Latitude
In
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
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
Since there are many different
In English texts, the latitude angle, defined below, is usually denoted by the Greek lower-case letter
This article relates to coordinate systems for the Earth: it may be adapted to cover the Moon, planets and other celestial objects (
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
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
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:
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
Normal Mercator | Transverse Mercator | |||
---|---|---|---|---|
\ |
Latitude on the ellipsoid
Ellipsoids
In 1687
Many different
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 (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
- 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 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.
- 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
The
For
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 | |
---|---|---|
0° | 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
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.
Auxiliary latitudes
There 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 latitude
The definitions given in this section all relate to locations on the reference ellipsoid but the first two auxiliary latitudes, like the geodetic latitude, can be extended to define a three-dimensional geographic coordinate system as discussed below. The remaining latitudes are not used in this way; they are used only as intermediate constructs in map projections of the reference ellipsoid to the plane 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 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
where N is the prime vertical radius of curvature. 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)
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 equal-area projection 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]
- When converting from isometric or conformal to geodetic, two iterations of Newton-Raphson gives
- 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] Orihuela[17] gives series for the conversions between all pairs of auxiliary latitudes in terms of the third flattening, n = (a - b)/(a + b). Karney[18] establishes that the truncation errors for such series are consistently smaller that the equivalent series in terms of the eccentricity. 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), θ ≤ χ ≤ μ ≤ ξ ≤ β ≤ ϕ; in the Southern hemisphere (negative latitudes), the inequalities are reversed, with equality at the equator and the poles. Although the graph appears symmetric about 45°, the minima of the curves actually lie between 45° 2′ and 45° 6′. Some representative data points are given in the table below. The conformal and geocentric latitudes are nearly indistinguishable, a fact that was exploited in the days of hand calculators to expedite the construction of map projections.[9]: 108
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 [1⁄2, 2⁄3, 3⁄4, 1, 1] for ζ = [β, ξ, μ, χ, θ].
ϕ | Parametric β − ϕ |
Authalic ξ − ϕ |
Rectifying μ − ϕ |
Conformal χ − ϕ |
Geocentric θ − ϕ |
---|---|---|---|---|---|
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 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
The geocentric latitude θ is the complement of the polar angle or
Ellipsoidal-harmonic coordinates
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[19] 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
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
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 geoid is an idealized, theoretical shape "at mean sea level". Points on land do not lie precisely on the geoid, and the vertical at a point at a specific time is influenced by tidal forces which the theoretical geoid averages out. 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][20]
Astronomical latitude is not to be confused with
See also
- Altitude (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 System
References
Footnotes
- ^ The value of this angle today is 23°26′10.1″ (or 23.43613°). 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
- ^ "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". Philosophiæ 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.
- ^ ISBN 3-11-017072-8.
- ^ . for LaTeX code and figures.
- ^ 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 tracés sur la surface d'un sphéroïde". Mém. Inst. Nat. Fr. 1st semester: 130–161.
- S2CID 118630614.
- .
- S2CID 119310141.
- ^ Lagrange, Joseph-Louis (1779). "Sur la Construction des Cartes Géographiques". Oevres (in French). Vol. IV. p. 667.
- S2CID 118619524.
- ^ Orihuela, Sebastián (2013). "Funciones de Latitud".
- .
- ^ Holfmann-Wellenfor & Moritz (2006) Physical Geodesy, p.240, eq. (6-6) to (6-10).
- ISBN 3-211-33544-7.
External links
- 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. Archived 2012-11-07 at the Wayback Machine – 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