of Defense whose primary mission is collecting, analyzing, and distributing geospatial intelligence (GEOINT) in support of nation ...
's (NGA) database of foreign geographic feature names.
Resources for determining your latitude and longitude


- Info about decimal to
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified for ...
conversion
Convert decimal degrees into degrees, minutes, secondsDistance calculation based on latitude and longitude
- JavaScript version
16th Century Latitude Survey
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