A graticule on a sphere or an ellipsoid. The lines from pole to pole are lines of constant longitude, or meridians. The circles parallel to the equator are lines of constant latitude, or parallels. The graticule determines the latitude and longitude of position on the surface.

In geography, latitude is a geographic coordinate that specifies the north-south position of a point on the Earth's surface. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is an angle (defined below) which ranges from 0° at the Equator to 90° (North or South) at the poles.

Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. Since the actual physical surface of the Earth is too complex for mathematical analysis two levels of abstraction are employed in the definition of these coordinates. In the first step the physical surface is modelled by the geoid, a surface which approximates the 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, but the geoid is more accurately modelled 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 constitute a geographic coordinate system as defined in the specification of the ISO 19111 standard.^[1]

Since there are many different reference ellipsoids the 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 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 letter phi (φ). It is measured in degrees, minutes and seconds or decimal degrees, north or south of the equator.

Measurement of latitude requires an understanding of the gravitational field of the Earth, either for setting up theodolites or for determination of GPS satellite orbits. The study of the Figure of the Earth together with its gravitational field is the science of Geodesy. These topics are not discussed in this article. (See for example the text books by Torge^[2] and Hofmann-Wellenhof and Moritz.^[3])

This article relates to coordinate systems for the Earth: it may be extended to cover the Moon, planets and other celestial objects by a simple change of nomenclature.

The following lists are available:

• List of countries by latitude
• List of cities by latitude

Latitude on the sphere[link]

A perspective view of the Earth showing how latitude (φ) and longitude (λ) are defined on a spherical model. The graticule spacing is 10 degrees.

The graticule on the sphere

The graticule, the mesh formed by the lines of constant latitude and constant longitude, is constructed by reference to the rotation axis of the Earth. The primary reference points are the poles where the axis of rotation of the Earth intersects the reference surface. Planes which contain the rotation axis intersect the surface in the meridians and the angle between any one meridian plane and that through Greenwich (the Prime Meridian) defines the longitude: meridians are lines of constant longitude. The plane through the centre of the Earth and orthogonal to the rotation axis intersects the surface in a great circle called the 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 has a latitude of 90° north (written 90° N or +90°), and the South pole 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 radius to that point.

The latitude defined in this way for the sphere is often termed the spherical latitude to avoid ambiguity with auxiliary latitudes defined in subsequent sections.

Named latitudes[link]

The orientation of the Earth at the December solstice.

Besides the equator, four other parallels are of significance:

Arctic Circle	66° 33′ 39″ N
Tropic of Cancer	23° 26′ 21″ N
Tropic of Capricorn	23° 26′ 21″ S
Antarctic Circle	66° 33′ 39" S

The plane of the Earth's orbit about the sun is called the ecliptic. 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 the inclination of the ecliptic, denoted by Failed to parse (Missing texvc executable; please see math/README to configure.): i

in the figure. The current value of this angle is 23° 26′ 21″.[4] It is also called the axial tilt of the Earth since it is equal to the angle between the axis of rotation and the normal to the ecliptic.

The figure shows the geometry of a cross section of the plane normal to the ecliptic and through the centres of the Earth and the Sun at the December solstice when the sun is overhead at some point of the Tropic of Capricorn. The south polar latitudes below the 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. The latitudes of the tropics are equal to the inclination of the ecliptic and the polar circles are at latitudes equal to its complement. Only at latitudes in between the two tropics is it possible for the sun to be directly overhead (at the zenith).

The named parallels are clearly indicated on the Mercator projections shown below.

Map projections from the sphere

On map projections there is no simple rule as to how meridians and parallels should appear. For example, on the spherical Mercator projection the parallels are horizontal and the meridians are vertical whereas on the Transverse Mercator projection there is no correlation of parallels and meridians with horizontal and vertical, both are complicated curves. The red lines are the named latitudes of the previous section.

	Normal Mercator			Transverse Mercator

For map projections of large regions, or the whole world, a spherical Earth model is completely satisfactory since the variations attributable to ellipticity are negligible on the final printed maps.

Meridian distance on the sphere

On the sphere the normal passes through the centre and the latitude (φ) is therefore equal to the angle subtended at the centre by the meridian arc from the equator to the point concerned. If the meridian distance is denoted by m(φ) then

Failed to parse (Missing texvc executable; please see math/README to configure.): m(\phi)=\frac{\pi}{180}R\phi_{\rm degrees}= R\phi_{\rm radians}.

where R denotes the mean radius of the Earth. R is equal to 6371 km or 3959 miles. No higher accuracy is appropriate for R since higher precision results necessitate an ellipsoid model. With this value for R the meridian length of 1 degree of latitude on the sphere is 111.2 km or 69 miles. The length of 1 minute of latitude is 1.853 km, or 1.15 miles. (See nautical mile).

Latitude on the ellipsoid[link]

Ellipsoids[link]

In 1687 Isaac Newton published the Principia in which he included a proof that a rotating self-gravitating fluid body in equilibrium takes the form of an oblate ellipsoid.^[5] (This article uses the term ellipsoid in preference to the older term spheroid). Newton's theoretical result was confirmed by precise geodetic measurements in the eighteenth century. (See Meridian arc). The definition of an oblate ellipsoid is the three dimensional surface generated by the rotation of a two dimensional ellipse about its shorter axis (minor axis). 'Oblate ellipsoid of revolution' is abbreviated to ellipsoid in the remainder of this article: this is the current practice in geodetic literature. (Ellipsoids which do not have an axis of symmetry are termed tri-axial).

A great many different reference ellipsoids have been used in the history of geodesy. In pre-satellite days they were devised to give a good fit to the geoid over the limited area of a survey but, with the advent of GPS, it has become natural to use reference ellipsoids (such as WGS84) with centres 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 100m of the geoid. Since latitude is defined with respect to an ellipsoid the position of a given feature is different on each ellipsoid: it is meaningless to specify the latitude and longitude of a geographical feature without specifying the ellipsoid used. The maps maintained by many national agencies are often based on the older ellipsoids so that is necessary to know how the latitude and longitude values may be transformed from one ellipsoid to another. For example GPS handsets must include software to carry out datum transformations which link WGS84 to the local reference ellipsoid with its associated grid reference system.

The geometry of the ellipsoid[link]

The overall shape of an ellipsoid of revolution is determined by the shape of the ellipse which is rotated about its minor (shorter) axis. Two parameters are required. One is invariably chosen to be the equatorial radius, which is the semi-major axis, a. The other parameter is usually taken as one of three possibilities: (1) the polar radius or semi-minor axis, b; (2) the (first) flattening, f; (3) the eccentricty, e. These parameters are not independent: they are related by

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} f&=\frac{a-b}{a}, \qquad e^2=2f-f^2,\qquad b=a(1-f)=a\sqrt{1-e^2}. \end{align}

A great many other parameters (see ellipse, ellipsoid) are used in the study of the ellipsoid in geodesy, geophysics and map projections but they can all may be expressed in terms of one or two members of the set a, b, f and e. Both f and e are small parameters which often appear in series expansions involved in practical calculations; they are of the order 1/300 and 0.08 respectively. Accurate values for a number of important ellipsoids are given in Figure of the Earth. It is conventional to define reference ellipsoids by giving the semi-major axis and the inverse flattening, 1/f. For example, the defining values for the WGS84 ellipsoid, used by all GPS devices, are^[6]

• a (equatorial radius): 6,378,137.0 m
• 1/f (inverse flattening): 298.257,223,563

from which are derived

• b (polar radius): 6,356,752.3142 m
• e² (eccentricity squared): 0.006,694,379,990,14

The difference of the major and minor semi-axes is approximately 21 km and as fraction of the semi-major axis it is given by the flattening. Thus the representation of the ellipsoid on a computer could be sized as 300px by 299px. Since this would be indistinguishable from a sphere shown as 300px by 300px illustrations invariably greatly exaggerate the flattening.

Geodetic and geocentric latitudes[link]

The definition of geodetic latitude (φ) and longitude (λ) on an ellipsoid. The normal to the surface does not pass through the centre.

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: the angle between the normal and the equatorial plane. The standard notation in English publications is φ. This is the definition assumed when the word latitude is used without qualification. The definition must be accompanied with a specification of the ellipsoid.

Geocentric latitude: 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 ψ, q, φ', φ_c, φ_g. This article uses ψ.

Spherical latitude: the angle between the normal to a spherical reference surface and the equatorial plane.

Geographic latitude must be used with care. Some authors use it as a synonym for geodetic latitude whilst others use it as an alternative to the astronomical latitude.

Latitude (unqualified) should normally refer to the geodetic latitude.

The importance of specifying the reference datum may be illustrated by a simple example. On the reference ellipsoid for WGS84, the centre of the Eiffel Tower has a geodetic latitude of 48° 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 define a point on the ground which is 140 m distant from Tower.^{[citation needed]} A web search may produce several different values for the latitude of the Tower; the reference ellipsoid is rarely specified.

The length of a degree of latitude[link]

In Meridian arc and standard texts^[2][7][8] it is shown that the meridian distance from a point at latitude φ to the equator is given by (φ in radians)

Failed to parse (Missing texvc executable; please see math/README to configure.): m(\phi) =\int_0^\phi M(\phi) d\phi = a(1 - e^2)\int_0^\phi \left (1 - e^2 \sin^2 \phi \right )^{-3/2} d\phi.

The function Failed to parse (Missing texvc executable; please see math/README to configure.): M(\phi)

in the first integral is the meridional radius of curvature. 

The distance from the equator to the pole is

Failed to parse (Missing texvc executable; please see math/README to configure.): m_p = m(\pi/2).\,

For WGS84 this distance is 10001.965729 km.

The evaluation of the meridian distance integral is central to many studies in geodesy and map projection. It is normally evaluated by expanding the integral by the binomial series and integrating term by term: see Meridian arc for details. The length of the meridian arc between two given latitudes is given by replacing the limits of the integral by the latitudes concerned. The length of a small meridian arc is given by^[7][8]

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \delta m(\phi) &= M(\phi) \delta\phi = a(1 - e^2) \left (1 - e^2 \sin^2 \phi \right )^{-3/2} \delta\phi\,. \end{align}

When the latitude difference is 1 degree, corresponding to Failed to parse (Missing texvc executable; please see math/README to configure.): \pi /180 radians, the arc distance is approximately

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta^1_{\rm LAT}= \frac{\pi a(1 - e^2)}{180(1 - e^2 \sin^2 \phi )^{3/2}} \,

The distance in metres between latitudes (Failed to parse (Missing texvc executable; please see math/README to configure.): \phi - 0.5

deg) and (Failed to parse (Missing texvc executable; please see math/README to configure.):  \phi + 0.5
deg) on the WGS84 spheroid is given to within one centimetre by

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta^1_{\rm LAT}= 111132.954 - 559.822\cos 2\phi + 1.175\cos 4\phi.

The variation of this distance with latitude (on WGS84) is shown in the table along with the length of a degree of longitude:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta^1_{\rm LONG}= \frac{\pi a\cos\phi}{180(1 - e^2 \sin^2 \phi)^{1/2}}\,.

A calculator for any latitude is provided by the U.S. government's National Geospatial-Intelligence Agency(NGA).^[9]

Auxiliary latitudes[link]

There are six auxiliary latitudes that have applications to special problems in geodesy, geophysics and the theory of map projections:

• geocentric latitude,
• reduced (or parametric) 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 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: their numerical values are not of interest. For example no one would need to calculate the authalic latitude of the Eiffel Tower.

The expressions below give the auxiliary latitudes in terms of the geodetic latitude, the semi-major axis, a, and the eccentricity, e. 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.^[10] Derivations of these expressions may be found in Adams^[11] and web publications by Rapp^[7] and Osborne^[8]

Geocentric latitude[link]

The definition of geodetic (or geographic) and geocentric latitudes.

The geocentric latitude is the angle between the equatorial plane and the radius from the centre to a point on the surface. The relation between the geocentric latitude (ψ) and the geodetic latitude (φ) is derived in the above references as

Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(\phi)=\tan^{-1}\left[(1-e^2)\tan\phi\right]\;\!.

The geodetic and geocentric latitudes are equal at the equator and poles. The value of the squared eccentricity is approximately 0.007 (depending on the choice of ellipsoid) and the maximum difference of (φ-ψ) is approximately 11.5 minutes of arc at a geodetic latitude of 45°5′.

Reduced (or parametric) latitude[link]

Definition of the reduced latitude (β) on the ellipsoid.

The reduced or parametric latitude, β, is defined by the radius drawn from the centre of the ellipsoid to that point Q on the surrounding sphere (of radius a) which is the projection parallel to the Earth's axis of a point P on the ellipsoid at latitude Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle\phi . It was introduced by Legendre^[12] and Bessel^[13] 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, Failed to parse (Missing texvc executable; please see math/README to configure.): u(\phi) , is also used in the current literature. The reduced latitude is related to the geodetic latitude by^[7][8]

Failed to parse (Missing texvc executable; please see math/README to configure.): \beta(\phi)=\tan^{-1}\left[\sqrt{1-e^2}\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 p, the distance from the minor axis, and z, the distance above the equatorial plane, the equation of the ellipse is

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{p^2}{a^2} + \frac{z^2}{b^2} =1 .

The Cartesian coordinates of the point are parameterized by

Failed to parse (Missing texvc executable; please see math/README to configure.): p=a\cos\beta, \qquad z=b\sin\beta;

Cayley^[14] suggested the term parametric latitude because of the form of these equations.

The reduced latitude is not used in the theory of map projections. Its most important application is in the theory of ellipsoid geodesics. (Vincenty, Karney^[15]).

Rectifying latitude[link]

The rectifying latitude, μ, is the meridian distance scaled so that its value at the poles is equal to 90 degrees or π/2 radians:

 Failed to parse (Missing texvc executable; please see math/README to configure.): \mu(\phi)=\frac{\pi}{2}\frac{m(\phi)}{m_p}\,\!

where the meridian distance from the equator to a latitude φ is (see Meridian arc)

Failed to parse (Missing texvc executable; please see math/README to configure.): m(\phi)= a(1 - e^2)\int_0^\phi \left (1 - e^2 \sin^2 \phi \right )^{-3/2} d\phi,

and the length of the meridian quadrant from the equator to the pole is

Failed to parse (Missing texvc executable; please see math/README to configure.): m_p = m(\pi/2).\,

Using the rectifying latitude to define a latitude on a sphere of radius

Failed to parse (Missing texvc executable; please see math/README to configure.): R=\frac{2m_p}{\pi}

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,^[10] Section 16). The rectifying latitude is also of great importance in the construction of the Transverse Mercator projection.

Authalic latitude[link]

The authalic (Greek for same area) latitude, ξ, gives an area-preserving transformation to a sphere.

Failed to parse (Missing texvc executable; please see math/README to configure.): \xi(\phi)=\sin^{-1}\left(\frac{q(\phi)}{q_p}\right)\,\!

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} q(\phi)&= \frac{(1 - e^2)\sin\phi}{1 - e^2 \sin^2 \phi} -\frac{1-e^2}{2e}\ln \left(\frac{1-e\sin\phi}{1+e\sin\phi}\right),\\ &= \frac{(1 - e^2)\sin\phi}{1 - e^2 \sin^2 \phi} +\frac{1-e^2}{e}\tanh^{-1}(e\sin\phi), \end{align}

and

Failed to parse (Missing texvc executable; please see math/README to configure.): q_p = q(\pi/2) =1-\frac{1-e^2}{2e}\ln \left(\frac{1-e}{1+e}\right) =1+\frac{1-e^2}{e}\tanh^{-1}e, \,

and the radius of the sphere is taken as

Failed to parse (Missing texvc executable; please see math/README to configure.): R_q=a\sqrt{q_p/2}.\,

An example of the use of the authalic latitude is the Albers equal-area conic projection. (Snyder,^[10] Section 14).

Conformal latitude[link]

The conformal latitude, χ, gives an angle-preserving (conformal) transformation to the sphere.

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \chi(\phi)&=2\tan^{-1}\left[ \left(\frac{1+\sin\phi}{1-\sin\phi}\right) \left(\frac{1-e\sin\phi}{1+e\sin\phi}\right)^{\!\textit{e}} \;\right]^{1/2} -\frac{\pi}{2}\\[2ex] &=2\tan^{-1}\left[ \tan\left(\frac{\phi}{2}+\frac{\pi}{4}\right) \left(\frac{1-e\sin\phi}{1+e\sin\phi}\right)^{\!\textit{e}/2} \;\right] -\frac{\pi}{2} \;\!\end{align}

. 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 on the ellipsoid is not a double projection. (It does, however, involve a generalisation of the conformal latitude to the complex plane).

Isometric latitude[link]

The isometric latitude is conventionally denoted by ψ (not to be confused with the geocentric latitude): it is used in the development of the ellipsoidal versions of the normal Mercator projection and the Transverse Mercator projection. The name "isometric" arises from the fact that at any point on the ellipsoid equal increments of ψ and longitude λ give rise to equal distance displacements along the meridians and parallels respectively. The graticule defined by the lines of constant ψ and constant λ, divides the surface of the ellipsoid into a mesh of squares (of varying size). The isometric latitude is zero at the equator but rapidly diverges from the geodetic latitude, tending to infinity at the poles. The conventional notation is given in Snyder^[10] (page 15):

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \psi(\phi) &=\ln\left[\tan\left( \frac{\pi}{4}+\frac{\phi}{2}\right) \right] + \frac{e}{2}\ln\left[ \frac{1-e\sin\phi}{1+e\sin\phi} \right]\\ &=\sinh^{-1}(\tan\phi) -e\tanh^{-1}(e\sin\phi). \end{align}

For the normal Mercator projection (on the ellipsoid) this function defines the spacing of the parallels: if the length of the equator on the projection is E (units of length or pixels) then the distance, y, of a parallel of latitude φ from the equator is

Failed to parse (Missing texvc executable; please see math/README to configure.): y(\phi)=\frac{E}{2\pi}\psi(\phi).

Numerical comparison of auxiliary latitudes[link]

The following plot shows the magnitude of the difference between the geodetic latitude, (denoted as the 'common' latitude on the plot), and the auxiliary latitudes other than the isometric latitude (which diverges to infinity at the poles). In every case the geodetic latitude is the greater. The differences shown on the plot are in arc minutes. The horizontal resolution of the plot fails to make clear that the maxima of the curves are not at 45° but calculation shows that they are within a few arc minutes of 45°. Some representative data points are given in the table following the plot. Note the closeness of the conformal and geocentric latitudes. This was exploited in the days of hand calculators to expedite the construction of map projections. (Snyder,^[10] page 108).

Approximate difference from geodetic latitude (Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,\! )
Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,\!	Reduced Failed to parse (Missing texvc executable; please see math/README to configure.): \phi-\beta\,\!	Authalic Failed to parse (Missing texvc executable; please see math/README to configure.): \phi-\xi\,\!	Rectifying Failed to parse (Missing texvc executable; please see math/README to configure.): \phi-\mu\,\!	Conformal Failed to parse (Missing texvc executable; please see math/README to configure.): \phi-\chi\,\!	Geocentric Failed to parse (Missing texvc executable; please see math/README to configure.): \phi-\psi\,\!
0°	0.00′	0.00′	0.00′	0.00′	0.00′
15°	2.91′	3.89′	4.37′	5.82′	5.82′
30°	5.05′	6.73′	7.57′	10.09′	10.09′
45°	5.84′	7.78′	8.76′	11.67′	11.67′
60°	5.06′	6.75′	7.59′	10.12′	10.13′
75°	2.92′	3.90′	4.39′	5.85′	5.85′
90°	0.00′	0.00′	0.00′	0.00′	0.00′

Latitude and coordinate systems[link]

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 reduced latitudes are used in geodetic coordinates, spherical polar coordinates and ellipsoidal coordinates respectively.

Geodetic coordinates[link]

Geodetic coordinates P(φ, λ,h)

At an arbitrary point P consider the line PN which is normal to the reference ellipsoid. The geodetic coordinates P(φ, λ,h) are the latitude and longitude of the point N on the ellipsoid and the distance PN. This height differs from the height above the geoid or a reference height such as that above mean sea level at a specified location. The direction of PN will also differ from the direction of a vertical plumb line. The relation of these different heights requires knowledge of the shape of the geoid and also the gravity field of the Earth.

Spherical polar coordinates[link]

Geocentric coordinate related to spherical polar coordinates P(r, θ, λ)

The geocentric latitude ψ is the complement of the polar angle θ in conventional spherical polar coordinates in which the coordinates of a point are P(r, θ, λ) where r is the distance of P from the centre O, θ is the angle between the radius vector and the polar axis andλ 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 geocentic latitudes. Spherical polar coordinate systems are used in the analysis of the gravity field.

Ellipsoidal coordinates[link]

Ellipsoidal coordinates P(u,β,λ)

The reduced latitude can also be extended to a three dimensional coordinate system. For a point P not on the reference ellipsoid (semi-axes OA and OB) construct an auxiliary ellipsoid which is confocal (same foci F, F') with the reference ellipsoid: the necessary condition is that the product ae of semi-major axis and eccentricity is the same for both ellipsoids. Let u be the semi-minor axis (OD) of the auxiliary ellipsoid. Further let β be the reduced latitude of P on the auxiliary ellipsoid. The set (u,β,λ) define the ellipsoid coordinates. (Torge^[2] Section 4.2.2). These coordinates are the natural choice in models of the gravity field for a uniform distribution of mass bounded by the reference ellipsoid.

Coordinate conversions[link]

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 Geodetic system. The relation of Cartesian and spherical polars is given in Spherical coordinate system. The relation of Cartesian and ellipsoidal coordinates is discussed in Torge.^[2]

Astronomical latitude[link]

The astronomical latitude (Φ) is defined as the angle between the equatorial plane and the true vertical at a point on the surface: the true vertical, the direction of a plumb line, is the direction of the gravity field at that point. (The gravity field is the resultant of the gravitational acceleration and the centrifugal acceleration at that point. See Torge.^[2]) The astronomic latitude is readily accessible by measuring the angle between the zenith and the celestial pole or alternatively the angle between the zenith and the sun when the elevation of the latter is obtained from the almanac.

In general the direction of the true vertical at a point on the surface does not coincide with the either the normal to the reference ellipsoid or the normal to the geoid. The deflections between the astronomic and geodetic normals are only of the order of a few seconds of arc but they are nevertheless important in high precision geodesy.^[2][3]

Astronomical latitude is not to be confused with declination, the coordinate astronomers used in a simlilar way to describe the locations of stars north/south of the celestial equator (see equatorial coordinates), nor with ecliptic latitude, the coordinate that astronomers use to describe the locations of stars north/south of the ecliptic (see ecliptic coordinates).

See also[link]

Footnotes[link]

External links[link]

Look up latitude in Wiktionary, the free dictionary.

Find more about Latitude on Wikipedia's sister projects:
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• A Comprehensive Library of Cartographic Projection Functions (Preliminary Draft) at the Internet ArchivePDF (1.88 MB)
• GEONets Names Server, access to the National Geospatial-Intelligence Agency's (NGA) database of foreign geographic feature names.
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Thom Yorke
Yorke at the Latitude Festival, Suffolk, 2009.
Background information
Birth name	Thomas Edward Yorke
Also known as	Tchock,[1] Tchocky,[2][3] Dr. Tchock,[4][5] Thom E. Yorke,[6] Zachariah Wildwood[citation needed]
Born	(1968-10-07) 7 October 1968 (age 43) Wellingborough, Northamptonshire, England
Genres	Alternative rock, electronic music
Occupations	Musician
Instruments	Vocals, guitar, piano, keyboards
Years active	1985—present
Labels	XL
Associated acts	Radiohead, Atoms for Peace

Thomas "Thom" Edward Yorke (born 7 October 1968) is an English musician and singer-songwriter who is the lead vocalist and principal songwriter of the rock band Radiohead. He mainly plays guitar and piano, but has also played drums and bass guitar (notably during the Kid A and Amnesiac sessions). In July 2006, he released his debut solo album, The Eraser.

Yorke has been cited among the most influential figures in the music industry; in 2002, Q Magazine named Yorke the 6th most powerful figure in music,^[7] and Radiohead were ranked #73 in Rolling Stone's "100 Greatest Artists of All Time" in 2005. Yorke has also been cited among the greatest singers in popular music; in 2005, a poll organised by Blender and MTV2 saw Yorke voted the 18th greatest singer of all time,^[8] and in 2008, he was ranked 66th in Rolling Stone's "100 Greatest Singers of All Time" list.^[9]

Early life[link]

Yorke was born on 7 October 1968, in Wellingborough, Northamptonshire. At birth, his left eye was fixed shut; the doctors determined that the eye was paralyzed and that the condition was permanent. Yorke's parents took him to an eye specialist, who suggested a muscle graft. Yorke underwent five eye operations before he was six years old.^[10] Yorke's father, a chemical equipment salesman, was hired by a firm in Scotland shortly after his son's birth and the family lived there until Yorke was seven. During this time Yorke had to wear a patch over his eye.^[11] He has stated that the last surgery was "botched", giving him a drooping eyelid.^[12]

Yorke's family moved frequently; Yorke would move from school to school, where classmates teased him because of his eye.^[13] The family finally settled in Oxfordshire in 1978.^[13] Yorke received his first guitar when he was seven, inspired by guitarist Brian May in a live performance with his band Queen.^[12] By 11, he had joined his first band and written his first song.^[14] He attended the all boys public school Abingdon where he met future band members Ed O'Brien, Phil Selway, Colin Greenwood and Colin's younger brother, Jonny Greenwood.^[15] Yorke and his friends formed a band named On A Friday, as Friday was the only day on which the members were allowed to rehearse.^[12] Yorke, in this early line up, played guitar and provided vocals, and was already developing his songwriting and lyrical skills. Yorke, speaking about music's influence on him as a schoolboy, said, "School was bearable for me because the music department was separate from the rest of the school. It had pianos in tiny booths, and I used to spend a lot of time hanging around there after school."^[16]

After leaving school, Yorke postponed going to university for a year. During that time he worked in a few jobs and was involved in a car accident that made him wary of any kind of mechanized transport.^[17] Yorke left Oxford to study at the University of Exeter in late 1988, which as a result put On a Friday on hiatus aside from holiday break rehearsals.^[18] While at Exeter, Yorke worked as a DJ at Guild nights in the Lemon Grove and played briefly with the band Headless Chickens.^[19] Yorke also met Rachel Owen, whom he began dating.^[20]

Career[link]

Radiohead[link]

On A Friday resumed activity in 1991 as the members were finishing their degree courses. Now relocated to Oxford, they signed to Parlophone and changed their name to Radiohead. Around this time, Yorke said he "hit the self-destruct button pretty quickly"; he would drink heavily, which resulted in him randomly cutting his hair off and being too drunk to perform onstage.^[21]

Radiohead first gained notice with the worldwide hit single "Creep", which later appeared on the band's 1993 debut album Pablo Honey. Yorke admitted later that the success had enlarged his ego; he tried to project himself as a rock star, which included bleaching his hair and wearing extensions. He said, "When I got back to Oxford I was unbearable . . . [A]s soon as you get any success you disappear up your own arse and lost it forever."^[22]

By the time of their second album, The Bends (1995), the band, through frequent touring and greater attention to detail in the recording studio, had picked up a large cult fan base and had begun to receive wider critical acclaim. After the album's release, the American group R.E.M. picked Radiohead as its opening act for the European leg of their tour.^[23] Whilst on tour Yorke and R.E.M. singer Michael Stipe became close friends; in particular, Stipe gave him advice on how to deal with the demands of being in a rock band.^[24] During the production of the band's third album, OK Computer (1997), all five members had differing opinions and equal production roles, with Yorke having "the loudest voice", according to guitarist Ed O'Brien.^[25] After the album was finished, Yorke and Jonny Greenwood contributed to the Velvet Goldmine soundtrack along with other musicians under the moniker Venus in Furs.^[26] Upon release, OK Computer was heralded as a landmark album by nearly every publication that reviewed it, establishing Radiohead as one of the leading alternative rock acts of the 1990s. But Yorke was ambivalent about this success. Some of these concerns were voiced in the documentary film Meeting People Is Easy, which focused on the period. Yorke has explained in various interviews that he dislikes the "mythology" within the rock genre, and hates the media's obsession with celebrities.^[27]

Yorke and the band adopted a more radical approach on 2000's Kid A and 2001's Amnesiac, processing vocals, obscuring lyrics, and departing from rock for a more varied musical landscape including electronic, jazz and avant-garde classical influences. Expanding Radiohead's sales whilst earning acclaim for experimentation, the albums also divided fans and critics. In 2003, Radiohead released their sixth album, Hail to the Thief, a blend of rock and electronica that Yorke described as a reaction to the events of the early 2000s and newfound fears for his children's future, though he denied political intent. The band has continued to tour, and in 2005 they undertook recording sessions for a seventh album, In Rainbows, released as a digital DRM-free download in October 2007. In 2011, The King of Limbs, the 8th studio album by Radiohead, was released.

Solo work[link]

Yorke released his solo album The Eraser in 2006. Produced by Nigel Godrich and featuring cover art by Stanley Donwood, it was released on the independent label XL Recordings. Yorke described the album as "more beats and electronics" and denied that it meant he was leaving Radiohead, saying, "I want no crap about me being a traitor or whatever splitting up blah blah... this was all done with their blessing."^[28] The Eraser reached number 3 in the UK in its first week and number 2 in the United States, Canada and Australia, as well as number 9 on the Irish charts. The album was on the prestigious Mercury Prize shortlist and was nominated for a Grammy Award for Best Alternative Music Album.

Yorke rarely plays as a solo act, having never embarked on a solo tour. He has sometimes played short acoustic sets of Radiohead songs in the band's webcasts and television appearances, and occasionally on his own at rallies. In 2006, he performed stripped-down versions of several songs from The Eraser ("Analyse", "The Clock", "Skip Divided" and "Cymbal Rush") on radio and TV programmes, and since then he has played and sung "Cymbal Rush" as an encore at some Radiohead concerts. In July 2009, Yorke played a rare solo performance at the Latitude Festival in England.^[29]

On September 21, 2009, Yorke released a new double-A side single, "Feeling Pulled Apart by Horses / The Hollow Earth".^[30] It was later announced that he has established an unnamed band with Flea of the Red Hot Chili Peppers, Joey Waronker of R.E.M., Mauro Refosco of Forro in the Dark and producer Nigel Godrich.^[31] They played two sold out shows at the Orpheum Theatre in Los Angeles on October 4 and 5, 2009.^[32] Two days before, Yorke also played a "warm-up" show at the Echoplex in Los Angeles.^[33] On February 25, 2010, Yorke officially gave the band the name Atoms for Peace^[34]

Collaborations[link]

Apart from his own solo work, Yorke has collaborated with several artists. He sang backing vocals on PJ Harvey's Mercury Prize-winning 2000 album Stories from the City, Stories from the Sea and duetted with Harvey on one of its songs, "This Mess We're In". In the same year, he also appeared on Björk's soundtrack album Selmasongs, singing "I've Seen It All" with her. The Oscar-nominated song was written for Dancer in the Dark, a film starring Björk, and Yorke's part is sung in the film by an actor; due to time constraints Björk performed it alone at the 2001 Oscars. The two worked together again in 2008 on a charity single named "Náttúra".

Yorke also sang covers of the Roxy Music songs "2HB", "Ladytron" and "Bitter-Sweet" for the 1998 film Velvet Goldmine, as part of Venus in Furs. The band existed solely for the film's soundtrack and also consisted of Radiohead's Jonny Greenwood, Suede's Bernard Butler, and Roxy Music's Andy Mackay. Yorke was duplicating the original vocals of Bryan Ferry. Two other cover songs were performed by Venus in Furs, with vocals by an actor in the film. Yorke never appeared on screen.

Yorke's other collaborations include the 1998 single "Rabbit in Your Headlights", which he sang and co-wrote with DJ Shadow and which closes Psyence Fiction, the debut album by the group UNKLE; "El President", a 1998 duet with Isabel Monteiro of the band Drugstore, which was also released as a single, and vocals on the 2010 track "...And The World Laughs with You" by experimental musician and producer Flying Lotus of Warp Records. Thom paired up with his brother Andy in 2009 to record the track "All for the Best" for inclusion on the 2009 compilation CD Ciao My Shining Star: The Songs of Mark Mulcahy. He has also collaborated on several tracks with the electronic music group Modeselektor, providing vocals on the 2007 track "The White Flash" from the album Happy Birthday, as well as 2011 tracks "Shipwreck" and "This" from the album Monkeytown, in addition to production contributions on the album's b-side "Dull Hull". Yorke has also contributed music and remix/production skills in collaboration with various electronic artists including Burial, Four Tet, as well as hip-hop artists such as MF DOOM.

Yorke has also collaborated with Stanley Donwood on a picture book titled Dead Children Playing.

Atoms for Peace[link]

On February 25, 2010 Yorke revealed the name of his new band: "Atoms for Peace", and announced a series of U.S. tour dates. Yorke first introduced the band late in 2009, with a series of concerts in Los Angeles. It includes Flea of the Red Hot Chili Peppers on bass, Beck, R.E.M., all-round session drummer Joey Waronker, percussionist Mauro Refosco of the musical collective Forro in the Dark and Radiohead producer Nigel Godrich on keyboards and treatments. When the band officially debuted in Los Angeles last October, it was billed with question marks due to it being unnamed. In a note on Radiohead's web site, Yorke wrote; "It has been decided that we call ourselves Atoms For Peace. hope you like the name. it seemed bleedin' obvious." Atoms for Peace is a track on Yorke's 2006 solo album, The Eraser, its title taken in turn by the Eisenhower speech of the same name. The band toured briefly, playing eight dates across the USA including the three-day Coachella music festival in Indio, California on April 18, 2010.^[35] Thom Yorke has also confirmed rumours that the band were recording material together, and has announced that an album is nearing completion, tentatively scheduled for release in 2012.

Personal life[link]

Yorke currently lives in Oxford with his girlfriend, Rachel Owen. They have two children, son Noah, born in 2001, and daughter Agnes, born 2004. Owen studied fine art printmaking at Exeter and painting at the Accademia di Belle Arti of Florence. She completed a PhD at the University of London researching the illustrations to Dante's Divine Comedy.^[36]

His only brother, Andy, is ex-vocalist of the band Unbelievable Truth.

Yorke is very active in creating public awareness for cultural and political issues. In 2005, Yorke became a spokesman for Friends of the Earth and their campaign to reduce carbon emissions, The Big Ask. He often pays tribute to his children whilst performing live; during songs at the 2003 Glastonbury Festival, he played with a photo of Noah resting on the top of the piano and at the 2006 Bonnaroo Music Festival, he used guitar picks that were marked with Agnes' fingerprints.

Musical approach[link]

Vocal characteristics[link]

Thom Yorke in 2006

As a singer, Yorke is recognisable by his distinctive tenor voice, vibrato, frequent use of falsetto and ability to reach, and sustain notes over a wide vocal range. In 1994, the band watched Jeff Buckley in concert; Yorke later said the concert had a direct effect on his vocal delivery on "Fake Plastic Trees".^[37] However, Yorke has said, "It annoys me how pretty my voice is... how polite it can sound when perhaps what I'm singing is deeply acidic."^[38] He has often adopted other styles of singing, such as an aggressive shouting style in the middle section of "Paranoid Android" and a semi-spoken style for 2003's "Myxomatosis" and "A Wolf at the Door".

Musicianship[link]

Aside from vocal duties and writing lyrics, Yorke's musical contributions to Radiohead include guitar, both acoustic and electric (usually rhythm parts, with band member Jonny Greenwood handling lead), and piano (including Rhodes piano, especially on Kid A). He also plays bass guitar on occasion (the bass line for "The National Anthem" was recorded by him) as well as drums; during the 2006 and 2008 tours he performed percussion on stage in tandem with drummer Phil Selway on the track "Bangers + Mash".

Yorke, unlike the other members of Radiohead, has never learned how to read music.^[39] He said, "If someone lays the notes on a page in front of me, it's meaningless... because to me you can't express the rhythms properly like that. It's a very ineffective way of doing it, so I've never really bothered picking it up."^[38]

Since Kid A, Radiohead, and in particular Yorke, have incorporated many elements of electronic music into their work. As a result, Yorke has taken an increased role in programming beats and samples and has been credited with playing "laptop" on recent albums. On a radio show in 2003 to publicise the release of Hail to the Thief, Yorke remarked that he would rather make a record just with a computer than with only an acoustic guitar.^[40] His solo effort The Eraser featured piano, bass and guitar, but was built primarily around electronics.

In interviews Yorke has cited a variety of personal musical heroes and influences, including jazz composer and bassist Charles Mingus, Neil Young, Miracle Legion, singer Scott Walker, electronic acts Aphex Twin and Autechre, and Krautrock band Can. Talking Heads, Queen, Joy Division, Magazine, Elvis Costello, The Smiths and Sonic Youth were early influences on Radiohead and Yorke. In 2004, at the Coachella Valley Music and Arts Festival, Yorke mentioned to the crowd, "When I was in college, the Pixies and R.E.M. changed my life",^[41] and he has often mentioned both bands as influential examples.

Activism[link]

Yorke has been outspoken on various contemporary political and social issues. Radiohead had read No Logo by Naomi Klein during the Kid A sessions ("No Logo" was also briefly considered as the album title) and all the members were reportedly heavily influenced by it, though Yorke said it "didn't teach him anything he didn't already know".^[42] Yorke's activism in support of fair trade practices, with an anti-WTO and anti-globalisation stance, garnered significant attention in the early 2000s.^[43] Yorke had previously referenced maquiladoras in the title of a Radiohead B-side in 1995, and decried the International Monetary Fund in 1997's "Electioneering". Yorke is also a professed fan of Noam Chomsky's political writings,^[44] and is a vegan.^[45]

Yorke is also notable as a political activist on behalf of other causes, including human rights and anti-war movements such as Jubilee 2000, Amnesty International and CND, and Friends of the Earth's Big Ask campaign.^[46] Radiohead played at the Free Tibet concert in both 1998 and 1999, and at an Amnesty International concert in 1998.^[47] In 2005, Yorke performed at an all-night vigil for the Trade Justice Movement.^[48] In 2006, Jonny Greenwood and Yorke performed a special benefit concert for Friends of the Earth. Yorke made headlines the same year for refusing Prime Minister Tony Blair's request to meet with him to discuss climate change, declaring Blair had "no environmental credentials".^[49] Yorke has subsequently been critical of his own energy use. He has said the music industry's use of air transport is dangerous and unsustainable, and that he would consider not touring if new carbon emissions standards do not force the situation to improve.^[50] Radiohead commissioned a study by the group Best Foot Forward which the band claims helped them choose venues and transport methods that will greatly reduce the carbon expended on their 2008 tour. The band also made use of a new low-energy LED lighting system and encouraged festivals to offer reusable plastics.^[51]

In December 2009, Yorke gained access to the COP 15 climate change talks in Copenhagen, posing as a member of the media.^[52] In the same year he lent his support to the 10:10 project, a movement encouraging people to take positive action on climate change by reducing their carbon emissions.^[53]

Relationship with celebrities and the media[link]

Yorke has had an uneasy relationship with other celebrities and the media. Following Radiohead's 1993 Pablo Honey tour of America, Yorke became disenchanted at being "right at the sharp end of the sexy, sassy, MTV eye-candy lifestyle" he felt he was helping to sell to the world.^[54] The 1998 documentary film Meeting People is Easy portrays Yorke's disaffection with the music industry and press during the 1997-8 "Against Demons" world tour.^[55]

A number of celebrities have been upset by Yorke's alleged rudeness. In 2001, Kelly Jones, the lead singer of the Welsh band Stereophonics, referred to Thom Yorke as a "miserable twat"^[56] (a comment he later retracted).^[57] In 2002, Jack Black claimed to have approached Yorke to congratulate him on his solo show at the Bridge School benefit concert in San Francisco, only for Yorke to ignore him and walk away. Referring to the incident, Black stated in an interview: "I heard later that he's famously cold, and it wasn't just me that he despises, but the whole world."^[58] After completing a trek of Kilimanjaro in 2009, Ronan Keating was asked by an interviewer which celebrity he would most like to throw off a mountain. Keating named Yorke, and referred to him as a "muppet", stating that Yorke was once rude to him.^[59] In the same year, Miley Cyrus and Kanye West also complained about Yorke's alleged rudeness.^[60] In 2010, it was reported that Yorke would be covering Peter Gabriel's song "Wallflower", from Gabriel's fourth solo album Peter Gabriel 4 (Security), with Gabriel claiming, "Thom wanted to do a version of Wallflower, which I'm very curious to hear. I think that was an important track for him when he was 14."^[61] However, it was subsequently reported that Yorke had withdrawn from the project and hadn't informed Gabriel why.^[62]

Discography[link]

• The Eraser (2006)

References[link]

• Randall, Mac. Exit Music: The Radiohead Story. Delta, 2000. ISBN 0-385-33393-5

Notes[link]

External links[link]

• Media related to Thom Yorke at Wikimedia Commons
• Quotations related to Thom Yorke at Wikiquote
• The Eraser: Official website for Yorke's solo album release
• Thom Yorke, Going Solo: Interview with Yorke, from NPR's "Fresh Air" radio program—July 2006