In geography, latitude is a geographic coordinate that specifies the
north–south position of a point on the Earth's surface.
Contents 1 Preliminaries
2
2.1 The graticule on the sphere 2.2 Named latitudes on the Earth 2.3 Meridian distance on the sphere 3
3.1 Ellipsoids 3.2 The geometry of the ellipsoid 3.3 Geodetic and geocentric latitudes 3.4 Length of a degree of latitude 4 Auxiliary latitudes 4.1 Geocentric latitude 4.2 Reduced (or parametric) latitude 4.3 Rectifying latitude 4.4 Authalic latitude 4.5 Conformal latitude 4.6 Isometric latitude 4.7 Inverse formulae and series 4.8 Numerical comparison of auxiliary latitudes 5
5.1 Geodetic coordinates 5.2 Spherical polar coordinates 5.3 Ellipsoidal coordinates 5.4 Coordinate conversions 6 Astronomical latitude 7 See also 8 Notes 9 References 10 External links Preliminaries[edit]
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.
A perspective view of the Earth showing how latitude ( ϕ displaystyle phi ) and longitude ( λ displaystyle lambda ) are defined on a spherical model. The graticule spacing is 10 degrees. The graticule on the sphere[edit]
The graticule is formed by the lines of constant latitude and constant
longitude, which are constructed with reference to the rotation axis
of the Earth. The primary reference points are the poles where the
axis of rotation of the Earth intersects the reference surface. Planes
which contain the rotation axis intersect the surface at the
meridians; and the angle between any one meridian plane and that
through Greenwich (the Prime Meridian) defines the longitude:
meridians are lines of constant longitude. The plane through the
centre of the Earth and perpendicular to the rotation axis intersects
the surface at a great circle called the Equator. Planes parallel to
the equatorial plane intersect the surface in circles of constant
latitude; these are the parallels. The
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
Normal Mercator Transverse Mercator Meridian distance on the sphere[edit] On the sphere the normal passes through the centre and the latitude (φ) is therefore equal to the angle subtended at the centre by the meridian arc from the equator to the point concerned. If the meridian distance is denoted by m(φ) then m ( ϕ ) = π 180 ∘ R ϕ d e g r e e s = R ϕ r a d i a n s displaystyle m(phi )= frac pi 180^ circ Rphi _ mathrm degrees =Rphi _ mathrm radians where R denotes the mean radius of the Earth. R is equal to
6,371 km or 3,959 miles. No higher accuracy is appropriate for R
since higher-precision results necessitate an ellipsoid model. With
this value for R the meridian length of 1 degree of latitude on the
sphere is 111.2 km or 69.1 miles. The length of 1 minute of
latitude is 1.853 km or 1.151 miles (see nautical mile).
f = a − b a , e 2 = 2 f − f 2 , b = a ( 1 − f ) = a 1 − e 2 . displaystyle f= frac a-b a ,qquad e^ 2 =2f-f^ 2 ,qquad b=a(1-f)=a sqrt 1-e^ 2 ,. Many other parameters (see ellipse, ellipsoid) appear in the study of
geodesy, geophysics and map projections but they can all be expressed
in terms of one or two members of the set a, b, f and e. Both f and e
are small and often appear in series expansions in calculations; they
are of the order 1/300 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
a (equatorial radius): 7006637813700000000♠6378137.0 m exactly 1/f (inverse flattening): 7002298257223563000♠298.257223563 exactly from which are derived b (polar radius): 7006635675231420000♠6356752.3142 m e2 (eccentricity squared): 6997669437999014000♠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[edit] 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 ψ, q,
φ′, φc, φg. This article uses ψ.
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.
The importance of specifying the reference datum may be illustrated by
a simple example. On the reference ellipsoid for WGS84, the centre of
the
m ( ϕ ) = ∫ 0 ϕ M ( ϕ ′ ) d ϕ ′ = a ( 1 − e 2 ) ∫ 0 ϕ ( 1 − e 2 sin 2 ϕ ′ ) − 3 2 d ϕ ′ displaystyle m(phi )=int _ 0 ^ phi M(phi '),dphi '=a(1-e^ 2 )int _ 0 ^ phi left(1-e^ 2 sin ^ 2 phi 'right)^ - frac 3 2 ,dphi ' where M(φ) is the meridional radius of curvature. The distance from the equator to the pole is m p = m ( π 2 ) displaystyle m_ mathrm p =mleft( frac pi 2 right), For
δ m ( ϕ ) = M ( ϕ ) δ ϕ = a ( 1 − e 2 ) ( 1 − e 2 sin 2 ϕ ) − 3 2 δ ϕ displaystyle delta m(phi )=M(phi ),delta phi =a(1-e^ 2 )left(1-e^ 2 sin ^ 2 phi right)^ - frac 3 2 ,delta phi ϕ displaystyle phi Δ1 lat Δ1 long 0° 110.574 km 111.320 km 15° 110.649 km 107.550 km 30° 110.852 km 96.486 km 45° 111.132 km 78.847 km 60° 111.412 km 55.800 km 75° 111.618 km 28.902 km 90° 111.694 km 0.000 km When the latitude difference is 1 degree, corresponding to π/180 radians, the arc distance is about Δ l a t 1 = π a ( 1 − e 2 ) 180 ∘ ( 1 − e 2 sin 2 ϕ ) 3 2 displaystyle Delta _ mathrm lat ^ 1 = frac pi aleft(1-e^ 2 right) 180^ circ left(1-e^ 2 sin ^ 2 phi right)^ frac 3 2 The distance in metres (correct to 0.01 metre) between latitudes ϕ displaystyle phi − 0.5 degrees and ϕ displaystyle phi + 0.5 degrees on the
Δ l a t 1 = 111 132.954 − 559.822 cos 2 ϕ + 1.175 cos 4 ϕ displaystyle Delta _ mathrm lat ^ 1 =111,132.954-559.822cos 2phi +1.175cos 4phi 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): Δ l o n g 1 = π a cos ϕ 180 ∘ 1 − e 2 sin 2 ϕ displaystyle Delta _ mathrm long ^ 1 = frac pi acos phi 180^ circ sqrt 1-e^ 2 sin ^ 2 phi , A calculator for any latitude is provided by the U.S. Government's
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 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.[7] Derivations of these expressions may be found in Adams[8] and online publications by Osborne[4] and Rapp.[5] Geocentric latitude[edit] The definition of geodetic (or geographic) and geocentric latitudes. 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 (ψ) and the geodetic latitude (φ) is derived in the above references as ψ ( ϕ ) = tan − 1 ( ( 1 − e 2 ) tan ϕ ) = tan − 1 ( ( 1 − f ) 2 tan ϕ ) . displaystyle psi (phi )=tan ^ -1 left((1-e^ 2 )tan phi right)=tan ^ -1 left((1-f)^ 2 tan phi right),. The geodetic and geocentric latitudes are equal at the equator and at the poles but at other latitudes they differ by a few minutes of arc. Taking the value of the squared eccentricity as 0.0067 (it depends on the choice of ellipsoid) the maximum difference of ϕ − ψ displaystyle phi - psi may be shown to be about 11.5 minutes of arc at a geodetic latitude of approximately 45° 6′.[c] Reduced (or parametric) latitude[edit] Definition of the reduced latitude (β) on the ellipsoid. The reduced or parametric 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[9] and Bessel[10] 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 reduced latitude is related to the geodetic latitude by:[4][5] β ( ϕ ) = tan − 1 ( 1 − e 2 tan ϕ ) = tan − 1 ( ( 1 − f ) tan ϕ ) displaystyle beta (phi )=tan ^ -1 left( sqrt 1-e^ 2 tan phi right)=tan ^ -1 left((1-f)tan phi right) The alternative name arises from the parameterization of the equation of the ellipse describing a meridian section. In terms of Cartesian coordinates p, the distance from the minor axis, and z, the distance above the equatorial plane, the equation of the ellipse is: p 2 a 2 + z 2 b 2 = 1 . displaystyle frac p^ 2 a^ 2 + frac z^ 2 b^ 2 =1,. The Cartesian coordinates of the point are parameterized by p = a cos β , z = b sin β ; displaystyle p=acos beta ,,qquad z=bsin beta ,; Cayley suggested the term parametric latitude because of the form of these equations.[11] The reduced latitude is not used in the theory of map projections. Its most important application is in the theory of ellipsoid geodesics, (Vincenty, Karney[12]). Rectifying latitude[edit] See also: Rectifying radius 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: μ ( ϕ ) = π 2 m ( ϕ ) m p displaystyle mu (phi )= frac pi 2 frac m(phi ) m_ mathrm p where the meridian distance from the equator to a latitude φ is (see Meridian arc) m ( ϕ ) = a ( 1 − e 2 ) ∫ 0 ϕ ( 1 − e 2 sin 2 ϕ ′ ) − 3 2 d ϕ ′ , displaystyle m(phi )=a(1-e^ 2 )int _ 0 ^ phi left(1-e^ 2 sin ^ 2 phi 'right)^ - frac 3 2 ,dphi ',, and the length of the meridian quadrant from the equator to the pole (the polar distance) is m p = m ( π 2 ) . displaystyle m_ mathrm p =mleft( frac pi 2 right),. Using the rectifying latitude to define a latitude on a sphere of radius R = 2 m p π displaystyle R= frac 2m_ mathrm p pi defines a projection from the ellipsoid to the sphere such that all meridians have true length and uniform scale. The sphere may then be projected to the plane with an equirectangular projection to give a double projection from the ellipsoid to the plane such that all meridians have true length and uniform meridian scale. An example of the use of the rectifying latitude is the Equidistant conic projection. (Snyder, Section 16).[7] The rectifying latitude is also of great importance in the construction of the Transverse Mercator projection. Authalic latitude[edit] See also: Authalic radius The authalic (Greek for same area) latitude, ξ, gives an area-preserving transformation to a sphere. ξ ( ϕ ) = sin − 1 ( q ( ϕ ) q p ) displaystyle xi (phi )=sin ^ -1 left( frac q(phi ) q_ mathrm p right) where q ( ϕ ) = ( 1 − e 2 ) sin ϕ 1 − e 2 sin 2 ϕ − 1 − e 2 2 e ln ( 1 − e sin ϕ 1 + e sin ϕ ) = ( 1 − e 2 ) sin ϕ 1 − e 2 sin 2 ϕ + 1 − e 2 e tanh − 1 ( e sin ϕ ) displaystyle begin aligned q(phi )&= frac (1-e^ 2 )sin phi 1-e^ 2 sin ^ 2 phi - frac 1-e^ 2 2e ln left( frac 1-esin phi 1+esin phi right)\&= frac (1-e^ 2 )sin phi 1-e^ 2 sin ^ 2 phi + frac 1-e^ 2 e tanh ^ -1 (esin phi )end aligned and q p = q ( π 2 ) = 1 − 1 − e 2 2 e ln ( 1 − e 1 + e ) = 1 + 1 − e 2 e tanh − 1 e displaystyle begin aligned q_ mathrm p =qleft( frac pi 2 right)&=1- frac 1-e^ 2 2e ln left( frac 1-e 1+e right)&=1+ frac 1-e^ 2 e tanh ^ -1 eend aligned and the radius of the sphere is taken as R q = a q p 2 . displaystyle R_ q =a sqrt frac q_ mathrm p 2 ,. An example of the use of the authalic latitude is the Albers equal-area conic projection.[7]:§14 Conformal latitude[edit] The conformal latitude, χ, gives an angle-preserving (conformal) transformation to the sphere. χ ( ϕ ) = 2 tan − 1 [ ( 1 + sin ϕ 1 − sin ϕ ) ( 1 − e sin ϕ 1 + e sin ϕ ) e ] 1 2 − π 2 = 2 tan − 1 [ tan ( ϕ 2 + π 4 ) ( 1 − e sin ϕ 1 + e sin ϕ ) e 2 ] − π 2 = sin − 1 [ tanh ( tanh − 1 ( sin ϕ ) − e tanh − 1 ( e sin ϕ ) ) ] = gd [ gd − 1 ( ϕ ) − e tanh − 1 ( e sin ϕ ) ] displaystyle begin aligned chi (phi )&=2tan ^ -1 left[left( frac 1+sin phi 1-sin phi right)left( frac 1-esin phi 1+esin phi right)^ e right]^ frac 1 2 - frac pi 2 \[2ex]&=2tan ^ -1 left[tan left( frac phi 2 + frac pi 4 right)left( frac 1-esin phi 1+esin phi right)^ frac e 2 right]- frac pi 2 \&=sin ^ -1 left[tanh left(tanh ^ -1 (sin phi )-etanh ^ -1 (esin phi )right)right]\&=operatorname gd left[operatorname gd ^ -1 (phi )-etanh ^ -1 (esin phi )right]end aligned where gd(x) is the Gudermannian function. (See also Mercator
projection.) The conformal latitude defines a transformation from the
ellipsoid to a sphere of arbitrary radius such that the angle of
intersection between any two lines on the ellipsoid is the same as the
corresponding angle on the sphere (so that the shape of small elements
is well preserved). A further conformal transformation from the sphere
to the plane gives a conformal double projection from the ellipsoid to
the plane. This is not the only way of generating such a conformal
projection. For example, the 'exact' version of the Transverse
ψ ( ϕ ) = ln [ tan ( π 4 + ϕ 2 ) ] + e 2 ln [ 1 − e sin ϕ 1 + e sin ϕ ] = tanh − 1 ( sin ϕ ) − e tanh − 1 ( e sin ϕ ) = gd − 1 ( ϕ ) − e tanh − 1 ( e sin ϕ ) . displaystyle begin aligned psi (phi )&=ln left[tan left( frac pi 4 + frac phi 2 right)right]+ frac e 2 ln left[ frac 1-esin phi 1+esin phi right]\&=tanh ^ -1 (sin phi )-etanh ^ -1 (esin phi )\&=operatorname gd ^ -1 (phi )-etanh ^ -1 (esin phi ).end aligned For the normal
y ( ϕ ) = E 2 π ψ ( ϕ ) . displaystyle y(phi )= frac E 2pi psi (phi ),. The isometric latitude ψ is closely related to the conformal latitude χ: ψ ( ϕ ) = gd − 1 χ ( ϕ ) . displaystyle psi (phi )=operatorname gd ^ -1 chi (phi ),. Inverse formulae and series[edit] The formulae in the previous sections give the auxiliary latitude in terms of the geodetic latitude. The expressions for the geocentric and reduced 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.[8] Osborne[4] derives series to arbitrary order by using the computer algebra package Maxima[13] 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[edit] 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.[7]:108 Approximate difference from geodetic latitude (φ) φ Reduced φ − β Authalic φ − ξ Rectifying φ − μ Conformal φ − χ Geocentric φ − ψ 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′
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[edit] Geocentric coordinate related to spherical polar coordinates P(r,θ,λ) The geocentric latitude ψ 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 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[edit] Ellipsoidal coordinates P(u,β,λ) 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. Let u 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.[3]:§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. Coordinate conversions[edit] The relations between the above coordinate systems, and also Cartesian coordinates are not presented here. The transformation between geodetic and Cartesian coordinates may be found in Geographic coordinate conversion. The relation of Cartesian and spherical polars is given in Spherical coordinate system. The relation of Cartesian and ellipsoidal coordinates is discussed in Torge.[3] Astronomical latitude[edit] 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][14] 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[edit]
Notes[edit] ^ 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 ^ The value of this angle today is 23°26′12.9″ (or 23.43692°). This figure is provided by Template:Circle of latitude. ^ An elementary calculation involves differentiation to find the maximum difference of the geodetic and geocentric latitudes. Full details may be found on page 90 of The Mercator projections References[edit] ^ Newton, Isaac. "Book III Proposition XIX Problem III". Philosophiæ
Naturalis Principia Mathematica. Translated by Motte, Andrew.
p. 407.
^ "TR8350.2".
External links[edit] Find more aboutLatitudeat's sister projects Definitions from Wiktionary Media from Wikimedia Commons News from Wikinews Quotations from Wikiquote Texts from Wikisource Textbooks from Wikibooks Learning resources from Wikiversity GEONets Names Server, access to the National Geospatial-Intelligence
Agency's (NGA) database of foreign geographic feature names.
Resources for determining your latitude and longitude
Convert decimal degrees into degrees, minutes, seconds - Info about
decimal to sexagesimal conversion
Convert decimal degrees into degrees, minutes, seconds
Distance calculation based on latitude and longitude - JavaScript
version
16th Century
v t e Circles of latitude / meridians Equator Tropic of Cancer Tropic of Capricorn Arctic Circle Antarctic Circle Equator Tropic of Cancer Tropic of Capricorn Arctic Circle Antarctic Circle Equator Tropic of Cancer Tropic of Capricorn Arctic Circle Antarctic Circle W 0° E 30° 60° 90° 120° 150° 180° 30° 60° 90° 120° 150° 180° 5° 15° 25° 35° 45° 55° 65° 75° 85° 95° 105° 115° 125° 135° 145° 155° 165° 175° 5° 15° 25° 35° 45° 55° 65° 75° 85° 95° 105° 115° 125° 135° 145° 155° 165° 175° 10° 20° 40° 50° 70° 80° 100° 110° 130° 140° 160° 170° 10° 20° 40° 50° 70° 80° 100° 110° 130° 140° 160° 170° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 10° 20° 30° 40° 50° 60° 70° 80° 90° 5° N 15° 25° 35° 45° 55° 65° 75° 85° 5° S 15° 25° 35° 45° 55° 65° 75° 85° 45x90 45x90 45x90 45x90 v t e Map projection History List Portal By surface Cylindrical Mercator-conformal Gauss–Krüger Transverse Mercator Equal-area Balthasart Behrmann Gall–Peters Hobo–Dyer Lambert Smyth equal-surface Trystan Edwards Cassini Central Equirectangular Gall stereographic Miller Space-oblique Mercator Web Mercator Pseudocylindrical Eckert II Eckert IV Eckert VI Goode homolosine Kavrayskiy VII Mollweide Sinusoidal Tobler hyperelliptical Wagner VI Conical Albers Equidistant Lambert conformal Pseudoconical Bonne Bottomley Polyconic Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert equal-area Pseudoazimuthal Aitoff Hammer Wiechel Winkel tripel By metric Conformal Adams hemisphere-in-a-square Gauss–Krüger Guyou hemisphere-in-a-square Lambert conformal conic Mercator Peirce quincuncial Stereographic Transverse Mercator Equal-area Bonne Sinusoidal Werner Bottomley Sinusoidal Werner Cylindrical Balthasart Behrmann Gall–Peters Hobo–Dyer Lambert cylindrical equal-area Smyth equal-surface Trystan Edwards Tobler hyperelliptical Collignon Mollweide Albers Briesemeister Eckert II Eckert IV Eckert VI Hammer Lambert azimuthal equal-area Quadrilateralized spherical cube Equidistant in some aspect Conic Equirectangular Sinusoidal Two-point Werner Gnomonic Gnomonic Loxodromic Loximuthal Mercator Retroazimuthal (Mecca or Qibla) Craig Hammer Littrow By construction Compromise Chamberlin trimetric Kavrayskiy VII Miller cylindrical Robinson Van der Grinten Wagner VI Winkel tripel Hybrid Goode homolosine HEALPix Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral Cahill Butterfly Dymaxion Quadrilateralized spherical cube Waterman butterfly See also Latitude Longitude T |