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Latitude

In geography, latitude is a coordinate that specifies the northsouth position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pole, with 0° at the Equator. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude and longitude are used together as a coordinate pair to specify a location on the surface of the Earth.

Earth's graticule. The vertical 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 shows the latitude and longitude of points on the surface. In this example meridians are spaced at 6° intervals and parallels at 4° intervals.

On its own, the term "latitude" normally refers to the geodetic latitude as defined below. Briefly, the geodetic latitude of a point is the angle formed between the vector perpendicular (or normal) to the ellipsoidal surface from the point, and the plane of the equator.

Background

Two levels of abstraction are employed in the definitions of latitude and longitude. In the first step the physical surface is modeled by the geoid, 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.[1]

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 (that is latitude and longitude) are ambiguous at best and meaningless at worst". This is of great importance in accurate applications, such as a Global Positioning System (GPS), but in common usage, where high accuracy is not required, the reference ellipsoid is not usually stated.

In English texts, the latitude angle, defined below, is usually denoted by the Greek lower-case letter phi (ϕ or φ). 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.[2]

This article relates to coordinate systems for the Earth: it may be adapted to cover the Moon, planets and other celestial objects (planetographic latitude).

For a brief history see History of latitude.

Determination

In celestial navigation, latitude is determined with the meridian altitude method. More precise measurement of latitude requires an understanding of the gravitational field of the Earth, either to set up 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.

Latitude on the sphere

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

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 (the Prime Meridian) defines the longitude: meridians are lines of constant longitude. The plane through the centre of the Earth and perpendicular to the rotation axis intersects the surface at a great circle called the Equator. 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 normal to the surface at that point: the normal to the surface of the sphere is along the radial vector.

The latitude, as defined in this way for the sphere, is often termed the spherical latitude, to avoid ambiguity with the geodetic latitude and the auxiliary latitudes defined in subsequent sections of this article.

Named latitudes on the Earth

 
The orientation of the Earth at the December solstice.

Besides the equator, four other parallels are of significance:

Arctic Circle 66° 34′ (66.57°) N
Tropic of Cancer 23° 26′ (23.43°) N
Tropic of Capricorn 23° 26′ (23.43°) S
Antarctic Circle 66° 34′ (66.57°) S

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 i. The latitude of the tropical circles is equal to i and the latitude of the polar circles is its complement (90° - i). The axis of rotation varies slowly over time and the values given here are those for the current epoch. The time variation is discussed more fully in the article on axial tilt.[a]

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 (at the zenith).

On map projections there is no universal rule as to how meridians and parallels should appear. The examples below show the named parallels (as red lines) on the commonly used Mercator projection 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.

Normal Mercator Transverse Mercator
 

\

 

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.[3] (This article uses the term ellipsoid in preference to the older term spheroid.) Newton's result was confirmed by geodetic measurements in the 18th century. (See Meridian arc.) An oblate ellipsoid is the three-dimensional surface generated by the rotation of an ellipse about its shorter axis (minor axis). "Oblate ellipsoid of revolution" is abbreviated to 'ellipsoid' in the remainder of this article. (Ellipsoids which do not have an axis of symmetry are termed triaxial.)

Many different reference ellipsoids have been used in the history of geodesy. 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 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 100 m (330 ft) 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

 
A sphere of radius a compressed along the z axis to form an oblate ellipsoid of revolution.

The 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 the equatorial radius, which is the semi-major axis, a. The other parameter is usually (1) the polar radius or semi-minor axis, b; or (2) the (first) flattening, f; or (3) the eccentricity, e. These parameters are not independent: they are related by

 

Many other parameters (see ellipse, ellipsoid) appear in the study of geodesy, geophysics and map projections but they can all be expressed in terms of one or two members of the set a, b, f and e. Both f and e are small and often appear in series expansions in calculations; they are of the order 1/298 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, 1/f. For example, the defining values for the WGS84 ellipsoid, used by all GPS devices, are[4]

  • a (equatorial radius): 6378137.0 m exactly
  • 1/f (inverse flattening): 298.257223563 exactly

from which are derived

  • b (polar radius): 6356752.31425 m
  • e2 (eccentricity squared): 0.00669437999014

The difference between the semi-major and semi-minor axes is about 21 km (13 miles) 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 definition of geodetic latitude ( ) and longitude ( ) on an ellipsoid. The normal to the surface does not pass through the centre, except at the equator and at the poles.

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 (also known as spherical latitude, after the 3D polar angle): the angle between the radius (from centre to the point on the surface) and the equatorial plane. (Figure below). There is no standard notation: examples from various texts include θ, ψ, q, ϕ′, ϕc, ϕg. This article uses θ.

Geographic latitude must be used with care, as some authors use it as a synonym for geodetic latitude whilst others use it as an alternative to the astronomical latitude. "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 metres (460 feet) distant from the tower.[citation needed] A web search may produce several different values for the latitude of the tower; the reference ellipsoid is rarely specified.

Meridian distance

The length of a degree of latitude depends on the figure of the Earth assumed.

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

 

where R denotes the mean radius of the Earth. R is equal to 6,371 km or 3,959 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 (69.1 statute miles) (60.0 nautical miles). The length of 1 minute of latitude is 1.853 km (1.151 statute miles) (1.00 nautical miles), while the length of 1 second of latitude is 30.8 m or 101 feet (see nautical mile).

Meridian distance on the ellipsoid

In Meridian arc and standard texts[5][6][7] it is shown that the distance along a meridian from latitude ϕ to the equator is given by (ϕ in radians)

 

where M(ϕ) is the meridional radius of curvature.

The quarter meridian distance from the equator to the pole is

 

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 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[6][7]

 
  Δ1
lat
Δ1
long
110.574 km 111.320 km
15° 110.649 km 107.550 km
30° 110.852 km 96.486 km
45° 111.132 km 78.847 km
60° 111.412 km 55.800 km
75° 111.618 km 28.902 km
90° 111.694 km 0.000 km

When the latitude difference is 1 degree, corresponding to π/180 radians, the arc distance is about

 

The distance in metres (correct to 0.01 metre) between latitudes   − 0.5 degrees and   + 0.5 degrees on the WGS84 spheroid is

 

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

 

A calculator for any latitude is provided by the U.S. Government's National Geospatial-Intelligence Agency (NGA).[8]

The following graph illustrates the variation of both a degree of latitude and a degree of longitude with latitude.

 
The definition of geodetic latitude (ϕ) and geocentric latitude (θ).

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, a, and the eccentricity, e. (For inverses see below.) The forms given are, apart from notational variants, those in the standard reference for map projections, namely "Map projections: a working manual" by J. P. Snyder.[9] Derivations of these expressions may be found in Adams[10] and online publications by Osborne[6] and Rapp.[7]

Geocentric latitude

 
The definition of geodetic latitude (ϕ) and geocentric latitude (θ).

The geocentric latitude is the angle between the equatorial plane and the radius from the centre to a point of interest.

When the point is on the surface of the ellipsoid, the relation between the geocentric latitude (θ) and the geodetic latitude (ϕ) is:

 

For points not on the surface of the ellipsoid, the relationship involves additionally the ellipsoidal height h:

 

The geodetic and geocentric latitudes are equal at the equator and at the poles but at other latitudes they differ by a few minutes of arc. Taking the value of the squared eccentricity as 0.0067 (it depends on the choice of ellipsoid) the maximum difference of   may be shown to be about 11.5 minutes of arc at a geodetic latitude of approximately 45° 6′.[b]

Parametric latitude (or reduced latitude)

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

The parametric latitude or reduced 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 ϕ. It was introduced by Legendre[11] and Bessel[12] 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, u(ϕ), is also used in the current literature. The parametric latitude is related to the geodetic latitude by:[6][7]

 

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:

 

The Cartesian coordinates of the point are parameterized by

 

Cayley suggested the term parametric latitude because of the form of these equations.[13]

The parametric latitude is not used in the theory of map projections. Its most important application is in the theory of ellipsoid geodesics, (Vincenty, Karney[14]).

Rectifying latitude

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:

 

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

 

and the length of the meridian quadrant from the equator to the pole (the polar distance) 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. (Snyder, Section 16).[9] The rectifying latitude is also of great importance in the construction of the Transverse Mercator projection.

Authalic latitude

The authalic latitude (after the Greek for "same area"), ξ, 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.[9]: §14 

Conformal latitude

The conformal latitude, χ, gives an angle-preserving (conformal) transformation to the sphere. [15]

 

where gd(x) is the Gudermannian function. (See also Mercator projection.)

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

The isometric latitude, ψ, 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 (page 15):[9]

 

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

 

The isometric latitude ψ is closely related to the conformal latitude χ:

 

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.
    • When converting from isometric or conformal to geodetic, two iterations of Newton-Raphson gives double precision accuracy.[16]
  • 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.[10] Osborne derives series to arbitrary order by using the computer algebra package Maxima and expresses the coefficients in terms of both eccentricity and flattening. The series method is not applicable to the isometric latitude and one must find the conformal latitude in an intermediate step.[6]

Numerical comparison of auxiliary latitudes

 

The plot to the right shows the difference between the geodetic latitude and the auxiliary latitudes other than the isometric latitude (which diverges to infinity at the poles) for the case of the WGS84 ellipsoid. The differences shown on the plot are in arc minutes. In the Northern hemisphere (positive latitudes), θχμξβϕ; in the Southern hemisphere (negative latitudes), the inequalities are reversed, with equality at the equator and the poles. Although the graph appears symmetric about 45°, the minima of the curves actually lie between 45° 2′ and 45° 6′. Some representative data points are given in the table below. The conformal and geocentric latitudes are nearly indistinguishable, a fact that was exploited in the days of hand calculators to expedite the construction of map projections.[9]: 108 

To first order in the flattening f, the auxiliary latitudes can be expressed as ζ = ϕCf sin 2ϕ where the constant C takes on the values [12, 23, 34, 1, 1] for ζ = [β, ξ, μ, χ, θ].

Approximate difference from geodetic latitude (ϕ)
ϕ Parametric
βϕ
Authalic
ξϕ
Rectifying
μϕ
Conformal
χϕ
Geocentric
θϕ
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

 
Geodetic coordinates P(ɸ,λ,h)

At an arbitrary point P consider the line PN which is normal to the reference ellipsoid. The geodetic coordinates P(ɸ,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

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

The geocentric latitude θ is the complement of the polar angle or colatitude 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 and λ is longitude. Since the normal at a general point on the ellipsoid does not pass through the centre it is clear that points P' on the normal, which all have the same geodetic latitude, will have differing geocentric latitudes. Spherical polar coordinate systems are used in the analysis of the gravity field.

Ellipsoidal-harmonic coordinates

 
Ellipsoidal coordinates P(u,β,λ)

The parametric 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 parametric latitude of P on the auxiliary ellipsoid. The set (u,β,λ) define the ellipsoidal-harmonic coordinates[17] or simply ellipsoidal coordinates[5]: §4.2.2  (although that term is also used to refer to geodetic coordinate). These coordinates are the natural choice in models of the gravity field for a rotating ellipsoidal body. The above applies to a biaxial ellipsoid (a spheroid, as in oblate spheroidal coordinates); for a generalization, see triaxial ellipsoidal coordinates.

Coordinate conversions

The relations between the above coordinate systems, and also Cartesian coordinates are not presented here. The transformation between geodetic and Cartesian coordinates may be found in geographic coordinate conversion. The relation of Cartesian and spherical polars is given in spherical coordinate system. The relation of Cartesian and ellipsoidal coordinates is discussed in Torge.[5]

Astronomical latitude

Astronomical latitude (Φ) is the angle between the equatorial plane and the true vertical direction at a point on the surface. The true vertical, the direction of a plumb line, is also the gravity direction (the resultant of the gravitational acceleration (mass-based) and the centrifugal acceleration) at that latitude.[5] 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 called vertical deflection and is usually a few seconds of arc but it is important in geodesy.[5][18] The reason why it differs from the normal to the geoid is, because the geoid is an idealized, theoretical shape "at mean sea level". Points on the real surface of the earth are usually above or below this idealized geoid surface and here the true vertical can vary slightly. Also, the true vertical at a point at a specific time is influenced by tidal forces, which the theoretical geoid averages out.

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

See also

References

Footnotes

  1. ^ The value of this angle today is 23°26′10.7″ (or 23.4363°). This figure is provided by Template:Circle of latitude.
  2. ^ An elementary calculation involves differentiation to find the maximum difference of the geodetic and geocentric latitudes.

Citations

  1. ^ "ISO 19111 Geographic information — Referencing by coordinates". ISO. 2021-06-01. Retrieved 2022-01-16.
  2. ^ The Corporation of Trinity House (10 January 2020). "1/2020 Needles Lighthouse". Notices to Mariners. Retrieved 24 May 2020.
  3. ^ Newton, Isaac. "Book III Proposition XIX Problem III". Philosophiæ Naturalis Principia Mathematica. Translated by Motte, Andrew. p. 407.
  4. ^ National Imagery and Mapping Agency (23 June 2004). "Department of Defense World Geodetic System 1984" (PDF). National Imagery and Mapping Agency. p. 3-1. TR8350.2. Retrieved 25 April 2020.
  5. ^ a b c d e Torge, W. (2001). Geodesy (3rd ed.). De Gruyter. ISBN 3-11-017072-8.
  6. ^ a b c d e Osborne, Peter (2013). "Chapters 5,6". The Mercator Projections. doi:10.5281/zenodo.35392. for LaTeX code and figures.
  7. ^ a b c d Rapp, Richard H. (1991). "Chapter 3". Geometric Geodesy, Part I. Columbus, OH: Dept. of Geodetic Science and Surveying, Ohio State Univ. hdl:1811/24333.
  8. ^ . National Geospatial-Intelligence Agency. Archived from the original on 2012-12-11. Retrieved 2011-02-08.
  9. ^ a b c d e Snyder, John P. (1987). . U.S. Geological Survey Professional Paper 1395. Washington, DC: United States Government Printing Office. Archived from the original on 2008-05-16. Retrieved 2017-09-02.
  10. ^ a b Adams, Oscar S. (1921). Latitude Developments Connected With Geodesy and Cartography (with tables, including a table for Lambert equal area meridional projection (PDF). Special Publication No. 67. US Coast and Geodetic Survey. (Note: Adams uses the nomenclature isometric latitude for the conformal latitude of this article (and throughout the modern literature).)
  11. ^ Legendre, A. M. (1806). "Analyse des triangles tracés sur la surface d'un sphéroïde". Mém. Inst. Nat. Fr. 1st semester: 130–161.
  12. ^ Bessel, F. W. (1825). "Über die Berechnung der geographischen Langen und Breiten aus geodatischen Vermessungen". Astron. Nachr. 4 (86): 241–254. arXiv:0908.1824. Bibcode:2010AN....331..852K. doi:10.1002/asna.201011352. S2CID 118760590.
    Translation: Karney, C. F. F.; Deakin, R. E. (2010). "The calculation of longitude and latitude from geodesic measurements". Astron. Nachr. 331 (8): 852–861. arXiv:0908.1824. Bibcode:1825AN......4..241B. doi:10.1002/asna.18260041601. S2CID 118630614.
  13. ^ Cayley, A. (1870). "On the geodesic lines on an oblate spheroid". Phil. Mag. 40 (4th ser): 329–340. doi:10.1080/14786447008640411.
  14. ^ Karney, C. F. F. (2013). "Algorithms for geodesics". Journal of Geodesy. 87 (1): 43–55. arXiv:1109.4448. Bibcode:2013JGeod..87...43K. doi:10.1007/s00190-012-0578-z. S2CID 119310141.
  15. ^ Lagrange, Joseph-Louis (1779). "Sur la Construction des Cartes Géographiques". Oevres (in French). Vol. IV. p. 667.
  16. ^ Karney, Charles F. F. (August 2011). "Transverse Mercator with an accuracy of a few nanometers". Journal of Geodesy. 85 (8): 475–485. arXiv:1002.1417. Bibcode:2011JGeod..85..475K. doi:10.1007/s00190-011-0445-3. S2CID 118619524.
  17. ^ Holfmann-Wellenfor & Moritz (2006) Physical Geodesy, p.240, eq. (6-6) to (6-10).
  18. ^ Hofmann-Wellenhof, B.; Moritz, H. (2006). Physical Geodesy (2nd ed.). ISBN 3-211-33544-7.

External links

  • GEONets Names Server 2008-03-09 at the Wayback Machine, access to the National Geospatial-Intelligence Agency's (NGA) database of foreign geographic feature names.
  • Resources for determining your latitude and longitude 2008-05-19 at the Wayback Machine
  • Convert decimal degrees into degrees, minutes, seconds - Info about decimal to sexagesimal conversion
  • Convert decimal degrees into degrees, minutes, seconds
  • Distance calculation based on latitude and longitude - JavaScript version
  • 16th Century Latitude Survey
  • Determination of Latitude by Francis Drake on the Coast of California in 1579

latitude, this, article, about, geographical, reference, system, other, uses, disambiguation, geography, latitude, coordinate, that, specifies, north, south, position, point, surface, earth, another, celestial, body, given, angle, that, ranges, from, south, po. This article is about the geographical reference system For other uses see Latitude disambiguation In geography latitude is a coordinate that specifies the north south position of a point on the surface of the Earth or another celestial body Latitude is given as an angle that ranges from 90 at the south pole to 90 at the north pole with 0 at the Equator Lines of constant latitude or parallels run east west as circles parallel to the equator Latitude and longitude are used together as a coordinate pair to specify a location on the surface of the Earth Earth s graticule The vertical 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 shows the latitude and longitude of points on the surface In this example meridians are spaced at 6 intervals and parallels at 4 intervals On its own the term latitude normally refers to the geodetic latitude as defined below Briefly the geodetic latitude of a point is the angle formed between the vector perpendicular or normal to the ellipsoidal surface from the point and the plane of the equator Contents 1 Background 1 1 Determination 2 Latitude on the sphere 2 1 The graticule on the sphere 2 2 Named latitudes on the Earth 3 Latitude on the ellipsoid 3 1 Ellipsoids 3 2 The geometry of the ellipsoid 3 3 Geodetic and geocentric latitudes 4 Meridian distance 4 1 Meridian distance on the sphere 4 2 Meridian distance on the ellipsoid 5 Auxiliary latitudes 5 1 Geocentric latitude 5 2 Parametric latitude or reduced latitude 5 3 Rectifying latitude 5 4 Authalic latitude 5 5 Conformal latitude 5 6 Isometric latitude 5 7 Inverse formulae and series 5 8 Numerical comparison of auxiliary latitudes 6 Latitude and coordinate systems 6 1 Geodetic coordinates 6 2 Spherical polar coordinates 6 3 Ellipsoidal harmonic coordinates 6 4 Coordinate conversions 7 Astronomical latitude 8 See also 9 References 9 1 Footnotes 9 2 Citations 10 External linksBackground EditTwo levels of abstraction are employed in the definitions 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 1 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 that is latitude and longitude are ambiguous at best and meaningless at worst This is of great importance in accurate applications such as a Global Positioning System GPS but in common usage where high accuracy is not required the reference ellipsoid is not usually stated In English texts the latitude angle defined below is usually denoted by the Greek lower case letter phi ϕ or f 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 2 This article relates to coordinate systems for the Earth it may be adapted to cover the Moon planets and other celestial objects planetographic latitude For a brief history see History of latitude Determination Edit See also Longitude determination Further information Celestial navigation Latitude In celestial navigation latitude is determined with the meridian altitude method More precise measurement of latitude requires an understanding of the gravitational field of the Earth either to set up 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 Latitude on the sphere Edit A perspective view of the Earth showing how latitude ϕ displaystyle phi and longitude l displaystyle lambda are defined on a spherical model The graticule spacing is 10 degrees The graticule on the sphere Edit 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 the Prime Meridian defines the longitude meridians are lines of constant longitude The plane through the centre of the Earth and perpendicular to the rotation axis intersects the surface at a great circle called the Equator 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 normal to the surface at that point the normal to the surface of the sphere is along the radial vector The latitude as defined in this way for the sphere is often termed the spherical latitude to avoid ambiguity with the geodetic latitude and the auxiliary latitudes defined in subsequent sections of this article Named latitudes on the Earth Edit The orientation of the Earth at the December solstice Besides the equator four other parallels are of significance Arctic Circle 66 34 66 57 NTropic of Cancer 23 26 23 43 NTropic of Capricorn 23 26 23 43 SAntarctic Circle 66 34 66 57 SThe 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 i The latitude of the tropical circles is equal to i and the latitude of the polar circles is its complement 90 i The axis of rotation varies slowly over time and the values given here are those for the current epoch The time variation is discussed more fully in the article on axial tilt a 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 at the zenith On map projections there is no universal rule as to how meridians and parallels should appear The examples below show the named parallels as red lines on the commonly used Mercator projection 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 Normal Mercator Transverse Mercator Latitude on the ellipsoid EditEllipsoids Edit Main article Ellipsoid In 1687 Isaac Newton published the Philosophiae Naturalis Principia Mathematica in which he proved that a rotating self gravitating fluid body in equilibrium takes the form of an oblate ellipsoid 3 This article uses the term ellipsoid in preference to the older term spheroid Newton s result was confirmed by geodetic measurements in the 18th century See Meridian arc An oblate ellipsoid is the three dimensional surface generated by the rotation of an ellipse about its shorter axis minor axis Oblate ellipsoid of revolution is abbreviated to ellipsoid in the remainder of this article Ellipsoids which do not have an axis of symmetry are termed triaxial Many different reference ellipsoids have been used in the history of geodesy 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 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 100 m 330 ft 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 Edit A sphere of radius a compressed along the z axis to form an oblate ellipsoid of revolution The 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 the equatorial radius which is the semi major axis a The other parameter is usually 1 the polar radius or semi minor axis b or 2 the first flattening f or 3 the eccentricity e These parameters are not independent they are related by f a b a e 2 2 f f 2 b a 1 f a 1 e 2 displaystyle f frac a b a qquad e 2 2f f 2 qquad b a 1 f a sqrt 1 e 2 Many other parameters see ellipse ellipsoid appear in the study of geodesy geophysics and map projections but they can all be expressed in terms of one or two members of the set a b f and e Both f and e are small and often appear in series expansions in calculations they are of the order 1 298 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 1 f For example the defining values for the WGS84 ellipsoid used by all GPS devices are 4 a equatorial radius 6378 137 0 m exactly 1 f inverse flattening 298 257223 563 exactlyfrom which are derived b polar radius 6356 752 31425 m e2 eccentricity squared 0 006694 379 990 14The difference between the semi major and semi minor axes is about 21 km 13 miles 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 Edit See also Geodetic coordinates Geodetic vs geocentric coordinates The definition of geodetic latitude ϕ displaystyle phi and longitude l displaystyle lambda on an ellipsoid The normal to the surface does not pass through the centre except at the equator and at the poles 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 also known as spherical latitude after the 3D polar angle the angle between the radius from centre to the point on the surface and the equatorial plane Figure below There is no standard notation examples from various texts include 8 ps q ϕ ϕc ϕg This article uses 8 Geographic latitude must be used with care as some authors use it as a synonym for geodetic latitude whilst others use it as an alternative to the astronomical latitude 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 metres 460 feet distant from the tower citation needed A web search may produce several different values for the latitude of the tower the reference ellipsoid is rarely specified Meridian distance EditMain article Meridian arc See also Length of a degree of longitude The length of a degree of latitude depends on the figure of the Earth assumed Meridian distance on the sphere Edit 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 m ϕ p 180 R ϕ d e g r e e s R ϕ r a d i a n s displaystyle m phi frac pi 180 circ R phi mathrm degrees R phi mathrm radians where R denotes the mean radius of the Earth R is equal to 6 371 km or 3 959 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 69 1 statute miles 60 0 nautical miles The length of 1 minute of latitude is 1 853 km 1 151 statute miles 1 00 nautical miles while the length of 1 second of latitude is 30 8 m or 101 feet see nautical mile Meridian distance on the ellipsoid Edit In Meridian arc and standard texts 5 6 7 it is shown that the distance along a meridian from latitude ϕ to the equator is given by ϕ in radians m ϕ 0 ϕ M ϕ d ϕ a 1 e 2 0 ϕ 1 e 2 sin 2 ϕ 3 2 d ϕ displaystyle m phi int 0 phi M phi d phi a left 1 e 2 right int 0 phi left 1 e 2 sin 2 phi right frac 3 2 d phi where M ϕ is the meridional radius of curvature The quarter meridian distance from the equator to the pole is m p m p 2 displaystyle m mathrm p m left frac pi 2 right 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 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 6 7 d m ϕ M ϕ d ϕ a 1 e 2 1 e 2 sin 2 ϕ 3 2 d ϕ displaystyle delta m phi M phi delta phi a left 1 e 2 right left 1 e 2 sin 2 phi right frac 3 2 delta phi ϕ displaystyle phi D1lat D1long0 110 574 km 111 320 km15 110 649 km 107 550 km30 110 852 km 96 486 km45 111 132 km 78 847 km60 111 412 km 55 800 km75 111 618 km 28 902 km90 111 694 km 0 000 kmWhen the latitude difference is 1 degree corresponding to p 180 radians the arc distance is about D lat 1 p a 1 e 2 180 1 e 2 sin 2 ϕ 3 2 displaystyle Delta text lat 1 frac pi a left 1 e 2 right 180 circ left 1 e 2 sin 2 phi right frac 3 2 The distance in metres correct to 0 01 metre between latitudes ϕ displaystyle phi 0 5 degrees and ϕ displaystyle phi 0 5 degrees on the WGS84 spheroid is D lat 1 111 132 954 559 822 cos 2 ϕ 1 175 cos 4 ϕ displaystyle Delta text lat 1 111 132 954 559 822 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 east west distance D long 1 p a cos ϕ 180 1 e 2 sin 2 ϕ displaystyle Delta text long 1 frac pi a cos phi 180 circ sqrt 1 e 2 sin 2 phi A calculator for any latitude is provided by the U S Government s National Geospatial Intelligence Agency NGA 8 The following graph illustrates the variation of both a degree of latitude and a degree of longitude with latitude The definition of geodetic latitude ϕ and geocentric latitude 8 Auxiliary latitudes EditThere are six auxiliary latitudes that have applications to special problems in geodesy geophysics and the theory of map projections Geocentric latitude Parametric or reduced latitude Rectifying latitude Authalic latitude Conformal latitude Isometric latitudeThe 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 a and the eccentricity e For inverses see below The forms given are apart from notational variants those in the standard reference for map projections namely Map projections a working manual by J P Snyder 9 Derivations of these expressions may be found in Adams 10 and online publications by Osborne 6 and Rapp 7 Geocentric latitude Edit See also Geodetic and geocentric latitudes The definition of geodetic latitude ϕ and geocentric latitude 8 The geocentric latitude is the angle between the equatorial plane and the radius from the centre to a point of interest When the point is on the surface of the ellipsoid the relation between the geocentric latitude 8 and the geodetic latitude ϕ is 8 ϕ tan 1 1 e 2 tan ϕ tan 1 1 f 2 tan ϕ displaystyle theta phi tan 1 left left 1 e 2 right tan phi right tan 1 left 1 f 2 tan phi right For points not on the surface of the ellipsoid the relationship involves additionally the ellipsoidal height h 8 ϕ h tan 1 N 1 f 2 h N h tan ϕ displaystyle theta phi h tan 1 left frac N 1 f 2 h N h tan phi right The geodetic and geocentric latitudes are equal at the equator and at the poles but at other latitudes they differ by a few minutes of arc Taking the value of the squared eccentricity as 0 0067 it depends on the choice of ellipsoid the maximum difference of ϕ 8 displaystyle phi theta may be shown to be about 11 5 minutes of arc at a geodetic latitude of approximately 45 6 b Parametric latitude or reduced latitude Edit Definition of the parametric latitude b on the ellipsoid The parametric latitude or reduced latitude b 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 ϕ It was introduced by Legendre 11 and Bessel 12 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 u ϕ is also used in the current literature The parametric latitude is related to the geodetic latitude by 6 7 b ϕ tan 1 1 e 2 tan ϕ tan 1 1 f tan ϕ displaystyle beta phi tan 1 left sqrt 1 e 2 tan phi right tan 1 left 1 f tan phi right The alternative name arises from the parameterization of the equation of the ellipse describing a meridian section In terms of Cartesian coordinates p the distance from the minor axis and z the distance above the equatorial plane the equation of the ellipse is p 2 a 2 z 2 b 2 1 displaystyle frac p 2 a 2 frac z 2 b 2 1 The Cartesian coordinates of the point are parameterized by p a cos b z b sin b displaystyle p a cos beta qquad z b sin beta Cayley suggested the term parametric latitude because of the form of these equations 13 The parametric latitude is not used in the theory of map projections Its most important application is in the theory of ellipsoid geodesics Vincenty Karney 14 Rectifying latitude Edit See also Rectifying radius The rectifying latitude m is the meridian distance scaled so that its value at the poles is equal to 90 degrees or p 2 radians m ϕ p 2 m ϕ m p displaystyle mu phi frac pi 2 frac m phi m mathrm p where the meridian distance from the equator to a latitude ϕ is see Meridian arc m ϕ a 1 e 2 0 ϕ 1 e 2 sin 2 ϕ 3 2 d ϕ displaystyle m phi a left 1 e 2 right int 0 phi left 1 e 2 sin 2 phi right frac 3 2 d phi and the length of the meridian quadrant from the equator to the pole the polar distance is m p m p 2 displaystyle m mathrm p m left frac pi 2 right Using the rectifying latitude to define a latitude on a sphere of radius R 2 m p p displaystyle R frac 2m mathrm 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 Section 16 9 The rectifying latitude is also of great importance in the construction of the Transverse Mercator projection Authalic latitude Edit See also Authalic radius The authalic latitude after the Greek for same area 3 gives an area preserving transformation to a sphere 3 ϕ sin 1 q ϕ q p displaystyle xi phi sin 1 left frac q phi q mathrm p right where q ϕ 1 e 2 sin ϕ 1 e 2 sin 2 ϕ 1 e 2 2 e ln 1 e sin ϕ 1 e sin ϕ 1 e 2 sin ϕ 1 e 2 sin 2 ϕ 1 e 2 e tanh 1 e sin ϕ displaystyle begin aligned q phi amp frac left 1 e 2 right 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 2pt amp frac left 1 e 2 right sin phi 1 e 2 sin 2 phi frac 1 e 2 e tanh 1 e sin phi end aligned and q p q p 2 1 1 e 2 2 e ln 1 e 1 e 1 1 e 2 e tanh 1 e displaystyle begin aligned q mathrm p q left frac pi 2 right amp 1 frac 1 e 2 2e ln left frac 1 e 1 e right amp 1 frac 1 e 2 e tanh 1 e end aligned and the radius of the sphere is taken as R q a q p 2 displaystyle R q a sqrt frac q mathrm p 2 An example of the use of the authalic latitude is the Albers equal area conic projection 9 14 Conformal latitude Edit The conformal latitude x gives an angle preserving conformal transformation to the sphere 15 x ϕ 2 tan 1 1 sin ϕ 1 sin ϕ 1 e sin ϕ 1 e sin ϕ e 1 2 p 2 2 tan 1 tan ϕ 2 p 4 1 e sin ϕ 1 e sin ϕ e 2 p 2 tan 1 sinh sinh 1 tan ϕ e tanh 1 e sin ϕ gd gd 1 ϕ e tanh 1 e sin ϕ displaystyle begin aligned chi phi amp 2 tan 1 left left frac 1 sin phi 1 sin phi right left frac 1 e sin phi 1 e sin phi right e right frac 1 2 frac pi 2 2pt amp 2 tan 1 left tan left frac phi 2 frac pi 4 right left frac 1 e sin phi 1 e sin phi right frac e 2 right frac pi 2 2pt amp tan 1 left sinh left sinh 1 tan phi e tanh 1 e sin phi right right amp operatorname gd left operatorname gd 1 phi e tanh 1 e sin phi right end aligned where gd x is the Gudermannian function See also Mercator projection 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 Edit The isometric latitude ps 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 ps and longitude l give rise to equal distance displacements along the meridians and parallels respectively The graticule defined by the lines of constant ps and constant l divides the surface of the ellipsoid into a mesh of squares of varying size The isometric latitude is zero at the equator but rapidly diverges from the geodetic latitude tending to infinity at the poles The conventional notation is given in Snyder page 15 9 ps ϕ ln tan p 4 ϕ 2 e 2 ln 1 e sin ϕ 1 e sin ϕ sinh 1 tan ϕ e tanh 1 e sin ϕ gd 1 ϕ e tanh 1 e sin ϕ displaystyle begin aligned psi phi amp 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 amp sinh 1 tan phi e tanh 1 e sin phi amp operatorname gd 1 phi e tanh 1 e sin phi end aligned 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 y ϕ E 2 p ps ϕ displaystyle y phi frac E 2 pi psi phi The isometric latitude ps is closely related to the conformal latitude x ps ϕ gd 1 x ϕ displaystyle psi phi operatorname gd 1 chi phi Inverse formulae and series Edit 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 When converting from isometric or conformal to geodetic two iterations of Newton Raphson gives double precision accuracy 16 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 10 Osborne derives series to arbitrary order by using the computer algebra package Maxima and expresses the coefficients in terms of both eccentricity and flattening The series method is not applicable to the isometric latitude and one must find the conformal latitude in an intermediate step 6 Numerical comparison of auxiliary latitudes Edit The plot to the right shows the difference between the geodetic latitude and the auxiliary latitudes other than the isometric latitude which diverges to infinity at the poles for the case of the WGS84 ellipsoid The differences shown on the plot are in arc minutes In the Northern hemisphere positive latitudes 8 x m 3 b ϕ in the Southern hemisphere negative latitudes the inequalities are reversed with equality at the equator and the poles Although the graph appears symmetric about 45 the minima of the curves actually lie between 45 2 and 45 6 Some representative data points are given in the table below The conformal and geocentric latitudes are nearly indistinguishable a fact that was exploited in the days of hand calculators to expedite the construction of map projections 9 108 To first order in the flattening f the auxiliary latitudes can be expressed as z ϕ Cf sin 2ϕ where the constant C takes on the values 1 2 2 3 3 4 1 1 for z b 3 m x 8 Approximate difference from geodetic latitude ϕ ϕ Parametricb ϕ Authalic3 ϕ Rectifyingm ϕ Conformalx ϕ Geocentric8 ϕ0 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 EditThe 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 Edit Main article Geodetic coordinates Geodetic coordinates P ɸ l h At an arbitrary point P consider the line PN which is normal to the reference ellipsoid The geodetic coordinates P ɸ l 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 Edit Geocentric coordinate related to spherical polar coordinates P r 8 l The geocentric latitude 8 is the complement of the polar angle or colatitude 8 in conventional spherical polar coordinates in which the coordinates of a point are P r 8 l where r is the distance of P from the centre O 8 is the angle between the radius vector and the polar axis and l is longitude Since the normal at a general point on the ellipsoid does not pass through the centre it is clear that points P on the normal which all have the same geodetic latitude will have differing geocentric latitudes Spherical polar coordinate systems are used in the analysis of the gravity field Ellipsoidal harmonic coordinates Edit Ellipsoidal coordinates P u b l The parametric 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 b be the parametric latitude of P on the auxiliary ellipsoid The set u b l define the ellipsoidal harmonic coordinates 17 or simply ellipsoidal coordinates 5 4 2 2 although that term is also used to refer to geodetic coordinate These coordinates are the natural choice in models of the gravity field for a rotating ellipsoidal body The above applies to a biaxial ellipsoid a spheroid as in oblate spheroidal coordinates for a generalization see triaxial ellipsoidal coordinates Coordinate conversions Edit 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 5 Astronomical latitude EditAstronomical latitude F is the angle between the equatorial plane and the true vertical direction at a point on the surface The true vertical the direction of a plumb line is also the gravity direction the resultant of the gravitational acceleration mass based and the centrifugal acceleration at that latitude 5 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 called vertical deflection and is usually a few seconds of arc but it is important in geodesy 5 18 The reason why it differs from the normal to the geoid is because the geoid is an idealized theoretical shape at mean sea level Points on the real surface of the earth are usually above or below this idealized geoid surface and here the true vertical can vary slightly Also the true vertical at a point at a specific time is influenced by tidal forces which the theoretical geoid averages out Astronomical latitude is not to be confused with declination the coordinate astronomers use in a similar way to specify the angular position of stars north south of the celestial equator see equatorial coordinates nor with ecliptic latitude the coordinate that astronomers use to specify the angular position of stars north south of the ecliptic see ecliptic coordinates See also EditAltitude mean sea level Bowditch s American Practical Navigator Cardinal direction Circle of latitude Colatitude Declination on celestial sphere Degree Confluence Project Geodesy Geodetic datum Geographic coordinate system Geographical distance Geomagnetic latitude Geotagging Great circle distance History of latitude Horse latitudes International Latitude Service List of countries by latitude Longitude Natural Area Code Navigation Orders of magnitude length World Geodetic SystemReferences EditFootnotes Edit The value of this angle today is 23 26 10 7 or 23 4363 This figure is provided by Template Circle of latitude An elementary calculation involves differentiation to find the maximum difference of the geodetic and geocentric latitudes Citations Edit ISO 19111 Geographic information Referencing by coordinates ISO 2021 06 01 Retrieved 2022 01 16 The Corporation of Trinity House 10 January 2020 1 2020 Needles Lighthouse Notices to Mariners Retrieved 24 May 2020 Newton Isaac Book III Proposition XIX Problem III Philosophiae Naturalis Principia Mathematica Translated by Motte Andrew p 407 National Imagery and Mapping Agency 23 June 2004 Department of Defense World Geodetic System 1984 PDF National Imagery and Mapping Agency p 3 1 TR8350 2 Retrieved 25 April 2020 a b c d e Torge W 2001 Geodesy 3rd ed De Gruyter ISBN 3 11 017072 8 a b c d e Osborne Peter 2013 Chapters 5 6 The Mercator Projections doi 10 5281 zenodo 35392 for LaTeX code and figures a b c d Rapp Richard H 1991 Chapter 3 Geometric Geodesy Part I Columbus OH Dept of Geodetic Science and Surveying Ohio State Univ hdl 1811 24333 Length of degree calculator National Geospatial Intelligence Agency Archived from the original on 2012 12 11 Retrieved 2011 02 08 a b c d e Snyder John P 1987 Map Projections A Working Manual U S Geological Survey Professional Paper 1395 Washington DC United States Government Printing Office Archived from the original on 2008 05 16 Retrieved 2017 09 02 a b Adams Oscar S 1921 Latitude Developments Connected With Geodesy and Cartography with tables including a table for Lambert equal area meridional projection PDF Special Publication No 67 US Coast and Geodetic Survey Note Adams uses the nomenclature isometric latitude for the conformal latitude of this article and throughout the modern literature Legendre A M 1806 Analyse des triangles traces sur la surface d un spheroide Mem Inst Nat Fr 1st semester 130 161 Bessel F W 1825 Uber die Berechnung der geographischen Langen und Breiten aus geodatischen Vermessungen Astron Nachr 4 86 241 254 arXiv 0908 1824 Bibcode 2010AN 331 852K doi 10 1002 asna 201011352 S2CID 118760590 Translation Karney C F F Deakin R E 2010 The calculation of longitude and latitude from geodesic measurements Astron Nachr 331 8 852 861 arXiv 0908 1824 Bibcode 1825AN 4 241B doi 10 1002 asna 18260041601 S2CID 118630614 Cayley A 1870 On the geodesic lines on an oblate spheroid Phil Mag 40 4th ser 329 340 doi 10 1080 14786447008640411 Karney C F F 2013 Algorithms for geodesics Journal of Geodesy 87 1 43 55 arXiv 1109 4448 Bibcode 2013JGeod 87 43K doi 10 1007 s00190 012 0578 z S2CID 119310141 Lagrange Joseph Louis 1779 Sur la Construction des Cartes Geographiques Oevres in French Vol IV p 667 Karney Charles F F August 2011 Transverse Mercator with an accuracy of a few nanometers Journal of Geodesy 85 8 475 485 arXiv 1002 1417 Bibcode 2011JGeod 85 475K doi 10 1007 s00190 011 0445 3 S2CID 118619524 Holfmann Wellenfor amp Moritz 2006 Physical Geodesy p 240 eq 6 6 to 6 10 Hofmann Wellenhof B Moritz H 2006 Physical Geodesy 2nd ed ISBN 3 211 33544 7 External links EditLatitude at Wikipedia s sister projects Definitions from Wiktionary Media from Commons News from Wikinews Quotations from Wikiquote Texts from Wikisource Textbooks from Wikibooks Resources from Wikiversity GEONets Names Server Archived 2008 03 09 at the Wayback Machine access to the National Geospatial Intelligence Agency s NGA database of foreign geographic feature names Resources for determining your latitude and longitude Archived 2008 05 19 at the Wayback Machine Convert decimal degrees into degrees minutes seconds Info about decimal to sexagesimal conversion Convert decimal degrees into degrees minutes seconds Distance calculation based on latitude and longitude JavaScript version 16th Century Latitude Survey Determination of Latitude by Francis Drake on the Coast of California in 1579 Retrieved from https en 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