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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Jimmy Buffett
Jimmy Buffett performing, January 2008
Background information
Birth name	James William Buffett
Born	(1946-12-25) December 25, 1946 (age 65) Pascagoula, Mississippi
Origin	Mobile, Alabama, U.S.
Genres	Gulf and Western
Occupations	singer-songwriter, businessman, film producer
Years active	1965–present
Labels	Barnaby, Dunhill, MCA, Margaritaville, Island, PolyGram
Associated acts	The Margartias, Alan Jackson, Glenn Frey, Eagles, Zac Brown Band
Website	Official Web Site

Jimmy Buffett aboard USS Harry S Truman, January 2008

James William "Jimmy" Buffett (born December 25, 1946) is an American singer–songwriter, film producer, businessman, and author. He is best known for his music, which often portrays an "island escapism" lifestyle. Together with his Coral Reefer Band, Buffett has recorded hit songs including "Margaritaville" (ranked 234th on the Recording Industry Association of America's list of "Songs of the Century"), and "Come Monday". He has a devoted base of fans known as "Parrotheads".

Aside from his career in music, Buffett is also a best-selling writer and is involved in two restaurant chains named after two of his best known songs, "Cheeseburger in Paradise" and "Margaritaville". He owns the Margaritaville Cafe restaurant chain and co-developed the Cheeseburger in Paradise restaurant concept with OSI Restaurant Partners (parent of Outback Steakhouse), which operates the chain under a licensing agreement with Buffett.

Personal life[link]

Buffett spent part of his childhood in Mobile, Alabama.^[1] In grade school years, he attended St. Ignatius School, where he played the trombone in the school band. He later lived in Fairhope, Alabama, mentioned by himself as his "Home Town" during a 2001 concert. He graduated high school from McGill Institute for Boys (now McGill-Toolen Catholic High School) in 1964. He began playing guitar during his college years at Pearl River Community College, The University of Southern Mississippi in Hattiesburg, Mississippi, and Auburn University where he received a bachelor's degree in journalism in 1969. He was initiated into the fraternity Kappa Sigma (ΚΣ) at the University of Southern Mississippi. After graduating from college, Buffett worked as a correspondent for Billboard magazine in Nashville, breaking the news of the separation of Flatt and Scruggs.

Buffett married Margie Washichek in 1969 and divorced in 1972. Buffett and his second wife Jane (née Slagsvol) have two daughters, Savannah Jane and Sarah Delaney, and an adopted son, Cameron Marley, and reside in Palm Beach, Florida. They were separated in the early 1980s, and later reconciled in 1991. Buffett also owns a home in St Barts, a Caribbean island where he lived on and off in the early 1980s while he was part-owner of the Autour de Rocher hotel and restaurant. An avid pilot, Buffett's flagship plane is a Dassault Falcon 900, call sign N908JB, that he often uses while on concert tour and traveling worldwide.

His father, John Delaney Buffett, Jr. died in 2003 at the age of 83.

Music[link]

Music career[link]

Buffett began his musical career in Nashville, Tennessee during the late 1960s as a country artist and recorded his first album, the folk rock Down to Earth, in 1970. During this time Buffett could be frequently found busking for tourists in New Orleans. Country music singer Jerry Jeff Walker took him to Key West on a busking expedition in November, 1971.^[2] Buffett then moved to Key West and began establishing the easy-going beach bum persona for which he is known. Following this move, Buffett combined country, folk, and pop music with coastal as well as tropical lyrical themes for a sound sometimes called "gulf and western". Today, he is a regular visitor to the Caribbean island of Saint Barts and other islands where he gets inspiration for many of his songs and some of the characters in his books.

Buffett's third album was the 1973 A White Sport Coat and a Pink Crustacean. Living & Dying in 3/4 Time and A1A both followed in 1974, Havana Daydreamin' appeared in 1976, followed by 1977's Changes in Latitudes, Changes in Attitudes, which featured the breakthrough hit song "Margaritaville".

With the untimely death of friend and mentor Jim Croce in September 1973, ABC Dunhill tapped Buffett to fill his space. Earlier, Buffett had visited Croce's farm in Pennsylvania and met with Croce in Florida.^[3][4]

During the 1980s, Buffett made far more money from his tours than his albums and became known as a popular concert draw. He released a series of albums during the following twenty years, primarily to his devoted audience, and also branched into writing and merchandising. In 1985, Buffett opened a "Margaritaville" retail store in Key West and then in 1987, the Margaritaville Cafe was opened. During the 1980s Buffett played at the Houston Livestock Show and Rodeo. He briefly changed the name of the band from "Coral Reefers" to the "Coral Reef Band" to suit the HLS&R's request as they thought "Reefers" was a drug related reference. HLS&R is a charity event that provides student grants to children and young adults that compete in agriculture contests (FFA).

Two of the more out-of-character albums were Christmas Island, a collection of Christmas songs, and Parakeets, a collection of Buffett songs sung by children and containing "cleaned-up" lyrics (like "a cold root beer" instead of "a cold draft beer").

In 1997, Buffett collaborated with novelist Herman Wouk to create a short-lived musical based on Wouk's novel, Don't Stop the Carnival. Broadway showed little interest in the play, (following the failure of Paul Simon's The Capeman) and it only ran for six weeks in Miami. He released an album of songs from the musical in 1998.

In August 2000 Buffett and the Coral Reefer Band played on the White House lawn for then President Bill Clinton.

In 2003, he partnered in a partial duet with Alan Jackson for the song "It's Five O'Clock Somewhere", a number one hit on the country charts. This song won the 2003 Country Music Association Award for Vocal Event of the Year.^[5] This was Buffett’s first award of any kind for his music in his 30 year career.

Buffett's album, License to Chill, released on July 13, 2004, sold 238,600 copies in its first week of release according to Nielsen SoundScan. With this, Buffett topped the U.S. pop albums chart for the first time in his three-decade career.

Buffett continues to tour throughout the year although he has shifted recently to a more relaxed schedule of around 20–30 dates, and rarely on back-to-back nights, preferring to play only on Tuesdays, Thursdays and Saturdays, thus the title of his 1999 live album Buffett Live: Tuesdays, Thursdays, Saturdays. Purchasing tickets is difficult with most of his concerts selling out in minutes.

In the summer of 2005 Buffett teamed up with Sirius radio and introduced Radio Margaritaville, and as of November 2008 is also on XM radio channel 24.^[6] Until this point Radio Margaritaville was solely an online channel. The channel broadcasts from the Margaritaville restaurant at Universal CityWalk in Orlando, Florida.

In August 2006, he released the album Take The Weather With You. The song "Breathe In, Breathe Out, Move On" on this album refers to 2005's Hurricane Katrina. Also on the album he pays tribute to Merle Haggard with his rendition of "Silver Wings" and covers, with Mark Knopfler playing on the track, "Whoop De Doo."

On 8 December 2009, Jimmy Buffett released his 28th studio album entitled Buffet Hotel.

On 20 April 2010, a double CD of performances recorded during the 2008 and 2009 tours called Encores was released exclusively at Walmart, Walmart.com and Margaritaville.com.

Buffett partnered in a duet with the Zac Brown Band on the song "Knee Deep": released on Brown's 2010 album You Get What You Give, it became a hit country and pop single in 2011.

Of the over 30 albums Jimmy Buffett has released, as of October 2007, 8 are Gold Albums and 9 Platinum or Multi Platinum.^[7] In 2003 Buffett won his first ever Country Music Award (CMA) for his song "It's 5 O'clock Somewhere" with Alan Jackson, and was nominated again in 2007 for the CMA Event of the Year Award for his song "Hey Good Lookin" which featured Alan Jackson and George Strait.

Musical style[link]

Buffett himself and others have used the term gulf and western to describe his musical style and that of other similar sounding performers.^{[8][9][10][11][12]} The name derives from elements in Buffett's early music including musical influence from country and western, along with folk music and lyrical themes from the Gulf of Mexico coast. A critic described Buffett's music as a combination of "tropical languor with country funkiness into what some [have] called the Key West sound, or Gulf-and-western."^[13] The term is a play on the name of the former megacorporation Gulf+Western (now part of National Amusements).

Other performers identified as gulf and western are often deliberately derivative of Buffett's musical style and some are tribute bands or, in the case of Greg "Fingers" Taylor, a former member of Jimmy Buffett's Coral Reefer Band.^[14] They can be heard on Buffett's online Radio Margaritaville and on the compilation album series Thongs in the Key of Life. Gulf and western performers include Jim Bowley, ^[15] Kenny Chesney,^[16] and Jim Morris.^[17]

Writing[link]

Jimmy Buffett at the Miami Book Fair International of 1989

Buffett has written three #1 best sellers. Tales from Margaritaville and Where Is Joe Merchant? both spent over seven months on The New York Times Best Seller fiction list. His book A Pirate Looks At Fifty went straight to No. 1 on the New York Times Best Seller non-fiction list, making him one of eight authors in that list's history to have reached No. 1 on both the fiction and non-fiction lists. The seven other authors who have accomplished this are Ernest Hemingway, John Steinbeck, William Styron, Irving Wallace, Dr. Seuss, Mitch Albom and Glenn Beck.

Buffett also co-wrote two children's books, The Jolly Mon and Trouble Dolls, with his eldest daughter, Savannah Jane Buffett. The original hard cover release of The Jolly Mon included a cassette tape recording of him and Savannah Jane reading the story accompanied by an original score written by Michael Utley.

Buffett's novel A Salty Piece of Land, was released on November 30, 2004, and the first edition of the book included a CD single of the song "A Salty Piece Of Land", which was recorded for License to Chill. The book was a New York Times best seller soon after its release.

Buffett's latest title, Swine Not?, was released on May 13, 2008.

Buffett is currently writing a follow-up to his autobiography A Pirate Looks at Fifty, which he says may take up to ten years to write and complete.

Jimmy Buffett quotation on Himank/BRO signboard in the Nubra Valley, Ladakh, Northern India

Buffett is one of several popular 'philosophers' whose quotations appear on the road signs of Project HIMANK in the Ladakh region of Northern India.

Film and television[link]

Buffett wrote the soundtrack for, co-produced and played a role in the 2006 film Hoot, directed by Wil Shriner and based on the book by Carl Hiaasen, which focuses on issues important to Buffett, such as conservation. The film was not a critical or commercial success. Among his other film music credits are the theme song to the short-lived 1993 CBS television series Johnny Bago; "Turning Around" for the 1985 film Summer Rental starring John Candy; "I Don't Know (Spicoli's Theme)" for the film Fast Times at Ridgemont High; "Hello, Texas" for the 1980 John Travolta film Urban Cowboy; and "If I Have To Eat Someone (It Might As Well Be You)" for the animated film FernGully: The Last Rainforest, which was sung in the film by rap artist Tone Loc.

In addition, Buffett has made several cameo appearances, including in Repo Man, Hook, Cobb, Hoot, Congo, and From the Earth to the Moon. He also made cameo appearances as himself in Rancho Deluxe (for which he also wrote the music) and in FM.^[18] He made a Guest appearance in the season two of Hawaii Five-0 on CBS in 2011. Buffett reportedly was offered a cameo role in Pirates of the Caribbean: The Curse of the Black Pearl, but declined the offer.^[19] In 1997, Buffett collaborated with novelist Herman Wouk on a musical production based on Wouk's 1965 novel Don't Stop the Carnival. In the South Park episode "Tonsil Trouble", an animated version of Buffett (but not voiced by Buffett) was seen singing "AIDSburger in Paradise" and "CureBurger in Paradise". Jimmy has also appeared on the Sesame Street special, Elmopalooza, singing "Caribbean Amphibian" with the popular Muppet, Kermit the Frog. Buffett appeared in an episode of Hawaii Five-O in November 2011. He played a helicopter pilot named Frank Bama, a character out of his novel, "Where is Joe Merchant". Another character mentioned that he preferred "margaritas"; Buffett's character replied, "You can't argue with that."

Business ventures[link]

Buffett has taken advantage of his name and the fan following for his music to launch several business ventures, usually with a tropical theme. He owns or licenses the Margaritaville Cafe and Cheeseburger in Paradise restaurant chains. As a baseball fan, he was part-owner of two minor league teams: the Fort Myers Miracle and the Madison Black Wolf. Between his restaurants, album sales, and tours, he earns an estimated US$100 million a year. He opened the Margaritaville store in Key West, Florida, in 1985.

In 1993, he launched Margaritaville Records, with distribution through MCA Records. His MCA record deal ended with the release of 1996's Christmas Island and he took Margaritaville Records over to Chris Blackwell's Island Records for a two record deal, 1998's Don't Stop The Carnival and 1999's Beach House On The Moon. In the fall of 1999, he started up Mailboat Records to release live albums. He partnered up with RCA Records for distribution in 2005 and 2006 for the two studio albums License To Chill and Take The Weather With You.

In 2006, Buffett launched a cooperative project with the Anheuser-Busch brewing company to produce his own beer under the Margaritaville Brewing label called Land Shark Lager.^[20]

Another Margaritaville Casino was slated to be opening in Atlantic City, New Jersey but has been put on hold indefinitely.^[21] Buffett has also licensed Margaritaville Tequila, Margaritaville Footwear and a Margaritaville Foods including Chips, Salsa, Guacamole, Shrimp, Chicken and more.

From May 8, 2009, through January 5, 2010, Sun Life Stadium] (formerly Dolphin Stadium) in Miami, the home of the Miami Dolphins, was named Landshark Stadium pursuant to an eight-month naming rights deal.^[22][23] Buffett also wrote new lyrics for the team to his 1979 song "Fins", which is played during Dolphins home games.^[24] Despite Buffet's partnership with the Dolphins, Buffet is a diehard New Orleans Saints fan, having attended the team's first game at Tulane Stadium in 1967 and later had Saints head coach Sean Payton serve as an honorary member of the Coral Reefer Band at a concert in New Orleans on April 1, 2012, in protest of Payton's suspension by the NFL as a result of the Saints' bounty scandal.^[25]

In May 2011, Buffet announced that he had partnered with game makers THQ to make a Margaritaville game for Facebook and iOS devices. The game opened in closed beta to what's being referred to as the Buffet Beta Club. The game, which requires users to download Unity Engine for use on the Facebook API, is expected to release on Facebook in December 2011 and to iOS devices by the end of January 2012.^[26]

Charity work[link]

Buffett has been involved in many charity efforts. In 1981 the Save the Manatee Club was founded by Buffett and former Florida governor Bob Graham.^[27] Save the Manatee Club is the world's leading manatee protection organization.^[28] West Indian Manatee In 1989, legislation was passed in Florida that introduced the "Save the Manatee" license plate, and earmarked funding for the Save the Manatee Club. One of the two manatees trained to interact with researchers at Mote Marine Laboratory is named Buffett after the singer.

The "Singing for Change" foundation was initially funded by proceeds from Buffett's 1995 concert tour, and provides grants to local charities in three main areas: children and family causes, environmental causes, and causes for disenfranchised groups.^[29][30]

On November 23, 2004, Buffett raised substantial money at his "Surviving the Storm" Hurricane Relief Concert in Orlando, Florida to provide relief for hurricane victims in Florida, Alabama and the Caribbean affected by the four major hurricanes that year.^[31]

Buffett performed in Hong Kong on January 18, 2008 for a concert that raised US$63,000 for the Foreign Correspondents' Club Charity Fund. This was his first concert in Hong Kong and it sold out within weeks. Not only did Buffett perform for the groundlings for free, but he also paid for the concertgoers' tequila and beer.^[32]

On July 11, 2010, Buffett, a Gulf Coast native, put on a free concert on the beach in Gulf Shores, Alabama. The concert was Buffett's response to the BP oil disaster in the Gulf. The concert was aired on CMT television. The 35,000 free tickets were given away within minutes to help draw people back to Alabama's beaches. Buffett played several popular songs including "Fins", "Son of a Son of a Sailor", "A Pirate Looks at Forty" and a modified version of "Margaritaville" where the lyrics were changed in the chorus to "now I know, it's all BP's fault." The concert featured Jesse Winchester and Allen Toussaint.

Controversy[link]

The earliest controversy with Buffett was his recording of "God's Own Drunk" found on the live recorded album You Had to Be There. In 1983 the son of the late entertainer Lord Buckley sued Buffett for $11 million for copyright infringement claiming that Buffett took parts of the monologue from Buckley's A Tribute to Buckley and claimed it as his own work in "God's Own Drunk". The suit also alleged that Buffett's "blasphemous" rendition presented to the public a distorted impression of Lord Buckley.^[33] A court injunction against Buffett prevented him from performing the song until the lawsuit was settled or resolved, so in 1986, when Buffett would get to the part of his show where he would normally perform "God's Own Drunk," he would say that he still wasn't allowed to play it because of the lawsuit and instead played a song he wrote called "The Lawyer and the Asshole" in which he accuses Buckley's son and lawyers as being greedy and tells them to "kiss his ass."

On October 6, 2006, it was reported that Buffett had been detained by French custom officials in Saint Tropez for allegedly carrying over 100 pills of ecstasy.^[34][35][36] Buffett’s luggage was searched after his Dassault Falcon 900 private jet landed at Toulon-Hyères International Airport. He paid a fine of $300 and was released. A spokesperson for Buffett stated the pills in question were prescription drugs, but declined to name the drug or the health problem for which he was being treated. Buffett released a statement that the "ecstasy" was in fact, a Vitamin B supplement known as Foltx.^[37]

In January 1996 Buffett's Grumman HU-16 airplane nicknamed "Hemisphere Dancer" was shot at by Jamaican police who believed the craft to be smuggling marijuana. The aircraft sustained minimal damage. The plane had been carrying Buffett as well as U2's Bono, and Island Records producer Chris Blackwell, and co-pilot Bill Dindy. They were not on board at the time.^[who?] The Jamaican government acknowledged the mistake and apologized to Buffett who penned the song "Jamaica Mistaica" for his Banana Wind album based on the experience. The plane from the incident is now at Orlando City Walk's Margaritaville.^{[citation needed]}

On February 4, 2001, he was ejected from the American Airlines Arena in Miami during a Miami Heat/New York Knicks basketball game for cursing. After the game, referee Joe Forte said that he ordered him moved during the fourth quarter because "there was a little boy sitting next to him and a lady sitting by him. He used some words he knows he shouldn't have used." Forte apparently didn't know who Buffett was, and censured Heat coach Pat Riley because he thought Riley—who was trying to explain to him who Buffett was—was insulting him by asking if he'd ever been a "Parrothead", the nickname for Buffett fans.^[38] Buffett didn't comment immediately after the incident, but discussed it with Matt Lauer on The Today Show three days later.

Jimmy Buffett was paid $250,000 to perform a private show for ex-Tyco CEO Dennis Kozlowski. This show was part of a multiple day event held to celebrate Kozlowski's girlfriend and her birthday. Video footage of this performance is included in a CNBC television series known as "American Greed."^[39]

Tour accident[link]

On January 26, 2011 (Australia Day), Buffett was performing a concert in Australia at Sydney's Hordern Pavilion and fell off the stage after an encore. A concert-goer said, "He just went over to the edge of the stage, like he had numerous times through the night, just to wave, and people were throwing stuffed toys and things at him. And he just took one step too many and just disappeared in a flash. He didn't have time to put his arms out to save himself or anything, he just dropped."^[40][41][42] Coincidentally, one of Australia's leading trauma surgeons was at the concert and close to the stage; Dr. Gordian Fulde treated Buffett at the scene. Fulde said, "I thought he'd broken his neck." "I heard the clunk of his head on a metal ledge; he has a deep gash on his scalp, which is all right now." "But at first I thought, This guy is going to be a spinal injury."^[40] Dr Fulde turned him on his side so he could breathe and administered first aid. Buffett regained consciousness within a few minutes. He was then transported to St Vincent's Hospital Emergency centre for treatment and was released the next day.^[43]

Discography[link]

See also[link]

• A Pirate Looks at Fifty
• List of best-selling music artists

References[link]

External links[link]

Wikiquote has a collection of quotations related to: Jimmy Buffett

Wikimedia Commons has media related to: Jimmy Buffett

• Official Jimmy Buffett Web Site
• Jimmy Buffett at the Internet Movie Database
• Jimmy Buffett Entry at the Encyclopedia of Alabama
• http://www.parrotheadconnection.com/

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]