Latitude


In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the geodetic latitude as defined [|below]. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six auxiliary latitudes that are used in special applications.

Preliminaries

Two levels of abstraction are employed in the definition of latitude and longitude. In the first step the physical surface is modeled 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 modeled by an ellipsoid. The definitions of latitude and longitude on such reference surfaces are detailed in the following sections. Lines of constant latitude and longitude together constitute a graticule on the reference surface. The latitude of a point on the actual surface is that of the corresponding point on the reference surface, the correspondence being along the normal to the reference surface which passes through the point on the physical surface. Latitude and longitude together with some specification of height constitute a geographic coordinate system as defined in the specification of the ISO 19111 standard.
Since there are many different reference ellipsoids, the precise latitude of a feature on the surface is not unique: this is stressed in the ISO standard which states that "without the full specification of the coordinate reference system, coordinates are ambiguous at best and meaningless at worst". This is of great importance in accurate applications, such as a Global Positioning System, but in common usage, where high accuracy is not required, the reference ellipsoid is not usually stated.
In English texts the latitude angle, defined below, is usually denoted by the Greek lower-case letter phi. It is measured in degrees, minutes and seconds or decimal degrees, north or south of the equator. For navigational purposes positions are given in degrees and decimal minutes. For instance, The Needles lighthouse is at 50°39.734′N 001°35.500′W.
The precise measurement of latitude requires an understanding of the gravitational field of the Earth, either to set up theodolites or to determine GPS satellite orbits. The study of the figure of the Earth together with its gravitational field is the science of geodesy.
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.

Latitude on the sphere

The graticule on the sphere

The graticule is formed by the lines of constant latitude and constant longitude, which are constructed with reference to the rotation axis of the Earth. The primary reference points are the poles where the axis of rotation of the Earth intersects the reference surface. Planes which contain the rotation axis intersect the surface at the meridians; and the angle between any one meridian plane and that through Greenwich defines the longitude: meridians are lines of constant longitude. The plane through the centre of the Earth and perpendicular to the rotation axis intersects the surface at a great circle called the Equator. 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, and the South Pole has a latitude of 90° South. The latitude of an arbitrary point is the angle between the equatorial plane and the normal to the surface at that point: the normal to the surface of the sphere is along the radius vector.
The latitude, as defined in this way for the sphere, is often termed the spherical latitude, to avoid ambiguity with the geodetic latitude and the auxiliary latitudes defined in subsequent sections of this article.

Named latitudes on the Earth

Besides the equator, four other parallels are of significance:
The plane of the Earth's orbit about the Sun is called the ecliptic, and the plane perpendicular to the rotation axis of the Earth is the equatorial plane. The angle between the ecliptic and the equatorial plane is called variously the axial tilt, the obliquity, or the inclination of the ecliptic, and it is conventionally denoted by. The latitude of the tropical circles is equal to and the latitude of the polar circles is its complement. The axis of rotation varies slowly over time and the values given here are those for the current epoch. The time variation is discussed more fully in the article on axial tilt.
The figure shows the geometry of a cross-section of the plane perpendicular to the ecliptic and through the centres of the Earth and the Sun at the December solstice 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. Only at latitudes in between the two tropics is it possible for the Sun to be directly overhead.
On map projections there is no universal rule as to how meridians and parallels should appear. The examples below show the named parallels on the commonly used Mercator projection and the Transverse Mercator projection. On the former the parallels are horizontal and the meridians are vertical, whereas on the latter there is no exact relationship of parallels and meridians with horizontal and vertical: both are complicated curves.

Meridian distance on the sphere

On the sphere the normal passes through the centre and the latitude is
therefore equal to the angle subtended at the centre by the meridian arc from the equator to the point concerned. If the meridian distance is denoted by then
where denotes the mean radius of the Earth. is equal to. No higher accuracy is appropriate for since higher-precision results necessitate an ellipsoid model. With this value for the meridian length of 1 degree of latitude on the sphere is . The length of 1 minute of latitude is , while the length of 1 second of latitude is .

Latitude on the ellipsoid

Ellipsoids

In 1687 Isaac Newton published the Philosophiæ Naturalis Principia Mathematica, in which he proved that a rotating self-gravitating fluid body in equilibrium takes the form of an oblate ellipsoid. Newton's result was confirmed by geodetic measurements in the 18th century. An oblate ellipsoid is the three-dimensional surface generated by the rotation of an ellipse about its shorter axis. "Oblate ellipsoid of revolution" is abbreviated to 'ellipsoid' in the remainder of this article.
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 with centre at the centre of mass of the Earth and minor axis aligned to the rotation axis of the Earth. These geocentric ellipsoids are usually within of the geoid. Since latitude is defined with respect to an ellipsoid, the position of a given point is different on each ellipsoid: one cannot exactly specify the latitude and longitude of a geographical feature without specifying the ellipsoid used. Many maps maintained by national agencies are based on older ellipsoids, so one must know how the latitude and longitude values are transformed from one ellipsoid to another. GPS handsets include software to carry out datum transformations which link WGS84 to the local reference ellipsoid with its associated grid.

The geometry of the ellipsoid

The shape of an ellipsoid of revolution is determined by the shape of the ellipse which is rotated about its minor axis. Two parameters are required. One is invariably the equatorial radius, which is the semi-major axis,. The other parameter is usually the polar radius or semi-minor axis, ; or the flattening, ; or the eccentricity,. These parameters are not independent: they are related by
Many other parameters appear in the study of geodesy, geophysics and map projections but they can all be expressed in terms of one or two members of the set,, and. Both and are small and often appear in series expansions in calculations; they are of the order and 0.0818 respectively. Values for a number of ellipsoids are given in Figure of the Earth. Reference ellipsoids are usually defined by the semi-major axis and the inverse flattening,. For example, the defining values for the WGS84 ellipsoid, used by all GPS devices, are
from which are derived
The difference between the semi-major and semi-minor axes is about and as fraction of the semi-major axis it equals the flattening; on a computer monitor the ellipsoid could be sized as 300 by 299 pixels. This would barely be distinguishable from a 300-by-300-pixel sphere, so illustrations usually exaggerate the flattening.

Geodetic and geocentric latitudes

The graticule on the ellipsoid is constructed in exactly the same way as on the sphere. The normal at a point on the surface of an ellipsoid does not pass through the centre, except for points on the equator or at the poles, but the definition of latitude remains unchanged as the angle between the normal and the equatorial plane. The terminology for latitude must be made more precise by distinguishing:
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 distant from the tower. A web search may produce several different values for the latitude of the tower; the reference ellipsoid is rarely specified.

Length of a degree of latitude

In Meridian arc and standard texts it is shown that the distance along a meridian from latitude to the equator is given by
where is the meridional radius of curvature.
The distance from the equator to the pole is
For WGS84 this distance is.
The evaluation of the meridian distance integral is central to many studies in geodesy and map projection. It can be evaluated by expanding the integral by the binomial series and integrating term by term: see Meridian arc 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
-
0°110.574 km111.320 km
15°110.649 km107.550 km
30°110.852 km96.486 km
45°111.132 km78.847 km
60°111.412 km55.800 km
75°111.618 km28.902 km
90°111.694 km0.000 km

When the latitude difference is 1 degree, corresponding to radians, the arc distance is about
The distance in metres between latitudes − 0.5 degrees and + 0.5 degrees on the WGS84 spheroid is
The variation of this distance with latitude is shown in the table along with the length of a degree of longitude :
A calculator for any latitude is provided by the U.S. Government's National Geospatial-Intelligence Agency.
The following graph illustrates the variation of both a degree of latitude and a degree of longitude with latitude.

The nautical mile

Historically a nautical mile was defined as the length of one minute of arc along a meridian of a spherical earth. An ellipsoid model leads to a variation of the nautical mile with latitude. This was resolved by defining the nautical mile to be exactly 1,852 metres. However, for all practical purposes distances are measured from the latitude scale of charts. As the Royal Yachting Association says in its manual for day skippers: "1 of Latitude = 1 sea mile", followed by "For most practical purposes distance is measured from the latitude scale, assuming that one minute of latitude equals one nautical mile".

Auxiliary latitudes

There are six auxiliary latitudes that have applications to special problems in geodesy, geophysics and the theory of map projections:
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 or in calculations of geodesics on the ellipsoid. Their numerical values are not of interest. For example, no one would need to calculate the authalic latitude of the Eiffel Tower.
The expressions below give the auxiliary latitudes in terms of the geodetic latitude, the semi-major axis,, and the eccentricity,. The forms given are, apart from notational variants, those in the standard reference for map projections, namely "Map projections: a working manual" by J. P. Snyder. Derivations of these expressions may be found in Adams and online publications by Osborne and Rapp.

Geocentric latitude

The geocentric latitude is the angle between the equatorial plane and the radius from the centre to a point on the surface. The relation between the geocentric latitude and the geodetic latitude is derived in the above references as
The geodetic and geocentric latitudes are equal at the equator and at the poles but at other latitudes they differ by a few minutes of arc. Taking the value of the squared eccentricity as 0.0067 the maximum difference of may be shown to be about 11.5 minutes of arc at a geodetic latitude of approximately 45° 6′.

Parametric (or reduced) latitude

The parametric or reduced latitude,, is defined by the radius drawn from the centre of the ellipsoid to that point on the surrounding sphere which is the projection parallel to the Earth's axis of a point on the ellipsoid at latitude. It was introduced by Legendre and Bessel who solved problems for geodesics on the ellipsoid by transforming them to an equivalent problem for spherical geodesics by using this smaller latitude. Bessel's notation,, is also used in the current literature. The parametric latitude is related to the geodetic latitude by:
The alternative name arises from the parameterization of the equation of the ellipse describing a meridian section. In terms of Cartesian coordinates, the distance from the minor axis, and, the distance above the equatorial plane, the equation of the ellipse is:
The Cartesian coordinates of the point are parameterized by
Cayley suggested the term parametric latitude because of the form of these equations.
The parametric latitude is not used in the theory of map projections. Its most important application is in the theory of ellipsoid geodesics,.

Rectifying latitude

The rectifying latitude,, is the meridian distance scaled so that its value at the poles is equal to 90 degrees or radians:
where the meridian distance from the equator to a latitude is
and the length of the meridian quadrant from the equator to the pole is
Using the rectifying latitude to define a latitude on a sphere of radius
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.. The rectifying latitude is also of great importance in the construction of the Transverse Mercator projection.

Authalic latitude

The authalic latitude,, gives an area-preserving transformation to a sphere.
where
and
and the radius of the sphere is taken as
An example of the use of the authalic latitude is the Albers equal-area conic projection.

Conformal latitude

The conformal latitude,, gives an angle-preserving transformation to the sphere.
where is the Gudermannian function.
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. 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..

Isometric latitude

The isometric latitude,, 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 and longitude give rise to equal distance displacements along the meridians and parallels respectively. The graticule defined by the lines of constant and constant, divides the surface of the ellipsoid into a mesh of squares. 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 :
For the normal Mercator projection this function defines the spacing of the parallels: if the length of the equator on the projection is then the distance,, of a parallel of latitude from the equator is
The isometric latitude is closely related to the conformal latitude :

Inverse formulae and series

The formulae in the previous sections give the auxiliary latitude in terms of the geodetic latitude. The expressions for the geocentric and parametric latitudes may be inverted directly but this is impossible in the four remaining cases: the rectifying, authalic, conformal, and isometric latitudes. There are two methods of proceeding. The first is a numerical inversion of the defining equation for each and every particular value of the auxiliary latitude. The methods available are fixed-point iteration and Newton–Raphson root finding. The other, more useful, approach is to express the auxiliary latitude as a series in terms of the geodetic latitude and then invert the series by the method of Lagrange reversion. Such series are presented by Adams who uses Taylor series expansions and gives coefficients in terms of the eccentricity. Osborne derives series to arbitrary order by using the computer algebra package Maxima and expresses the coefficients in terms of both eccentricity and flattening. The series method is not applicable to the isometric latitude and one must use the conformal latitude in an intermediate step.

Numerical comparison of auxiliary latitudes

The plot to the right shows the difference between the geodetic latitude and the auxiliary latitudes other than the isometric latitude for the case of the WGS84 ellipsoid. The differences shown on the plot are in arc minutes.
In the Northern hemisphere, θ ≤ χ ≤ μ ≤ ξ ≤ β ≤ φ; in the Southern hemisphere, the inequalities are reversed, with equality at the equator and the poles. Although the graph appears symmetric about 45°, the minima of the curves actually lie between 45° 2′ and 45° 6′. Some representative data points are given in the table below. The conformal and geocentric latitudes are nearly indistinguishable, a fact that was exploited in the days of hand calculators to expedite the construction of map projections.
To first order in the flattening f, the auxiliary latitudes can be expressed as
ζ = φ − Cf sin 2φ
where the constant C takes on the values
for
ζ = .
Parametric
Authalic
Rectifying
Conformal
Geocentric
0°0.00′0.00′0.00′0.00′0.00′
15°−2.88′−3.84′−4.32′−5.76′−5.76′
30°−5.00′−6.66′−7.49′−9.98′−9.98′
45°−5.77′−7.70′−8.66′−11.54′−11.55′
60°−5.00′−6.67′−7.51′−10.01′−10.02′
75°−2.89′−3.86′−4.34′−5.78′−5.79′
90°0.00′0.00′0.00′0.00′0.00′

Latitude and coordinate systems

The geodetic latitude, or any of the auxiliary latitudes defined on the reference ellipsoid, constitutes with longitude a two-dimensional coordinate system on that ellipsoid. To define the position of an arbitrary point it is necessary to extend such a coordinate system into three dimensions. Three latitudes are used in this way: the geodetic, geocentric and parametric latitudes are used in geodetic coordinates, spherical polar coordinates and ellipsoidal coordinates respectively.

Geodetic coordinates

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

Spherical polar coordinates

The geocentric latitude is the complement of the polar angle in conventional spherical polar coordinates in which the coordinates of a point are where is the distance of from the centre, is the angle between the radius vector and the polar axis and is longitude. Since the normal at a general point on the ellipsoid does not pass through the centre it is clear that points on the normal, which all have the same geodetic latitude, will have differing geocentric latitudes. Spherical polar coordinate systems are used in the analysis of the gravity field.

Ellipsoidal coordinates

The parametric latitude can also be extended to a three-dimensional coordinate system. For a point not on the reference ellipsoid construct an auxiliary ellipsoid which is confocal with the reference ellipsoid: the necessary condition is that the product of semi-major axis and eccentricity is the same for both ellipsoids. Let be the semi-minor axis of the auxiliary ellipsoid. Further let be the parametric latitude of on the auxiliary ellipsoid. The set define the ellipsoid coordinates. These coordinates are the natural choice in models of the gravity field for a rotating ellipsoidal body.

Coordinate conversions

The relations between the above coordinate systems, and also Cartesian coordinates are not presented here. The transformation between geodetic and Cartesian coordinates may be found in Geographic coordinate conversion. The relation of Cartesian and spherical polars is given in Spherical coordinate system. The relation of Cartesian and ellipsoidal coordinates is discussed in Torge.

Astronomical latitude

Astronomical latitude is the angle between the equatorial plane and the true vertical at a point on the surface. The true vertical, the direction of a plumb line, is also the direction of the gravity acceleration, the resultant of the gravitational acceleration and the centrifugal acceleration at that latitude. Astronomic latitude is calculated from angles measured between the zenith and stars whose declination is accurately known.
In general the true vertical at a point on the surface does not exactly coincide with either the normal to the reference ellipsoid or the normal to the geoid. The angle between the astronomic and geodetic normals is usually a few seconds of arc but it is important in geodesy. The reason why it differs from the normal to the geoid is, because the geoid is an idealized, theoretical shape "at mean sea level". Points on the real surface of the earth are usually above or below this idealized geoid surface and here the true vertical can vary slightly. Also, the true vertical at a point at a specific time is influenced by tidal forces, which the theoretical geoid averages out.
Astronomical latitude is not to be confused with declination, the coordinate astronomers use in a similar way to specify the angular position of stars north/south of the celestial equator, nor with ecliptic latitude, the coordinate that astronomers use to specify the angular position of stars north/south of the ecliptic.

Footnotes

Citations