User:Peter Mercator/Draft for Latitude
In geography, Latitude is a geographic coordinatethat specifies the north-south position of a point on the Earth's surface. Latitude ranges from 0° at the Equator to 90° (North or South) at the poles. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. Since the actual physical surface of the Earth is too complex for mathematical analysis two levels of abstraction are employed in the definition of these coordinates. In the first step the physical surface is modelled by the geoid, a surface which approximates themean 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 asphere, but the geoid is more accurately modelled by an ellipsoid. The definitions of latitude and longitude on such reference surfaces are detailed in the following sections. Lines of constant latitude and longitude together constitute a graticule on the reference surface. The latitude of a point on the actual surface is that of the corresponding point on the reference surface, the correspondence being along the normal to the reference surface which passes through the point on the physical surface. Latitude and longitude together with some specification of height constitute a geographic coordinate system as defined in the specification of the ISO 19111 standard.
Since there are many different reference ellipsoids the latitude of a feature on the surface is not unique: this is stressed in the ISO standard which states that "without the full specification of the coordinate reference system, coordinates (that is latitude and longitude) are ambiguous at best and meaningless at worst". This is of great importance in accurate applications, such as GPS, but in common usage, where high accuracy is not required, the reference ellipsoid is not usually stated.
Measurement of latitude requires an understanding of the gravitational field of the Earth, either for setting uptheodolites or for determination of GPSsatellite orbits. The study of the Figure of the Earthtogether with its gravitational field is the science ofGeodesy. These topics are not discussed in this article. (See for example the text books by Torge and Hofmann-Wellenhof and Moritz.)
This article relates to coordinate systems for the Earth: it may be extended to cover the Moon, planets and other celestial objects by a simple change of nomenclature.
The following lists are available:
- 1 Latitude on the sphere
- 2 Latitude on the ellipsoid
- 3 Auxiliary latitudes
- 4 Numerical comparison of auxiliary latitudes
- 5 Latitude and coordinate systems
- 6 Astronomical latitude
- 7 See also
- 8 Footnotes
- 9 External links
Latitude on the sphere
The graticule on the sphere
The graticule, the mesh formed by the lines of constant latitude and constant longitude, is constructed by reference to the rotation axis of the Earth. The primary reference points are thepoles where the axis of rotation of the Earth intersects the reference surface. Planes which contain the rotation axis intersect the surface in the meridians and the angle between any one meridian plane and that through Greenwich (the Prime Meridian) defines the longitude: meridians are lines of constant longitude. The plane through the centre of the Earth and orthogonal to the rotation axis intersects the surface in a great circle called the equator. Planes parallel to the equatorial plane intersect the surface in circles of constant latitude; these are the parallels. The equator has a latitude of 0°, the North pole has a latitude of 90° north (written 90° N or +90°), and the South pole has a latitude of 90° south (written 90° S or −90°). The latitude of an arbitrary point is the angle between the equatorial plane and the radius to that point.
The latitude defined in this way for the sphere is often termed the spherical latitude to avoid ambiguity with auxiliary latitudes defined in subsequent sections.
Besides the equator, four other parallels are of significance:
The plane of the Earth's orbit about the sun is called theecliptic. The plane perpendicular to the rotation axis of the Earth is the equatorial plane. The angle between the ecliptic and the equatorial plane is called the inclination of the ecliptic, denoted by in the figure. The current value of this angle is 23° 26′ 21″. It is also called the axial tilt of the Earth since it is equal to the angle between the axis of rotation and the normal to the ecliptic.
The figure shows the geometry of a cross section of the plane normal to the ecliptic and through the centres of the Earth and the Sun at the December solstice when the sun is overhead at some point of the Tropic of Capricorn. The south polar latitudes below the Antarctic Circle are in daylight whilst the north polar latitudes above the Arctic Circle are in night. The situation is reversed at the June solstice when the sun is overhead at the Tropic of Cancer. The latitudes of the tropics are equal to the inclination of the ecliptic and the polar circles are at latitudes equal to its complement. Only at latitudes in between the two tropics is it possible for thesun to be directly overhead (at the zenith).
The named parallels are clearly indicated on the Mercator projections shown below.
Map projections from the sphere
On map projections there is no simple rule as to how meridians and parallels should appear. For example, on the spherical Mercator projection the parallels are horizontal and the meridians are vertical whereas on the Transverse Mercator projection there is no correlation of parallels and meridians with horizontal and vertical, both are complicated curves. The red lines are the named latitudes of the previous section.
|Normal Mercator||Transverse Mercator|
For map projections of large regions, or the whole world, a spherical Earth model is completely satisfactory since the variations attributable to ellipticity are negligible on the final printed maps.
Meridian distance on the sphere
On the sphere the normal passes through the centre and the latitude (φ) is therefore equal to the angle subtended at the centre by the meridian arc from the equator to the point concerned. If the meridian distance is denoted by m(φ) then
where R denotes the mean radius of the Earth. R is equal to 6371 km or 3959 miles. No higher accuracy is appropriate for R since higher precision results necessitate an ellipsoid model. With this value for R the meridian length of 1 degree of latitude on the sphere is 111.2 km or 69 miles. The length of 1 minute of latitude is 1.853 km, or 1.15 miles. (See nautical mile).
Latitude on the ellipsoid
In 1687 Isaac Newton published the Principia in which he included a proof that a rotating self-gravitating fluid body in equilibrium takes the form of an oblate ellipsoid. (This article uses the term ellipsoid in preference to the older term spheroid). Newton's theoretical result was confirmed by precise geodetic measurements in the eighteenth century. (See Meridian arc). The definition of an oblate ellipsoid is the three dimensional surface generated by the rotation of a two dimensional ellipse about its shorter axis (minor axis). 'Oblate ellipsoid of revolution' is abbreviated to ellipsoid in the remainder of this article: this is the current practice in geodetic literature. (Ellipsoids which do not have an axis of symmetry are termed tri-axial).
A great many different reference ellipsoids have been used in the history of geodesy. In pre-satellite days they were devised to give a good fit to thegeoid over the limited area of a survey but, with the advent of GPS, it has become natural to use reference ellipsoids (such as WGS84) with centres at the centre of mass of the Earth and minor axis aligned to the rotation axis of the Earth. These geocentric ellipsoids are usually within 100m of the geoid. Since latitude is defined with respect to an ellipsoid the position of a given feature is different on each ellipsoid: it is meaningless to specify the latitude and longitude of a geographical feature without specifying the ellipsoid used. The maps maintained by many national agencies are often based on the older ellipsoids so that is necessary to know how the latitude and longitude values may be transformed from one ellipsoid to another. For example GPS handsets must include software to carry out datum transformations which link WGS84 to the local reference ellipsoid with its associated grid reference system.
The geometry of the ellipsoid
The overall shape of an ellipsoid of revolution is determined by the shape of the ellipse which is rotated about its minor (shorter) axis. Two parameters are required. One is invariably chosen to be the equatorial radius, which is thesemi-major axis, a. The other parameter is usually taken as one of three possibilities: (1) the polar radius or semi-minor axis, b; (2) the (first) flattening, f; (3) theeccentricty, e. These parameters are not independent for they are related by
There are great many other parameters (seeellipse,ellipsoid) used in the study of the ellipsoid in geodesy, geophysics and map projections but they can all may be expressed in terms of one or two members of the set a, b, f and e. Both f and e are small parameters which often appear in series expansions involved in practical calculations; they are of the order 1/300 and 0.08 respectively. Accurate values for a number of important ellipsoids are given in Figure of the Earth. It is conventional to define reference ellipsoids by giving the semi-major axis and the inverse flattening, 1/f. For example, the defining values for the WGS84 ellipsoid, used by all GPS devices, are
- a (equatorial radius): 6,378,137.0 m
- 1/f (inverse flattening): 298.257,223,563
from which are derived
- b (polar radius): 6,356,752.3142 m
- e2 (eccentricity squared): 0.006,694,379,990,14
The difference of the major and minor semi-axes is approximately 21 km and as fraction of the semi-major axis it is given by the flattening. Thus the representation of the ellipsoid on a computer could be sized as 300px by 299px. Since this would be indistinguishable from a sphere shown as 300px by 300px illustrations invariably greatly exaggerate the flattening.
Geodetic and geocentric latitudes
The graticule on the ellipsoid is constructed in exactly the same way as on the sphere. The normal at a point on the surface of an ellipsoid does not pass through the centre, except for points on the equator or at the poles, but the definition of latitude remains unchanged as the angle between the normal and the equatorial plane. The terminology for latitude must be made more precise by distinguishing
- Geodetic latitude: the angle between the normal and the equatorial plane. The standard notation in English publications is φ. This is the definition assumed when the word latitude is used without qualification. The definition must be accompanied with a specification of the ellipsoid.
- Geocentric latitude: the angle between the radius (from centre to the point on the surface) and the equatorial plane. (Figure below). There is no standard notation: examples from various texts include ψ, q,φ', φc, φg. This article usesψ.
- Spherical latitude: the angle between the normal to a spherical reference surface and the equatorial plane.
- Geographic latitude must be used with care. Some authors use it as a synonym for geodetic latitude whilst others use it as an alternative to the astronomical latitude.
- Latitude (unqualified) should normally refer to the geodetic latitude.
The importance of specifying the reference datum may be illustrated by a simple example. On the reference ellipsoid for WGS84, the centre of the Eiffel Tower has a geodetic latitude of 48° 51′ 29″ N, or 48.8583° N and longitude of 2° 17′ 40″ E or 2.2944°E. The same coordinates on the datum ED50 define a point on the ground which is 140 m distant from Tower.A web search may produce several different values for the latitude of the Tower; the reference ellipsoid is rarely specified.
The length of a degree of latitude
The function in the first integral is the meridional radius of curvature.
The 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 is normally evaluated by expanding the integral by the binomial series and integrating term by term: see Meridian arc for details. The length of the meridian arc between two given latitudes is given by replacing the limits of the integral by the latitudes concerned. The length of a small meridian arc is given by
and when the latitude difference is 1 degree, corresponding to/180 radians, the arc distance is approximately
Within a centimeter, the distance between latitudes ( deg) and ( deg) on the WGS84 spheroid is
|0°||110.574 km||111.320 km|
|15°||110.649 km||107.551 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|
The variation of this distance with latitude is shown in the table along with the length of a degree of longitude. The latter follows immediately from the expression for the radiusp(φ) of a parallel circle at latitude φ:
The approximate length of a degree of longitude at a latitude is therefore
This is measured along a parallel of latitude; the distance along the shortest route is within 0.6 meter of that. A calculator for any latitude is provided by the U.S. government'sNational Geospatial-Intelligence Agency(NGA).
There are six auxiliary latitudes that have applications to special problems in geodesy, geophysics and the theory of map projections:
- geocentric latitude,
- reduced (or parametric) latitude,
- rectifying latitude,
- authalic latitude,
- conformal latitude,
- isometric latitude.
The definitions given in this section all relate to locations on the reference ellipsoid but the first two auxiliary latitudes, like the geodetic latitude, can be extended to define a three dimensional geographic coordinate system as discussedbelow. The remaining latitudes are not used in this way; they are used only as intermediate constructs in map projections of the reference ellipsoid to the plane: their numerical values are not of interest. For example no one would need to calculate the authalic latitude of the Eiffel Tower.
The expression below give the auxilliary latitudes in terms of the geodetic latitude, the semi-major axis, a, and the eccentricity, e. The forms given are, apart from notational variants, those in the standard reference for map projections, namely "Map projections — a working manual" by J.P.Snyder.Derivations of these expressions may be found in Adams and web publications by Rapp and Osborne
The geocentric latitude is the angle between the the equatorial plane and the radius from the centre to a point on the surface. The relation between the geocentric latitude (ψ) and the geodetic latitude (φ) is derived in the above references as
The geodetic and geocentric latitudes are equal at the equator and poles. The value of the squared eccentricity is approximately 0.007 (depending on the choice of ellipsoid) and the maximum difference of (φ-ψ) is approximately 11.5 minutes of arc at a geodetic latitude of 45°5′.
Reduced (or parametric) latitude
The reduced or parametric latitude, β, is defined by the radius drawn from the centre of the ellipsoid to that point on the surrounding sphere (of radius a) which is the vertical projection of a point on the ellipsoid at latitude. It was introduced by Legendre and Bessel who solved problems for geodesics on the ellipsoid by transforming them to an equivalent problem for spherical geodesics by using this smaller latitude. Bessel's notation, , is also used in the current literature. The reduced latitude is related to the geodetic latitude by
The alternative name arises from the parameterization of the equation of the ellipse describing a meridian section. In terms of Cartesian coordinates 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
Cayleysuggested the term parametric latitude because of the form of these equations.
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 latitudeφis (see Meridian arc)
and the length of the meridian quadrant from the equator to the pole is
Using the rectifying latitude to define a latitude on a sphere of radius
defines a projection from the ellipsoid to the sphere such that all meridians have true length and uniform scale. The sphere may then be projected to the plane with anequirectangular 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). The rectifying latitude is also of great importance in the construction of the Transverse Mercator projection.
The authalic latitude, ξ, gives an area-preserving transformation to a sphere.
and the radius of the sphere is taken as
The conformal latitude, χ, gives an angle-preserving (conformal) transformation to the sphere.
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).
The isometric latitude is conventionally denoted by ψ(not to be confused with the geocentric latitude): it is used in the development of the ellipsoidal versions of the normalMercator 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. Thegraticule 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):
For the normal Mercator projection (on the 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 latitudeφ from the equator is
Numerical comparison of auxiliary latitudes
The following plot shows the magnitude of the difference between the geodetic latitude, (denoted as the 'common' latitude on the plot), and the auxiliary latitudes other than the isometric latitude (which diverges to infinity at the poles). In every case the geodetic latitude is the greater. The differences shown on the plot are in arc minutes. The horizontal resolution of the plot fails to make clear that the maxima of the curves are not at 45° but calculation shows that they are within a few arc minutes of 45°. Some representative data points are given in the table following the plot. Note the closeness of the conformal and geocentric latitudes. This was exploited in the days of hand calculators to expedite the construction of map projections. (Snyder, page 108).
Approximate difference from geodetic latitude ( ) Reduced
0° 0.00′ 0.00′ 0.00′ 0.00′ 0.00′ 15° 2.91′ 3.89′ 4.37′ 5.82′ 5.82′ 30° 5.05′ 6.73′ 7.57′ 10.09′ 10.09′ 45° 5.84′ 7.78′ 8.76′ 11.67′ 11.67′ 60° 5.06′ 6.75′ 7.59′ 10.12′ 10.13′ 75° 2.92′ 3.90′ 4.39′ 5.85′ 5.85′ 90° 0.00′ 0.00′ 0.00′ 0.00′ 0.00′
Latitude and coordinate systems
The geodetic latitude, or any of the auxiliary latitudes defined on the reference ellipsoid, constitutes with longitude a two-dimensional coordinate system on that ellipsoid. To define the position of an arbitrary point it is necessary to extend such a coordinate system into three dimensions. Three latitudes are used in this way: the geodetic, geocentric and reduced latitudes are used in geodetic coordinates, spherical polar coordinates and ellipsoidal coordinates respectively.
At an arbitrary point P consider the line PN which is normal to the reference ellipsoid. The geodetic coordinates P(φ,λ,h) are the latitude and longitude of the point N on the ellipsoid and the distance PN. This height differs from the height above the geoid or a reference height such as that above mean sea level at a specified location. The direction of PN will also differ from the direction of a vertical plumb line. The relation of these different heights requires knowledge of the the shape of the geoid and also the gravity field of the Earth.
Spherical polar coordinates
The geocentric latitude ψ is the complement of the polar angle θ in conventional spherical polar coordinatesin which the coordinates of a point are P(r, θ,λ) where r is the distance of P from the centre O,θ is the angle between the radius vector and the polar axis andλ is longitude. Since the normal at a general point on the ellipsoid does not pass through the centre it is clear that points on the normal, which all have the same geodetic latitude, will have differing geocentic latitudes. Spherical polar coordinate systems are used in the analysis of the gravity field.
The reduced latitude can also be extended to a three dimensional coordinate system. For a point P not on the reference ellipsoid (semi-axes OA and OB) construct an auxiliary ellipsoid which is confocal (same foci F, F') with the reference ellipsoid: the necessary condition is that the product ae of semi-major axis and eccentricity is the same for both ellipsoids. Letu be the semi-minor axis (OD) of the auxiliary ellipsoid. Further let β be the reduced latitude of P on the auxiliary ellipsoid. The set (u,β,λ) define the ellipsoid coordinates. (Torge Section 4.2.2). These coordinates are the natural choice in models of the gravity field for a uniform distribution of mass bounded by the reference ellipsoid.
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 inGeodetic system. The relation of Cartesian and spherical polars is given in Spherical coordinate system. The relation of Cartesian and ellipsoidal coordinates is discussed in Torge.
The astronomical latitude (Φ) is defined as the angle between the equatorial plane and the true vertical at a point on the surface: the true vertical, the direction of a plumb line, is the direction of the gravity field at that point. (The gravity field is the resultant of the gravitational acceleration and the centrifugal acceleration at that point. See Torge.) The astronomic latitude is readily accessible by measuring the angle between the zenith and the celestial pole or alternatively the angle between the zenth and the sun when the elevation of the latter is obtained from the almanac.
In general the direction of the true vertical at a point on the surface does not coincide with the either the normal to the reference ellipsoid or the normal to the geoid. The deflections between the astronomic and geodetic normals are only of the order of a few seconds of arc but they are nevertheless important in high precision geodesy.
Astronomical latitude is not to be confused withdeclination, the coordinate astronomers used in a simlilar way to describe the locations of stars north/south of the celestial equator (see equatorial coordinates), nor with ecliptic latitude, the coordinate that astronomers use to describe the locations of stars north/south of theecliptic (see ecliptic coordinates).
- American Practical Navigator
- Cardinal direction
- Geographic coordinate system
- Geodetic system
- Geographical distance
- Great-circle distance
- Horse latitudes
- List of countries by latitude
- List of cities by latitude
- List of cities by longitude
- Natural Area Code
- World Geodetic System
- Orders of magnitude (length)
- The current full documentation of ISO 19111 may be purchased fromhttp://www.iso.org but drafts of the final standard are freely available at many web sites, one such is available at 
- Torge, W (2001) Geodesy (3rd edition), published by de Gruyter, isbn=3-11-017072-8
- Hofmann-Wellenhof, B and Moritz, H (2006). 'Physical Geodesy (second edition)' ISBN-103211-33544-7.
- The Astronomical Almanac published annually by the National Almanac Office in the United States (http://asa.usno.navy.mil/) and the United Kingdom (http://astro.ukho.gov.uk/nao/publicat/asa.html).
- Isaac Newton:Principia Book III Proposition XIX Problem III, p. 407 in Andrew Motte translation, available on line at 
- WGS84 parameters are listed in the National Geospatial-Intelligence Agency publication TR8350.2 page 3-1.
- Rapp, Richard H. (1991). Geometric Geodesy, Part I, Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio.(Chapter 3)
- Osborne, P (2008)The Mercator Projections (Chapters 5,6)
- of degree calculator - National Geospatial-Intelligence Agency
- Snyder, John P. (1987). Map Projections – A Working Manual. U.S. Geological Survey Professional Paper 1395. United States Government Printing Office, Washington, D.C. This paper can be downloaded fromUSGS pages.
- Adams, Oscar S (1921). Latitude Developments Connected With Geodesy and Cartography, (with tables, including a table for Lambert equal area meridional projection). Special Publication No. 67 of the US Coast and Geodetic Survey. A facsimile of this publication is available from the US National Oceanic and Atmospheric Administration (NOAA) athttp://docs.lib.noaa.gov/rescue/cgs_specpubs/QB275U35no671921.pdfWarning: Adams uses the nomenclature isometric latitude for the conformal latitude of this article.
- A. M. Legendre, 1806, Analyse des triangles tracés sur la surface d'un sphéroïde, Mém. de l'Inst. Nat. de France, 130--161 (1st semester).
- F. W. Bessel, 1825, Uber die Berechnung der geographischen Langen und Breiten aus geodatischen Vermessungen, Astron.Nachr., 4(86), 241-254, doi:10.1002/asna.201011352, translated into English by C. F. F. Karney and R. E. Deakin as The calculation of longitude and latitude from geodesic measurements, Astron. Nachr. 331(8), 852-861 (2010), E-print , http://adsabs.harvard.edu/abs/1825AN......4..241B.
- A. Cayley, 1870, On the geodesic lines on an oblate spheroid, Phil. Mag. 40 (4th ser.), 329-340.
- C. F. F. Karney, 2011, Geodesics on an ellipsoid of revolution, Technical report, E-printarXiv:1102.1215.
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