Latitude

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

Division of the Earth into Gauss-Krueger zones - Globe
A graticule on the Earth as a sphere or an ellipsoid. The 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.

Preliminaries

Two levels of abstraction are employed in the definition of latitude and longitude. In the first step the physical surface is modeled by the geoid, a surface which approximates the mean sea level over the oceans and its continuation under the land masses. The second step is to approximate the geoid by a mathematically simpler reference surface. The simplest choice for the reference surface is a sphere, but the geoid is more accurately modeled by an ellipsoid. The definitions of latitude and longitude on such reference surfaces are detailed in the following sections. Lines of constant latitude and longitude together constitute a graticule on the reference surface. The latitude of a point on the actual surface is that of the corresponding point on the reference surface, the correspondence being along the normal to the reference surface which passes through the point on the physical surface. Latitude and longitude together with some specification of height constitute a geographic coordinate system as defined in the specification of the ISO 19111 standard.[a]

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.

The precise measurement of latitude requires an understanding of the gravitational field of the Earth, either to set up theodolites or to determine GPS satellite orbits. The study of the figure of the Earth together with its gravitational field is the science of geodesy.

This article relates to coordinate systems for the Earth: it may be extended to cover the Moon, planets and other celestial objects by a simple change of nomenclature.

Latitude on the sphere

Latitude and longitude graticule on a 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 radius vector.

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

Named latitudes on the Earth

December solstice geometry
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.[b]

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
MercNormSph enhanced

\

MercTranSph enhanced

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).

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.[1] (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

Ellipsoid parametric euler mono
A sphere of radius a compressed to 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 1/300 and 0.08 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[2]

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

from which are derived

  • b (polar radius): 6356752.3142 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

Latitude and longitude graticule on an ellipsoid
The definition of geodetic latitude (φ) and longitude (λ) 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: 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.
  • 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 metres (460 feet) distant from the tower. A web search may produce several different values for the latitude of the tower; the reference ellipsoid is rarely specified.

Length of a degree of latitude

In Meridian arc and standard texts[3][4][5] 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 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[4][5]

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).[6]

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

WGS84 angle to distance conversion
The definition of geodetic latitude (φ) and geocentric latitude (θ).

The nautical mile

Historically a nautical mile was defined as the length of one minute of arc along a meridian of a spherical earth. An ellipsoid model leads to a variation of the nautical mile with latitude. This was resolved by defining the nautical mile to be exactly 1,852 metres. However for all practical purposes distances are measured from the latitude scale of charts. As the Royal Yachting Association says in its manual for day skippers: "1 (minute) of Latitude = 1 sea mile", followed by "For most practical purposes distance is measured from the latitude scale, assuming that one minute of latitude equals one nautical mile".[7]

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.[8] Derivations of these expressions may be found in Adams[9] and online publications by Osborne[4] and Rapp.[5]

Geocentric latitude

Geocentric coords 03
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 on the surface. The relation between the geocentric latitude (θ) and the geodetic latitude (φ) is derived in the above references as

The geodetic and geocentric latitudes are equal at the equator and at the poles but at other latitudes they differ by a few minutes of arc. Taking the value of the squared eccentricity as 0.0067 (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′.[c]

Parametric (or reduced) latitude

Ellipsoid reduced angle definition
Definition of the parametric latitude (β) on the ellipsoid.

The parametric 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[10] and Bessel[11] 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:[4][5]

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.[12]

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[13]).

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).[8] The rectifying latitude is also of great importance in the construction of the Transverse Mercator projection.

Authalic latitude

The authalic (Greek for same area) latitude, 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.[8]:§14

Conformal latitude

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

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):[8]

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. 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.[9] Osborne[4] derives series to arbitrary order by using the computer algebra package Maxima[14] and expresses the coefficients in terms of both eccentricity and flattening. The series method is not applicable to the isometric latitude and one must use the conformal latitude in an intermediate step.

Numerical comparison of auxiliary latitudes

The following plot 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. In every case the auxiliary latitude is the less (in magnitude) than the geodetic latitude. 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.[8]: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].

Types of latitude difference
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
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 coords 02
Geocentric coordinate related to spherical polar coordinates P(r,θ′,λ)

The geocentric latitude θ is the complement of the polar angle 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 coordinates

Ellipsoidal 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 ellipsoid coordinates.[3]:§4.2.2 These coordinates are the natural choice in models of the gravity field for a rotating ellipsoidal body.

Coordinate conversions

The relations between the above coordinate systems, and also Cartesian coordinates are not presented here. The transformation between geodetic and Cartesian coordinates may be found in Geographic coordinate conversion. The relation of Cartesian and spherical polars is given in Spherical coordinate system. The relation of Cartesian and ellipsoidal coordinates is discussed in Torge.[3]

Astronomical latitude

Astronomical latitude (Φ) is the angle between the equatorial plane and the true vertical at a point on the surface. The true vertical, the direction of a plumb line, is also the direction of the gravity acceleration, the resultant of the gravitational acceleration (mass-based) and the centrifugal acceleration at that latitude.[3] Astronomic latitude is calculated from angles measured between the zenith and stars whose declination is accurately known.

In general the true vertical at a point on the surface does not exactly coincide with either the normal to the reference ellipsoid or the normal to the geoid. The angle between the astronomic and geodetic normals is usually a few seconds of arc but it is important in geodesy.[3][15] 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 current full documentation of ISO 19111 may be purchased from http://www.iso.org but drafts of the final standard are freely available at many web sites, one such is available at the following CSIRO
  2. ^ The value of this angle today is 23°26′12.4″ (or 23.43678°). This figure is provided by Template:Circle of latitude.
  3. ^ An elementary calculation involves differentiation to find the maximum difference of the geodetic and geocentric latitudes.

Citations

  1. ^ Newton, Isaac. "Book III Proposition XIX Problem III". Philosophiæ Naturalis Principia Mathematica. Translated by Motte, Andrew. p. 407.
  2. ^ "TR8350.2". National Geospatial-Intelligence Agency publication. p. 3-1.
  3. ^ a b c d e Torge, W. (2001). Geodesy (3rd ed.). De Gruyter. ISBN 3-11-017072-8.
  4. ^ a b c d e Osborne, Peter (2013). "Chapters 5,6". The Mercator Projections. doi:10.5281/zenodo.35392. for LaTeX code and figures.
  5. ^ 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.
  6. ^ "Length of degree calculator". National Geospatial-Intelligence Agency.
  7. ^ Hopkinson, Sara (2012). RYA day skipper handbook - sail. Hamble: The Royal Yachting Association. p. 76. ISBN 9781-9051-04949.
  8. ^ 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.
  9. ^ 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).)
  10. ^ Legendre, A. M. (1806). "Analyse des triangles tracés sur la surface d'un sphéroïde". Mém. Inst. Nat. Fr. 1st semester: 130–161.
  11. ^ 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.
    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.
  12. ^ Cayley, A. (1870). "On the geodesic lines on an oblate spheroid". Phil. Mag. 40 (4th ser.): 329–340.
  13. ^ Karney, C. F. F. (2013). "Algorithms for geodesics". J. Geodesy. 87 (1): 43–55. arXiv:1109.4448. Bibcode:2013JGeod..87...43K. doi:10.1007/s00190-012-0578-z.
  14. ^ "Maxima computer algebra system". Sourceforge.
  15. ^ Hofmann-Wellenhof, B.; Moritz, H. (2006). Physical Geodesy (2nd ed.). ISBN 3-211-33544-7.

External links

30th parallel north

The 30th parallel north is a circle of latitude that is 30 degrees north of the Earth's equatorial plane. It stands one-third of the way between the equator and the North Pole and crosses Africa, Asia, the Pacific Ocean, North America and the Atlantic Ocean.

It is the approximate southern border of the horse latitudes in the Northern Hemisphere, meaning that much of the land area touching the 30th parallel is arid or semi-arid. If there is a source of wind from a body of water the area would more likely be subtropical.

At this latitude the sun is visible for 14 hours, 5 minutes during the summer solstice and 10 hours, 13 minutes during the winter solstice. On 21 June, the maximum altitude of the sun is 83.83 degrees and 36.17 degrees on 21 December.At this latitude:

One degree of longitude = 96.49 km or 59.95 mi

One minute of longitude = 1.61 km or 1.00 mi

One second of longitude = 26.80 m or 87.93 ft

55th parallel north

The 55th parallel north is a circle of latitude that is 55 degrees north of the Earth's equatorial plane. It crosses Europe, Asia, the Pacific Ocean, North America, and the Atlantic Ocean.

At this latitude the sun is visible for 17 hours, 22 minutes during the summer solstice and 7 hours, 10 minutes during the winter solstice.

Antarctica

Antarctica (UK: or , US: (listen)) is Earth's southernmost continent. It contains the geographic South Pole and is situated in the Antarctic region of the Southern Hemisphere, almost entirely south of the Antarctic Circle, and is surrounded by the Southern Ocean. At 14,200,000 square kilometres (5,500,000 square miles), it is the fifth-largest continent. For comparison, Antarctica is nearly twice the size of Australia. About 98% of Antarctica is covered by ice that averages 1.9 km (1.2 mi; 6,200 ft) in thickness, which extends to all but the northernmost reaches of the Antarctic Peninsula.

Antarctica, on average, is the coldest, driest, and windiest continent, and has the highest average elevation of all the continents. Most of Antarctica is a polar desert, with annual precipitation of only 200 mm (8 in) along the coast and far less inland. The temperature in Antarctica has reached −89.2 °C (−128.6 °F) (or even −94.7 °C (−135.8 °F) as measured from space), though the average for the third quarter (the coldest part of the year) is −63 °C (−81 °F). Anywhere from 1,000 to 5,000 people reside throughout the year at research stations scattered across the continent. Organisms native to Antarctica include many types of algae, bacteria, fungi, plants, protista, and certain animals, such as mites, nematodes, penguins, seals and tardigrades. Vegetation, where it occurs, is tundra.

Antarctica is noted as the last region on Earth in recorded history to be discovered, unseen until 1820 when the Russian expedition of Fabian Gottlieb von Bellingshausen and Mikhail Lazarev on Vostok and Mirny sighted the Fimbul ice shelf. The continent, however, remained largely neglected for the rest of the 19th century because of its hostile environment, lack of easily accessible resources, and isolation. In 1895, the first confirmed landing was conducted by a team of Norwegians.

Antarctica is a de facto condominium, governed by parties to the Antarctic Treaty System that have consulting status. Twelve countries signed the Antarctic Treaty in 1959, and thirty-eight have signed it since then. The treaty prohibits military activities and mineral mining, prohibits nuclear explosions and nuclear waste disposal, supports scientific research, and protects the continent's ecozone. Ongoing experiments are conducted by more than 4,000 scientists from many nations.

Arctic Circle

The Arctic Circle is one of the two polar circles and the most northerly of the five major circles of latitude as shown on maps of Earth. It marks the northernmost point at which the centre of the noon sun is just visible on the December solstice and the southernmost point at which the centre of the midnight sun is just visible on the June solstice. The region north of this circle is known as the Arctic, and the zone just to the south is called the Northern Temperate Zone.

As seen from the Arctic, the Sun is above the horizon for 24 continuous hours at least once per year (and therefore visible at midnight) and below the horizon for 24 continuous hours at least once per year (and therefore not visible at noon). This is also true in the Antarctic region, south of the equivalent Antarctic Circle.

The position of the Arctic Circle is not fixed; as of 23 April 2019, it runs 66°33′47.6″ north of the Equator. Its latitude depends on the Earth's axial tilt, which fluctuates within a margin of more than 2° over a 41,000-year period, due to tidal forces resulting from the orbit of the Moon. Consequently, the Arctic Circle is currently drifting northwards at a speed of about 15 m (49 ft) per year.

Arctic Ocean

The Arctic Ocean is the smallest and shallowest of the world's five major oceans. The International Hydrographic Organization (IHO) recognizes it as an ocean, although some oceanographers call it the Arctic Mediterranean Sea or simply the Arctic Sea, classifying it a mediterranean sea or an estuary of the Atlantic Ocean. It is also seen as the northernmost part of the all-encompassing World Ocean.

Located mostly in the Arctic north polar region in the middle of the Northern Hemisphere, the Arctic Ocean is almost completely surrounded by Eurasia and North America. It is partly covered by sea ice throughout the year and almost completely in winter. The Arctic Ocean's surface temperature and salinity vary seasonally as the ice cover melts and freezes; its salinity is the lowest on average of the five major oceans, due to low evaporation, heavy fresh water inflow from rivers and streams, and limited connection and outflow to surrounding oceanic waters with higher salinities. The summer shrinking of the ice has been quoted at 50%. The US National Snow and Ice Data Center (NSIDC) uses satellite data to provide a daily record of Arctic sea ice cover and the rate of melting compared to an average period and specific past years.

Circle of latitude

A circle of latitude on Earth is an abstract east–west circle connecting all locations around Earth (ignoring elevation) at a given latitude.

Circles of latitude are often called parallels because they are parallel to each other; that is, any two circles are always the same distance apart. A location's position along a circle of latitude is given by its longitude. Circles of latitude are unlike circles of longitude, which are all great circles with the centre of Earth in the middle, as the circles of latitude get smaller as the distance from the Equator increases. Their length can be calculated by a common sine or cosine function. The 60th parallel north or south is half as long as the Equator (disregarding Earth's minor flattening by 0.3%). A circle of latitude is perpendicular to all meridians.

The latitude of the circle is approximately the angle between the Equator and the circle, with the angle's vertex at Earth's centre. The equator is at 0°, and the North Pole and South Pole are at 90° north and 90° south, respectively. The Equator is the longest circle of latitude and is the only circle of latitude which also is a great circle.

There are 89 integral (whole degree) circles of latitude between the equator and the Poles in each hemisphere, but these can be divided into more precise measurements of latitude, and are often represented as a decimal degree (e.g. 34.637°N) or with minutes and seconds (e.g. 22°14'26"S). There is no limit to how precisely latitude can be measured, and so there are an infinite number of circles of latitude on Earth.

On a map, the circles of latitude may or may not be parallel, and their spacing may vary, depending on which projection is used to map the surface of the Earth onto a plane. On an equirectangular projection, centered on the equator, the circles of latitude are horizontal, parallel, and equally spaced. On other cylindrical and pseudocylindrical projections, the circles of latitude are horizontal and parallel, but may be spaced unevenly to give the map useful characteristics. For instance, on a Mercator projection the circles of latitude are more widely spaced near the poles to preserve local scales and shapes, while on a Gall–Peters projection the circles of latitude are spaced more closely near the poles so that comparisons of area will be accurate. On most non-cylindrical and non-pseudocylindrical projections, the circles of latitude are neither straight nor parallel.

Arcs of circles of latitude are sometimes used as boundaries between countries or regions where distinctive natural borders are lacking (such as in deserts), or when an artificial border is drawn as a "line on a map", which was made in massive scale during the 1884 Berlin Conference, regarding huge parts of the African continent. North American nations and states have also mostly been created by straight lines, which are often parts of circles of latitudes. For instance, the northern border of Colorado is at 41°N while the southern border is at 37°N. Roughly half the length of border between the United States and Canada follows 49°N.

Decimal degrees

Decimal degrees (DD) express latitude and longitude geographic coordinates as decimal fractions and are used in many geographic information systems (GIS), web mapping applications such as OpenStreetMap, and GPS devices. Decimal degrees are an alternative to using degrees, minutes, and seconds (DMS). As with latitude and longitude, the values are bounded by ±90° and ±180° respectively.

Positive latitudes are north of the equator, negative latitudes are south of the equator. Positive longitudes are east of Prime meridian, negative longitudes are west of the Prime Meridian. Latitude and longitude are usually expressed in that sequence, latitude before longitude.

Declination

In astronomy, declination (abbreviated dec; symbol δ) is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of the celestial equator, along the hour circle passing through the point in question.

The root of the word declination (Latin, declinatio) means "a bending away" or "a bending down". It comes from the same root as the words incline ("bend toward") and recline ("bend backward").In some 18th and 19th century astronomical texts, declination is given as North Pole Distance (N.P.D.), which is equivalent to 90 - (declination). For instance an object marked as declination -5 would have a NPD of 95, and a declination of -90 (the south celestial pole) would have a NPD of 180.

Equator

An equator of a rotating spheroid (such as a planet) is its zeroth circle of latitude (parallel). It is the imaginary line on the spheroid, equidistant from its poles, dividing it into northern and southern hemispheres. In other words, it is the intersection of the spheroid with the plane perpendicular to its axis of rotation and midway between its geographical poles.

On Earth, the Equator is about 40,075 km (24,901 mi) long, of which 78.8% lies across water and 21.3% over land. Indonesia is the country straddling the greatest length of the equatorial line across both land and sea.

Extratropical cyclone

Extratropical cyclones, sometimes called mid-latitude cyclones or wave cyclones, are low-pressure areas which, along with the anticyclones of high-pressure areas, drive the weather over much of the Earth. Extratropical cyclones are capable of producing anything from cloudiness and mild showers to heavy gales, thunderstorms, blizzards, and tornadoes. These types of cyclones are defined as large scale (synoptic) low pressure weather systems that occur in the middle latitudes of the Earth. In contrast with tropical cyclones, extratropical cyclones produce rapid changes in temperature and dew point along broad lines, called weather fronts, about the center of the cyclone.

Geographic coordinate system

A geographic coordinate system is a coordinate system that enables every location on Earth to be specified by a set of numbers, letters or symbols. The coordinates are often chosen such that one of the numbers represents a vertical position and two or three of the numbers represent a horizontal position; alternatively, a geographic position may be expressed in a combined three-dimensional Cartesian vector.

A common choice of coordinates is latitude, longitude and elevation.

To specify a location on a plane requires a map projection.

Knot (unit)

The knot () is a unit of speed equal to one nautical mile per hour, exactly 1.852 km/h (approximately 1.15078 mph). The ISO standard symbol for the knot is kn. The same symbol is preferred by the Institute of Electrical and Electronics Engineers (IEEE); kt is also common, especially in aviation where it is the form recommended by the International Civil Aviation Organization (ICAO). The knot is a non-SI unit. Worldwide, the knot is used in meteorology, and in maritime and air navigation—for example, a vessel travelling at 1 knot along a meridian travels approximately one minute of geographic latitude in one hour.

Etymologically, the term derives from counting the number of knots in the line that unspooled from the reel of a chip log in a specific time.

Longitude

Longitude (, AU and UK also ), is a geographic coordinate that specifies the east–west position of a point on the Earth's surface, or the surface of a celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter lambda (λ). Meridians (lines running from pole to pole) connect points with the same longitude. By convention, one of these, the Prime Meridian, which passes through the Royal Observatory, Greenwich, England, was allocated the position of 0° longitude. The longitude of other places is measured as the angle east or west from the Prime Meridian, ranging from 0° at the Prime Meridian to +180° eastward and −180° westward. Specifically, it is the angle between a plane through the Prime Meridian and a plane through both poles and the location in question. (This forms a right-handed coordinate system with the z-axis (right hand thumb) pointing from the Earth's center toward the North Pole and the x-axis (right hand index finger) extending from the Earth's center through the Equator at the Prime Meridian.)

A location's north–south position along a meridian is given by its latitude, which is approximately the angle between the local vertical and the equatorial plane.

If the Earth were perfectly spherical and radially homogeneous, then the longitude at a point would be equal to the angle between a vertical north–south plane through that point and the plane of the Greenwich meridian. Everywhere on Earth the vertical north–south plane would contain the Earth's axis. But the Earth is not radially homogeneous and has rugged terrain, which affect gravity and so can shift the vertical plane away from the Earth's axis. The vertical north–south plane still intersects the plane of the Greenwich meridian at some angle; that angle is the astronomical longitude, calculated from star observations. The longitude shown on maps and GPS devices is the angle between the Greenwich plane and a not-quite-vertical plane through the point; the not-quite-vertical plane is perpendicular to the surface of the spheroid chosen to approximate the Earth's sea-level surface, rather than perpendicular to the sea-level surface itself.

Nautical mile

A nautical mile is a unit of measurement used in both air and marine navigation, and for the definition of territorial waters. Historically, it was defined as one minute (1/60) of a degree of latitude. Today it is defined as exactly 1852 metres. The derived unit of speed is the knot, one nautical mile per hour.

Polar regions of Earth

The polar regions, also called the frigid zones, of Earth are the regions of the planet that surround its geographical poles (the North and South Poles), lying within the polar circles. These high latitudes are dominated by Earth's polar ice caps: the northern resting on the Arctic Ocean and the southern on the continent of Antarctica.

Temperate climate

In geography, the temperate or tepid climates of Earth occur in the middle latitudes, which span between the tropics and the polar regions of Earth. These zones generally have wider temperature ranges throughout the year and more distinct seasonal changes compared to tropical climates, where such variations are often small. They typically feature four distinct seasons, Summer the warmest, Autumn the transitioning season to Winter, the colder season, and Spring the transitioning season from winter back into summer. On the northern hemisphere the year starts with winter, transitions in the first halfyear through spring into summer which is in mid-year, then at the second halfyear through autumn into winter at year-end. On the southern hemisphere seasons are swapped with summer in between years and winter in mid-year.

The temperate zones (latitudes from 23.5° to the polar circles at about 66.5°, north and south) are where the widest seasonal changes occur, with most climates found in it having some influence from both the tropics and the poles. The subtropics (latitudes from about 23.5° to 35°, north and south) have temperate climates that have the least seasonal change and the warmest in winter, while at the other end, Boreal climates located from 55 to 65 north latitude have the most seasonal changes and long and severe winters.

In temperate climates, not only due latitudinal positions influence temperature changes, but sea currents, prevailing wind direction, continentality (how large a landmass is), and altitude also shape temperate climates.

The Köppen climate classification defines a climate as "temperate" when the mean temperature is above −3 °C (26.6 °F) but below 18 °C (64.4 °F) in the coldest month. However, in more recent climate classifications climatologists use the 0 °C (32.0 °F) line .

Tropic of Cancer

The Tropic of Cancer, which is also referred to as the Northern Tropic, is the most northerly circle of latitude on Earth at which the Sun can be directly overhead. This occurs on the June solstice, when the Northern Hemisphere is tilted toward the Sun to its maximum extent. It is currently 23°26′12.4″ (or 23.43678°) north of the Equator.

Its Southern Hemisphere counterpart, marking the most southerly position at which the Sun can be directly overhead, is the Tropic of Capricorn. These tropics are two of the five major circles of latitude that mark maps of Earth; the others being the Arctic and Antarctic Circles and the Equator. The positions of these two circles of latitude (relative to the Equator) are dictated by the tilt of Earth's axis of rotation relative to the plane of its orbit.

Tropic of Capricorn

The Tropic of Capricorn (or the Southern Tropic) is the circle of latitude that contains the subsolar point on the December (or southern) solstice. It is thus the southernmost latitude where the Sun can be directly overhead. Its northern equivalent is the Tropic of Cancer.

The Tropic of Capricorn is one of the five major circles of latitude that mark maps of Earth. As of 24 April 2019, its latitude is 23°26′12.4″ (or 23.43678°) south of the Equator, but it is very gradually moving northward, currently at the rate of 0.47 arcseconds, or 15 metres, per year.

Tropics

The tropics are the region of the Earth surrounding the Equator. They are delimited in latitude by The Tropic of Cancer in the Northern Hemisphere at 23°26′12.4″ (or 23.43678°) N and the Tropic of Capricorn in

the Southern Hemisphere at 23°26′12.4″ (or 23.43678°) S; these latitudes correspond to the axial tilt of the Earth. The tropics are also referred to as the tropical zone and the torrid zone (see geographical zone). The tropics include all the areas on the Earth where the Sun contacts a point directly overhead at least once during the solar year (which is a subsolar point) - thus the latitude of the tropics is roughly equal to the angle of the Earth's axial tilt.

The tropics are distinguished from the other climatic and biomatic regions of Earth, which are the middle latitudes and the polar regions on either side of the equatorial zone.

The tropics comprise 40% of the Earth's surface area and contain 36% of the Earth's landmass. As of 2014, the region is home to 40% of the world population, and this figure is projected to reach 50% by the late 2030s.

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