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### Latitude

2. 2. by an ellipsoid. The definitions of latitude and longitude on such referencesurfaces are detailed in the following sections. Lines of constant latitudeand longitude together constitute a graticule on the reference surface. Thelatitude of a point on the actual surface is that of the corresponding pointon the reference surface, the correspondence being along the normal tothe reference surface which passes through the point on the physicalsurface. Latitude and longitude together with some specification of heightconstitute a geographic coordinate system as defined in the specificationof the ISO 19111 standard.[1]Since there are many different reference ellipsoids the latitude of a featureon the surface is not unique: this is stressed in the ISO standard whichstates that "without the full specification of the coordinate referencesystem, coordinates (that is latitude and longitude) are ambiguous at bestand meaningless at worst". This is of great importance in accurateapplications, such as GPS, but in common usage, where high accuracy isnot required, the reference ellipsoid is not usually stated.In English texts the latitude angle, defined below, is usually denoted by theGreek lower-case letter phi (φ or ɸ ). It is measured in degrees, minutesand seconds or decimal degrees, north or south of the equator.Measurement of latitude requires an understanding of the gravitational fieldof the Earth, either for setting up theodolites or for determination of GPSsatellite orbits. The study of the figure of the Earth together with itsgravitational field is the science of Geodesy. These topics are notdiscussed in this article. (See for example the textbooks by Torge[2] andHofmann-Wellenhof and Moritz.[3])This article relates to coordinate systems for the Earth: it may be extendedto cover the Moon, planets and other celestial objects by a simple changeof nomenclature.The following lists are available:• List of countries by latitude• List of cities by latitude Contents [hide] 1 Latitude on the sphere 1.1 The graticule on the sphere 1.2 Named latitudes 1.3 Map projections from the sphere 1.4 Meridian distance on the sphere 2 Latitude on the ellipsoid
3. 3. 2.1 Ellipsoids 2.2 The geometry of the ellipsoid 2.3 Geodetic and geocentric latitudes 2.4 The length of a degree of latitude 3 Auxiliary latitudes 3.1 Geocentric latitude 3.2 Reduced (or parametric) latitude 3.3 Rectifying latitude 3.4 Authalic latitude 3.5 Conformal latitude 3.6 Isometric latitude 4 Numerical comparison of auxiliary latitudes 5 Latitude and coordinate systems 5.1 Geodetic coordinates 5.2 Spherical polar coordinates 5.3 Ellipsoidal coordinates 5.4 Coordinate conversions 6 Astronomical latitude 7 See also 8 Footnotes 9 External linksLatitude on the sphere
4. 4. A perspective view of the Earth showing how latitude (φ) and longitude (λ) are definedon a spherical model. The graticule spacing is 10 degrees.The graticule on the sphereThe graticule, the mesh formed by the lines of constant latitude andconstant longitude, is constructed by reference to the rotation axis of theEarth. The primary reference points are the poles where the axis ofrotation of the Earth intersects the reference surface. Planes which containthe rotation axis intersect the surface in the meridians and the anglebetween any one meridian plane and that through Greenwich (the PrimeMeridian) defines the longitude: meridians are lines of constant longitude.The plane through the centre of the Earth and orthogonal to the rotationaxis intersects the surface in a great circle called the equator. Planesparallel to the equatorial plane intersect the surface in circles of constantlatitude; these are the parallels. The equator has a latitude of 0°, the Northpole has a latitude of 90° north (written 90° N or +90°), and the South polehas a latitude of 90° south (written 90° S or −90°). The latitude of anarbitrary point is the angle between the equatorial plane and the radius tothat point.The latitude that is defined in this way for the sphere is often termed thespherical latitude to avoid ambiguity with auxiliary latitudes defined insubsequent sections.Named latitudes
5. 5. The orientation of the Earth at the December solstice.Besides the equator, four other parallels are of significance: 66° 33′ 39″Arctic Circle N 23° 26′ 21″Tropic of Cancer NTropic of 23° 26′ 21″Capricorn S 66° 33′ 39"Antarctic Circle SThe plane of the Earths orbit about the sun is called the ecliptic. The planeperpendicular to the rotation axis of the Earth is the equatorial plane. Theangle between the ecliptic and the equatorial plane is called the inclinationof the ecliptic, denoted by in the figure. The current value of this angle is23° 26′ 21″.[4] It is also called the axial tilt of the Earth since it is equal tothe 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 tothe ecliptic and through the centres of the Earth and the Sun at theDecember solstice when the sun is overhead at some point of the Tropic ofCapricorn. The south polar latitudes below the Antarctic Circle are indaylight 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 atthe Tropic of Cancer. The latitudes of the tropics are equal to theinclination of the ecliptic and the polar circles are at latitudes equal to itscomplement. Only at latitudes in between the two tropics is it possible forthe sun to be directly overhead (at the zenith).The named parallels are clearly indicated on the Mercator projections
6. 6. shown below.Map projections from the sphereOn map projections there is no simple rule as to how meridians andparallels should appear. For example, on the spherical Mercator projectionthe parallels are horizontal and the meridians are vertical whereas on theTransverse Mercator projection there is no correlation of parallels andmeridians with horizontal and vertical, both are complicated curves. Thered lines are the named latitudes of the previous section. Normal Mercator Transverse MercatorFor map projections of large regions, or the whole world, a spherical Earthmodel is completely satisfactory since the variations attributable toellipticity are negligible on the final printed maps.Meridian distance on the sphereOn the sphere the normal passes through the centre and the latitude (φ) istherefore equal to the angle subtended at the centre by the meridian arcfrom the equator to the point concerned. If the meridian distance isdenoted by m(φ) thenwhere R denotes the mean radius of the Earth. R is equal to 6371 km or3959 miles. No higher accuracy is appropriate for R since higher precisionresults necessitate an ellipsoid model. With this value for R the meridianlength of 1 degree of latitude on the sphere is 111.2 km or 69 miles. The
7. 7. length of 1 minute of latitude is 1.853 km, or 1.15 miles. (See nauticalmile).Latitude on the ellipsoidEllipsoidsIn 1687 Isaac Newton published the Principia in which he included a proofthat a rotating self-gravitating fluid body in equilibrium takes the form of anoblate ellipsoid.[5] (This article uses the term ellipsoid in preference to theolder term spheroid). Newtons theoretical result was confirmed by precisegeodetic measurements in the eighteenth century. (See Meridian arc). Thedefinition of an oblate ellipsoid is the three dimensional surface generatedby the rotation of a two dimensional ellipse about its shorter axis (minoraxis). Oblate ellipsoid of revolution is abbreviated to ellipsoid in theremainder 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 ofgeodesy. 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 hasbecome natural to use reference ellipsoids (such as WGS84) with centresat the centre of mass of the Earth and minor axis aligned to the rotationaxis of the Earth. These geocentric ellipsoids are usually within 100m ofthe geoid. Since latitude is defined with respect to an ellipsoid the positionof a given feature is different on each ellipsoid: it is meaningless to specifythe latitude and longitude of a geographical feature without specifying theellipsoid used. The maps maintained by many national agencies are oftenbased on the older ellipsoids so that is necessary to know how the latitudeand longitude values may be transformed from one ellipsoid to another.For example GPS handsets must include software to carry out datumtransformations which link WGS84 to the local reference ellipsoid with itsassociated grid reference system.The geometry of the ellipsoidThe overall shape of an ellipsoid of revolution is determined by the shapeof the ellipse which is rotated about its minor (shorter) axis. Twoparameters are required. One is invariably chosen to be the equatorialradius, which is the semi-major axis, a. The other parameter is usuallytaken as one of three possibilities: (1) the polar radius or semi-minor axis,b; (2) the (first) flattening, f; (3) the eccentricty, e. These parameters are
8. 8. not independent: they are related byA great many other parameters (see ellipse, ellipsoid) are used in thestudy of the ellipsoid in geodesy, geophysics and map projections but theycan all may be expressed in terms of one or two members of the set a, b, fand e. Both f and e are small parameters which often appear in seriesexpansions involved in practical calculations; they are of the order 1/300and 0.08 respectively. Accurate values for a number of important ellipsoidsare given in Figure of the Earth. It is conventional to define referenceellipsoids by giving the semi-major axis and the inverse flattening, 1/f. Forexample, the defining values for the WGS84 ellipsoid, used by all GPSdevices, are[6]• a (equatorial radius): 6,378,137.0 m• 1/f (inverse flattening): 298.257,223,563from which are derived• b (polar radius): 6,356,752.3142 m• e2 (eccentricity squared): 0.006,694,379,990,14The difference of the major and minor semi-axes is approximately 21 kmand as fraction of the semi-major axis it is given by the flattening. Thus therepresentation of the ellipsoid on a computer could be sized as 300px by299px. Since this would be indistinguishable from a sphere shown as300px by 300px illustrations invariably greatly exaggerate the flattening.Geodetic and geocentric latitudesThe definition of geodetic latitude (φ) and longitude (λ) on an ellipsoid. The normal to thesurface does not pass through the centre.
9. 9. The graticule on the ellipsoid is constructed in exactly the same way as onthe sphere. The normal at a point on the surface of an ellipsoid does notpass through the centre, except for points on the equator or at the poles,but the definition of latitude remains unchanged as the angle between thenormal and the equatorial plane. The terminology for latitude must bemade more precise by distinguishingGeodetic latitude: the angle between the normal and the equatorial plane.The standard notation in English publications is φ. This is the definitionassumed when the word latitude is used without qualification. Thedefinition must be accompanied with a specification of the ellipsoid.Geocentric latitude: the angle between the radius (from centre to thepoint on the surface) and the equatorial plane. (Figure below). There is nostandard notation: examples from various texts include ψ, q, φ, φc, φg. Thisarticle uses ψ.Spherical latitude: the angle between the normal to a spherical referencesurface and the equatorial plane.Geographic latitude must be used with care. Some authors use it as asynonym for geodetic latitude whilst others use it as an alternative to theastronomical latitude.Latitude (unqualified) should normally refer to the geodetic latitude.The importance of specifying the reference datum may be illustrated by asimple example. On the reference ellipsoid for WGS84, the centre of theEiffel Tower has a geodetic latitude of 48° 51′ 29″ N, or 48.8583° N andlongitude of 2° 17′ 40″ E or 2.2944°E. The same coordinates on the datumED50 define a point on the ground which is 140 m distant fromTower.[citation needed] A web search may produce several different valuesfor the latitude of the Tower; the reference ellipsoid is rarely specified.The length of a degree of latitudeIn Meridian arc and standard texts[2][7][8] it is shown that the distance alonga meridian from latitude φ to the equator is given by (φ in radians)The function in the first integral is the meridional radius ofcurvature.The distance from the equator to the pole isFor WGS84 this distance is 10001.965729 km.
10. 10. The evaluation of the meridian distance integral is central to many studiesin geodesy and map projection. It can be evaluated by expanding theintegral by the binomial series and integrating term by term: see Meridianarc for details. The length of the meridian arc between two given latitudesis given by replacing the limits of the integral by the latitudes concerned.The length of a small meridian arc is roughly given by[7][8]0 110.574 k 111.320 k ° m m1 110.649 k 107.551 k5 m m °3 110.852 k 96.486 k0 m m °4 111.132 k 78.847 k5 m m °6 111.412 k 55.800 k0 m m °7 111.618 k 28.902 k5 m m °9 111.694 k0 0.000 km m °When the latitude difference is 1 degree, corresponding to /180 radians,the arc distance is aboutThe distance in metres (correct to 0.01 metre) between latitudes ( deg) and ( deg) on the WGS84 spheroid isThe variation of this distance with latitude (on WGS84) is shown in thetable along with the length of a degree of longitude:
11. 11. A calculator for any latitude is provided by the U.S. governments NationalGeospatial-Intelligence Agency(NGA).[9]Auxiliary latitudesThere are six auxiliary latitudes that have applications to specialproblems 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 referenceellipsoid but the first two auxiliary latitudes, like the geodetic latitude, canbe extended to define a three dimensional geographic coordinate systemas discussed below. The remaining latitudes are not used in this way; theyare used only as intermediate constructs in map projections of thereference ellipsoid to the plane: their numerical values are not of interest.For example no one would need to calculate the authalic latitude of theEiffel Tower.The expressions below give the auxiliary latitudes in terms of the geodeticlatitude, the semi-major axis, a, and the eccentricity, e. The forms givenare, apart from notational variants, those in the standard reference for mapprojections, namely "Map projections — a working manual" byJ.P.Snyder.[10] Derivations of these expressions may be found inAdams[11] and web publications by Rapp[7] and Osborne[8]Geocentric latitude
12. 12. The definition of geodetic (or geographic) and geocentric latitudes.The geocentric latitude is the angle between the equatorial plane and theradius from the centre to a point on the surface. The relation between thegeocentric latitude (ψ) and the geodetic latitude (φ) is derived in the abovereferences asThe geodetic and geocentric latitudes are equal at the equator and poles.The value of the squared eccentricity is approximately 0.007 (dependingon the choice of ellipsoid) and the maximum difference of (φ-ψ) isapproximately 11.5 minutes of arc at a geodetic latitude of 45°5′.Reduced (or parametric) latitudeDefinition of the reduced latitude (β) on the ellipsoid.The reduced or parametric latitude, β, is defined by the radius drawnfrom the centre of the ellipsoid to that point Q on the surrounding sphere
13. 13. (of radius a) which is the projection parallel to the Earths axis of a point Pon the ellipsoid at latitude . It was introduced by Legendre[12] andBessel[13] who solved problems for geodesics on the ellipsoid bytransforming them to an equivalent problem for spherical geodesics byusing this smaller latitude. Bessels notation, , is also used in thecurrent literature. The reduced latitude is related to the geodetic latitudeby[7][8]The alternative name arises from the parameterization of the equation ofthe ellipse describing a meridian section. In terms of Cartesian coordinatesp, the distance from the minor axis, and z, the distance above theequatorial plane, the equation of the ellipse isThe Cartesian coordinates of the point are parameterized byCayley[14] suggested the term parametric latitude because of the form ofthese equations.The reduced latitude is not used in the theory of map projections. Its mostimportant application is in the theory of ellipsoid geodesics. (Vincenty,Karney[15]).Rectifying latitudeThe rectifying latitude, μ, is the meridian distance scaled so that its valueat the poles is equal to 90 degrees or π/2 radians:where the meridian distance from the equator to a latitude φ is (seeMeridian arc)and the length of the meridian quadrant from the equator to the pole isUsing the rectifying latitude to define a latitude on a sphere of radius
14. 14. defines a projection from the ellipsoid to the sphere such that all meridianshave true length and uniform scale. The sphere may then be projected tothe plane with an equirectangular projection to give a double projectionfrom the ellipsoid to the plane such that all meridians have true length anduniform meridian scale. An example of the use of the rectifying latitude isthe Equidistant conic projection. (Snyder,[10] Section 16). The rectifyinglatitude is also of great importance in the construction of the TransverseMercator projection.Authalic latitudeThe authalic (Greek for same area) latitude, ξ, gives an area-preservingtransformation to a sphere.whereandand the radius of the sphere is taken asAn example of the use of the authalic latitude is the Albers equal-areaconic projection. (Snyder,[10] Section 14).Conformal latitudeThe conformal latitude, χ, gives an angle-preserving (conformal)transformation to the sphere.
15. 15. .The conformal latitude defines a transformation from the ellipsoid to asphere of arbitrary radius such that the angle of intersection between anytwo lines on the ellipsoid is the same as the corresponding angle on thesphere (so that the shape of small elements is well preserved). A furtherconformal transformation from the sphere to the plane gives a conformaldouble projection from the ellipsoid to the plane. This is not the only way ofgenerating such a conformal projection. For example, the exact version ofthe Transverse Mercator projection on the ellipsoid is not a doubleprojection. (It does, however, involve a generalisation of the conformallatitude to the complex plane).Isometric latitudeThe isometric latitude is conventionally denoted by ψ (not to be confusedwith the geocentric latitude): it is used in the development of the ellipsoidalversions of the normal Mercator projection and the Transverse Mercatorprojection. The name "isometric" arises from the fact that at any point onthe ellipsoid equal increments of ψ and longitude λ give rise to equaldistance displacements along the meridians and parallels respectively. Thegraticule defined by the lines of constant ψ and constant λ, divides thesurface of the ellipsoid into a mesh of squares (of varying size). Theisometric latitude is zero at the equator but rapidly diverges from thegeodetic latitude, tending to infinity at the poles. The conventional notationis given in Snyder[10] (page 15):For the normal Mercator projection (on the ellipsoid) this function definesthe spacing of the parallels: if the length of the equator on the projection isE (units of length or pixels) then the distance, y, of a parallel of latitude φ
16. 16. from the equator isNumerical comparison of auxiliary latitudesThe following plot shows the magnitude of the difference between thegeodetic latitude, (denoted as the common latitude on the plot), and theauxiliary latitudes other than the isometric latitude (which diverges toinfinity at the poles). In every case the geodetic latitude is the greater. Thedifferences shown on the plot are in arc minutes. The horizontal resolutionof 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°. Somerepresentative data points are given in the table following the plot. Note thecloseness of the conformal and geocentric latitudes. This was exploited inthe days of hand calculators to expedite the construction of mapprojections. (Snyder,[10] page 108). Approximate difference from geodetic latitude ( ) Reduc Autha Rectifyi Conform Geocentric ed lic ng al
17. 17. 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 systemsThe geodetic latitude, or any of the auxiliary latitudes defined on thereference ellipsoid, constitutes with longitude a two-dimensional coordinatesystem on that ellipsoid. To define the position of an arbitrary point it isnecessary to extend such a coordinate system into three dimensions.Three latitudes are used in this way: the geodetic, geocentric and reducedlatitudes are used in geodetic coordinates, spherical polar coordinates andellipsoidal coordinates respectively.Geodetic coordinates
18. 18. Geodetic coordinates P(ɸ ,λ,h)At an arbitrary point P consider the line PN which is normal to thereference ellipsoid. The geodetic coordinates P(ɸ ,λ,h) are the latitude andlongitude of the point N on the ellipsoid and the distance PN. This heightdiffers from the height above the geoid or a reference height such as thatabove mean sea level at a specified location. The direction of PN will alsodiffer from the direction of a vertical plumb line. The relation of thesedifferent heights requires knowledge of the shape of the geoid and also thegravity field of the Earth.Spherical polar coordinatesGeocentric coordinate related to spherical polar coordinates P(r, θ, λ)The geocentric latitude ψ is the complement of the polar angle θ inconventional spherical polar coordinates in which the coordinates of apoint are P(r, θ, λ) where r is the distance of P from the centre O, θ is theangle between the radius vector and the polar axis andλ is longitude. Since
19. 19. the normal at a general point on the ellipsoid does not pass through thecentre it is clear that points on the normal, which all have the samegeodetic latitude, will have differing geocentic latitudes. Spherical polarcoordinate systems are used in the analysis of the gravity field.Ellipsoidal coordinatesEllipsoidal coordinates P(u,β,λ)The reduced latitude can also be extended to a three dimensionalcoordinate system. For a point P not on the reference ellipsoid (semi-axesOA and OB) construct an auxiliary ellipsoid which is confocal (same foci F,F) with the reference ellipsoid: the necessary condition is that the productae of semi-major axis and eccentricity is the same for both ellipsoids. Let ube the semi-minor axis (OD) of the auxiliary ellipsoid. Further let β be thereduced latitude of P on the auxiliary ellipsoid. The set (u,β,λ) define theellipsoid coordinates. (Torge[2] Section 4.2.2). These coordinates are thenatural choice in models of the gravity field for a uniform distribution ofmass bounded by the reference ellipsoid.Coordinate conversionsThe relations between the above coordinate systems, and also Cartesiancoordinates are not presented here. The transformation between geodeticand Cartesian coordinates may be found in Geodetic system. The relationof Cartesian and spherical polars is given in Spherical coordinate system.The relation of Cartesian and ellipsoidal coordinates is discussed inTorge.[2]
20. 20. Astronomical latitudeThe astronomical latitude (Φ) is defined as the angle between theequatorial plane and the true vertical at a point on the surface: the truevertical, the direction of a plumb line, is the direction of the gravity field atthat point. (The gravity field is the resultant of the gravitational accelerationand the centrifugal acceleration at that point. See Torge.[2]) Theastronomic latitude is readily accessible by measuring the angle betweenthe zenith and the celestial pole or alternatively the angle between thezenith and the sun when the elevation of the latter is obtained from thealmanac.In general the direction of the true vertical at a point on the surface doesnot coincide with the either the normal to the reference ellipsoid or thenormal to the geoid. The deflections between the astronomic and geodeticnormals are only of the order of a few seconds of arc but they arenevertheless important in high precision geodesy.[2][3]Astronomical latitude is not to be confused with declination, the coordinateastronomers used in a simlilar way to describe the locations of starsnorth/south of the celestial equator (see equatorial coordinates), nor withecliptic latitude, the coordinate that astronomers use to describe thelocations of stars north/south of the ecliptic (see ecliptic coordinates).