Latitude - Knowledge

27: 595:) 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 998: 263: 2846: 2382: 5639: 608: 353: 6112: 5671:; in the Southern hemisphere (negative latitudes), the inequalities are reversed, with equality at the equator and the poles. Although the graph appears symmetric about 45°, the minima of the curves actually lie between 45° 2′ and 45° 6′. Some representative data points are given in the table below. The conformal and geocentric latitudes are nearly indistinguishable, a fact that was exploited in the days of hand calculators to expedite the construction of map projections. 6020: 5953: 2477: 4117: 516: 502: 3765: 6231: 5404: 4993: 165:, 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 4327: 1560: 4112:{\displaystyle {\begin{aligned}q(\phi )&={\frac {\left(1-e^{2}\right)\sin \phi }{1-e^{2}\sin ^{2}\phi }}-{\frac {1-e^{2}}{2e}}\ln \left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)\\&={\frac {\left(1-e^{2}\right)\sin \phi }{1-e^{2}\sin ^{2}\phi }}+{\frac {1-e^{2}}{e}}\tanh ^{-1}(e\sin \phi )\end{aligned}}} 2808:

is the prime vertical radius of curvature. The geodetic and geocentric latitudes are equal at the equator and at the poles but at other latitudes they differ by a few minutes of arc. Taking the value of the squared eccentricity as 0.0067 (it depends on the choice of ellipsoid) the maximum difference

5577:

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.

1045:

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

982:

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

6294:

In general the true vertical at a point on the surface does not exactly coincide with either the normal to the reference ellipsoid or the normal to the geoid. The geoid is an idealized, theoretical shape "at mean sea level". Points on land do not lie precisely on the geoid, and the vertical at a

5646:

The plot to the right shows the difference between the geodetic latitude and the auxiliary latitudes other than the isometric latitude (which diverges to infinity at the poles) for the case of the WGS84 ellipsoid. The differences shown on the plot are in arc minutes. In the Northern hemisphere

5937:

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

568:.) 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 5080: 1830: 2123: 3500: 4426: 1335:

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

5629:. Karney establishes that the truncation errors for such series are consistently smaller that the equivalent series in terms of the eccentricity. The series method is not applicable to the isometric latitude and one must find the conformal latitude in an intermediate step. 2665: 5025:

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

4128: 2462:.) The forms given are, apart from notational variants, those in the standard reference for map projections, namely "Map projections: a working manual" by J. P. Snyder. Derivations of these expressions may be found in Adams and online publications by Osborne and Rapp. 415:, 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 2446:

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.

5074:, 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): 3037: 2366: 1310: 339:

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 radial vector.

5606:. Such series are presented by Adams who uses Taylor series expansions and gives coefficients in terms of the eccentricity. Orihuela gives series for the conversions between all pairs of auxiliary latitudes in terms of the third flattening, 795: 2799: 6198:(although that term is also used to refer to geodetic coordinate). These coordinates are the natural choice in models of the gravity field for a rotating ellipsoidal body. The above applies to a biaxial ellipsoid (a spheroid, as in 2247: 1366: 3754: 3330: 454:

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

5399:{\displaystyle {\begin{aligned}\psi (\phi )&=\ln \left+{\frac {e}{2}}\ln \left\\&=\sinh ^{-1}(\tan \phi )-e\tanh ^{-1}(e\sin \phi )\\&=\operatorname {gd} ^{-1}(\phi )-e\tanh ^{-1}(e\sin \phi ).\end{aligned}}} 478:. 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. 2885:. It was introduced by Legendre and Bessel who solved problems for geodesics on the ellipsoid by transforming them to an equivalent problem for spherical geodesics by using this smaller latitude. Bessel's notation, 343:

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.

4431: 1683: 5567: 4988:{\displaystyle {\begin{aligned}\chi (\phi )&=2\tan ^{-1}\left^{\frac {1}{2}}-{\frac {\pi }{2}}\\&=2\tan ^{-1}\left-{\frac {\pi }{2}}\\&=\tan ^{-1}\left\\&=\operatorname {gd} \left\end{aligned}}} 3567: 1668:

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

1139:

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.

3133: 1989: 1644: 4391: 3770: 2433:

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

129:

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

5489: 327:) 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 4322:{\displaystyle {\begin{aligned}q_{\mathrm {p} }=q\left({\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}e\end{aligned}}} 3201: 2513: 3349: 5085: 4133: 3620: 7285: 315:

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

2902: 6011:

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.

6214:

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

3629:

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

2266: 6103:

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.

1197: 2835: 138:. 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 3625:

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

1135:

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

1039: 304: 1673:

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

2166: 2146: 1855: 1019: 665: 284: 2683: 6295:

point at a specific time is influenced by tidal forces which the theoretical geoid averages out. The angle between the astronomic and geodetic normals is called

1555:{\displaystyle m(\phi )=\int _{0}^{\phi }M(\phi ')\,d\phi '=a\left(1-e^{2}\right)\int _{0}^{\phi }\left(1-e^{2}\sin ^{2}\phi '\right)^{-{\frac {3}{2}}}\,d\phi '} 5602:

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

1059:. This is the definition assumed when the word latitude is used without qualification. The definition must be accompanied with a specification of the ellipsoid. 331:. 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 2174: 52:. 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. 8372: 5021:

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

3673: 470:

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

6007:. 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 7259: 3263: 3042:

The alternative name arises from the parameterization of the equation of the ellipse describing a meridian section. In terms of Cartesian coordinates

5582:

The first is a numerical inversion of the defining equation for each and every particular value of the auxiliary latitude. The methods available are

6711: 1131:

The importance of specifying the reference datum may be illustrated by a simple example. On the reference ellipsoid for WGS84, the centre of the

3213:

The parametric latitude is not used in the theory of map projections. Its most important application is in the theory of ellipsoid geodesics, (

1825:{\displaystyle \delta m(\phi )=M(\phi )\,\delta \phi =a\left(1-e^{2}\right)\left(1-e^{2}\sin ^{2}\phi \right)^{-{\frac {3}{2}}}\,\delta \phi } 548: 319:

where the axis of rotation of the Earth intersects the reference surface. Planes which contain the rotation axis intersect the surface at the

121:

Two levels of abstraction are employed in the definitions of latitude and longitude. In the first step the physical surface is modeled by the

5413:

Mercator projection (on the ellipsoid) this function defines the spacing of the parallels: if the length of the equator on the projection is

150:

to the reference surface, which passes through the point on the physical surface. Latitude and longitude together with some specification of

2118:{\displaystyle \Delta _{\text{lat}}^{1}={\frac {\pi a\left(1-e^{2}\right)}{180^{\circ }\left(1-e^{2}\sin ^{2}\phi \right)^{\frac {3}{2}}}}} 5933:

The geodetic latitude, or any of the auxiliary latitudes defined on the reference ellipsoid, constitutes with longitude a two-dimensional

5508: 5030:

on the ellipsoid is not a double projection. (It does, however, involve a generalisation of the conformal latitude to the complex plane).

80:

or another celestial body. Latitude is given as an angle that ranges from −90° at the south pole to 90° at the north pole, with 0° at the

6782:

Latitude Developments Connected With Geodesy and Cartography (with tables, including a table for Lambert equal area meridional projection

3515: 808:) 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 3060: 1595: 6352: 9114: 8646: 8152: 8029: 4338: 7202: 6794:: Adams uses the nomenclature isometric latitude for the conformal latitude of this article (and throughout the modern literature).) 627:

which is rotated about its minor (shorter) axis. Two parameters are required. One is invariably the equatorial radius, which is the

203:

This article relates to coordinate systems for the Earth: it may be adapted to cover the Moon, planets and other celestial objects (

7189: 2660:{\displaystyle \theta (\phi )=\tan ^{-1}\left(\left(1-e^{2}\right)\tan \phi \right)=\tan ^{-1}\left((1-f)^{2}\tan \phi \right)\,.} 8732: 8528: 8518: 8438: 7180: 2372: 242:

method. More precise measurement of latitude requires an understanding of the gravitational field of the Earth, either to set up

8656: 8651: 8626: 8498: 8162: 8157: 8132: 5431: 3495:{\displaystyle m(\phi )=a\left(1-e^{2}\right)\int _{0}^{\phi }\left(1-e^{2}\sin ^{2}\phi '\right)^{-{\frac {3}{2}}}\,d\phi '\,,} 1124:

must be used with care, as some authors use it as a synonym for geodetic latitude whilst others use it as an alternative to the

7252: 3144: 992: 6519: 1176:) is therefore equal to the angle subtended at the centre by the meridian arc from the equator to the point concerned. If the 196:, north or south of the equator. For navigational purposes positions are given in degrees and decimal minutes. For instance, 8523: 8104: 6099:

is longitude. Since the normal at a general point on the ellipsoid does not pass through the centre it is clear that points

6499:

An elementary calculation involves differentiation to find the maximum difference of the geodetic and geocentric latitudes.

6751:. U.S. Geological Survey Professional Paper 1395. Washington, DC: United States Government Printing Office. Archived from 3578: 8533: 8334: 3032:{\displaystyle \beta (\phi )=\tan ^{-1}\left({\sqrt {1-e^{2}}}\tan \phi \right)=\tan ^{-1}\left((1-f)\tan \phi \right)} 1577: 1041:) on an ellipsoid. The normal to the surface does not pass through the centre, except at the equator and at the poles. 8666: 8618: 8279: 8205: 8124: 7245: 7150: 6632: 103:

as defined below. Briefly, the geodetic latitude of a point is the angle formed between the vector perpendicular (or

6746: 2378:

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

9057: 8854: 8781: 8737: 8433: 6215: 6203: 2361:{\displaystyle \Delta _{\text{long}}^{1}={\frac {\pi a\cos \phi }{180^{\circ }{\sqrt {1-e^{2}\sin ^{2}\phi }}}}\,} 8902: 8849: 7172: 6819:

Bessel, F. W. (1825). "Über die Berechnung der geographischen Langen und Breiten aus geodatischen Vermessungen".

5011: 2257: 1155: 6868:

Karney, C. F. F.; Deakin, R. E. (2010). "The calculation of longitude and latitude from geodesic measurements".

169:(GPS), but in common usage, where high accuracy is not required, the reference ellipsoid is not usually stated. 9010: 8979: 8553: 8402: 8180: 8109: 6431: 5051: 5027: 3634: 475: 6545: 9094: 9062: 8912: 8543: 8367: 8200: 8190: 8022: 6436: 6199: 1305:{\displaystyle m(\phi )={\frac {\pi }{180^{\circ }}}R\phi _{\mathrm {degrees} }=R\phi _{\mathrm {radians} }} 1075:): the angle between the radius (from centre to the point on the surface) and the equatorial plane. (Figure 442:

of the plane perpendicular to the ecliptic and through the centres of the Earth and the Sun at the December

9139: 9052: 8766: 8420: 8329: 6456: 6396: 5063: 3630: 2450:

The expressions below give the auxiliary latitudes in terms of the geodetic latitude, the semi-major axis,

2435: 1046:

normal and the equatorial plane. The terminology for latitude must be made more precise by distinguishing:

155: 139: 65: 2896:, is also used in the current literature. The parametric latitude is related to the geodetic latitude by: 1055:: the angle between the normal and the equatorial plane. The standard notation in English publications is 9042: 8992: 8955: 8722: 8415: 8264: 8114: 6487: 6219: 6060: 26: 6715: 146:

surface is that of the corresponding point on the reference surface, the correspondence being along the

8970: 8927: 8771: 8636: 8362: 8195: 8185: 8142: 6276: 3633:. (Snyder, Section 16). The rectifying latitude is also of great importance in the construction of the 3626: 172:

In English texts, the latitude angle, defined below, is usually denoted by the Greek lower-case letter

8894: 8074: 6780: 6589: 8907: 8292: 6381: 2812: 2406:

that have applications to special problems in geodesy, geophysics and the theory of map projections:

552:, in which he proved that a rotating self-gravitating fluid body in equilibrium takes the form of an 166: 20: 427:). The axis of rotation varies slowly over time and the values given here are those for the current 229: 8997: 8937: 8917: 8697: 8641: 8548: 8510: 8475: 8147: 8015: 6280: 6134:

The parametric latitude can also be extended to a three-dimensional coordinate system. For a point

5054:. The name "isometric" arises from the fact that at any point on the ellipsoid equal increments of 997: 439: 6486:

The value of this angle today is 23°26′09.9″ (or 23.43608°). This figure is provided by

6324:, the coordinate that astronomers use to specify the angular position of stars north–south of the 2837:

may be shown to be about 11.5 minutes of arc at a geodetic latitude of approximately 45° 6′.

8224: 8210: 8054: 7394: 7227: 7023:

Karney, Charles F. F. (August 2011). "Transverse Mercator with an accuracy of a few nanometers".

2499:

When the point is on the surface of the ellipsoid, the relation between the geocentric latitude (

2496:

is the angle between the equatorial plane and the radius from the centre to a point of interest.

537: 223: 204: 135: 31: 5638: 5593:

When converting from isometric or conformal to geodetic, two iterations of Newton-Raphson gives

9109: 8742: 8717: 8452: 8259: 8049: 6317: 3214: 262: 8882: 8837: 7005: 5062:

give rise to equal distance displacements along the meridians and parallels respectively. The

1024: 790:{\displaystyle f={\frac {a-b}{a}},\qquad e^{2}=2f-f^{2},\qquad b=a(1-f)=a{\sqrt {1-e^{2}}}\,.} 289: 9032: 8822: 8776: 8603: 8580: 8563: 8344: 8274: 7001: 6461: 6416: 6401: 5583: 5007: 3664: 2794:{\displaystyle \theta (\phi ,h)=\tan ^{-1}\left({\frac {N(1-f)^{2}+h}{N+h}}\tan \phi \right)} 9037: 8932: 8756: 8712: 8707: 8702: 8679: 8674: 8595: 8357: 8297: 8269: 8254: 8249: 8244: 8239: 8086: 7272: 7042: 6967: 6887: 6838: 6406: 6329: 5947: 3239:, is the meridian distance scaled so that its value at the poles is equal to 90 degrees or 2670:

For points not on the surface of the ellipsoid, the relationship involves additionally the

2151: 2131: 1840: 1004: 320: 269: 235: 9024: 8814: 7216: 3506: 96:

are used together as a coordinate pair to specify a location on the surface of the Earth.

8: 9149: 8987: 8922: 8827: 8804: 8631: 8538: 8410: 8389: 8137: 8095: 7962: 7957: 7952: 7945: 7940: 7935: 7930: 7924: 7915: 7910: 7905: 7900: 7893: 7888: 7883: 7878: 7872: 7858: 7853: 7848: 7843: 7838: 7833: 7828: 7823: 7813: 7808: 7803: 7798: 7793: 7788: 7783: 7778: 7766: 7761: 7756: 7751: 7746: 7741: 7706: 7701: 7696: 7691: 7686: 7681: 7646: 7641: 7636: 7631: 7626: 7621: 7616: 7611: 7556: 7551: 7546: 7541: 7536: 7531: 7526: 7521: 7459: 7454: 7425: 7420: 7375: 7342: 7305: 7221: 7198: 6421: 6297: 5047: 2410: 2242:{\displaystyle \Delta _{\text{lat}}^{1}=111\,132.954-559.822\cos 2\phi +1.175\cos 4\phi } 1161: 1063: 849: 576: 471: 447: 388: 247: 211: 162: 85: 8796: 7186: 7046: 6971: 6891: 6842: 8859: 8470: 8316: 8175: 7920: 7868: 7736: 7731: 7726: 7721: 7716: 7711: 7676: 7671: 7666: 7661: 7656: 7651: 7606: 7601: 7596: 7591: 7586: 7581: 7576: 7571: 7566: 7516: 7511: 7506: 7501: 7496: 7491: 7486: 7481: 7476: 7447: 7442: 7437: 7413: 7408: 7403: 7397: 7391: 7268: 7100: 7058: 7032: 6983: 6957: 6903: 6877: 6854: 6828: 6362: 6357: 6283:) at that latitude. Astronomic latitude is calculated from angles measured between the 6264: 5603: 5587: 3749:{\displaystyle \xi (\phi )=\sin ^{-1}\left({\frac {q(\phi )}{q_{\mathrm {p} }}}\right)} 2671: 832:

are small and often appear in series expansions in calculations; they are of the order

569: 8947: 6804:

Legendre, A. M. (1806). "Analyse des triangles tracés sur la surface d'un sphéroïde".

1320: 8786: 8727: 8692: 8608: 8585: 8465: 8460: 8379: 8324: 8302: 7561: 7471: 7146: 7062: 6987: 6907: 6858: 6628: 6446: 6321: 6313: 6272: 5934: 4397: 3226: 2845: 2421: 2381: 1051: 428: 316: 239: 147: 105: 564:.) Newton's result was confirmed by geodetic measurements in the 18th century. (See 9144: 8572: 8352: 7385: 7370: 7356: 7335: 7319: 7298: 7110: 7050: 6975: 6930: 6895: 6846: 6693: 6662: 6376: 5594: 1585: 1331:

since higher-precision results necessitate an ellipsoid model. With this value for

451: 398: 378: 110: 7237: 7114: 5017:

The conformal latitude defines a transformation from the ellipsoid to a sphere of

3656: 607: 7206: 7193: 7176: 6574: 6426: 6312:

use in a similar way to specify the angular position of stars north–south of the

3646: 3325:{\displaystyle \mu (\phi )={\frac {\pi }{2}}{\frac {m(\phi )}{m_{\mathrm {p} }}}} 596: 553: 467: 193: 7993: 7986: 7979: 7972: 2869:, is defined by the radius drawn from the centre of the ellipsoid to that point 1352:

and standard texts it is shown that the distance along a meridian from latitude

1327:

is equal to 6,371 km or 3,959 miles. No higher accuracy is appropriate for

8038: 7464: 7430: 6776: 6391: 6222:. The relation of Cartesian and ellipsoidal coordinates is discussed in Torge. 1072: 324: 323:; and the angle between any one meridian plane and that through Greenwich (the 185: 7054: 6979: 6934: 9133: 7380: 7349: 7312: 7233:

Determination of Latitude by Francis Drake on the Coast of California in 1579

6899: 4417: 1337: 368: 352: 19:

This article is about the geographical reference system. For other uses, see

6154:) with the reference ellipsoid: the necessary condition is that the product 599:

which link WGS84 to the local reference ellipsoid with its associated grid.

9047: 6850: 3340: 1670: 1349: 1177: 1149: 1132: 1128:. "Latitude" (unqualified) should normally refer to the geodetic latitude. 565: 543: 173: 6752: 6692:. Columbus, OH: Dept. of Geodetic Science and Surveying, Ohio State Univ. 6111: 3505:

and the length of the meridian quadrant from the equator to the pole (the

852:. Reference ellipsoids are usually defined by the semi-major axis and the 623:

The shape of an ellipsoid of revolution is determined by the shape of the

8059: 7210: 6666: 6372: 6305: 6288: 6158:

of semi-major axis and eccentricity is the same for both ellipsoids. Let

3572:

Using the rectifying latitude to define a latitude on a sphere of radius

197: 6568: 848:

and 0.0818 respectively. Values for a number of ellipsoids are given in

306:) are defined on a spherical model. The graticule spacing is 10 degrees. 7967: 7863: 7818: 7284: 6451: 6411: 6367: 6309: 6268: 6052: 6019: 5952: 5046:, is used in the development of the ellipsoidal versions of the normal 2476: 644: 515: 501: 432: 336: 332: 243: 7168: 6697: 1079:). There is no standard notation: examples from various texts include 9104: 6441: 6346: 5983:

which is normal to the reference ellipsoid. The geodetic coordinates

1172:

On the sphere the normal passes through the centre and the latitude (

805: 189: 126: 93: 77: 57: 37: 6301:

and is usually a few seconds of arc but it is important in geodesy.

5562:{\displaystyle \psi (\phi )=\operatorname {gd} ^{-1}\chi (\phi )\,.} 583:. In pre-satellite days they were devised to give a good fit to the 7232: 7105: 7077: 6587: 6342: 6325: 3562:{\displaystyle m_{\mathrm {p} }=m\left({\frac {\pi }{2}}\right)\,.} 2371:

A calculator for any latitude is provided by the U.S. Government's

443: 412: 151: 7037: 6962: 6921:

Cayley, A. (1870). "On the geodesic lines on an oblate spheroid".

6882: 6833: 3128:{\displaystyle {\frac {p^{2}}{a^{2}}}+{\frac {z^{2}}{b^{2}}}=1\,.} 2877:) which is the projection parallel to the Earth's axis of a point 2442:. The remaining latitudes are not used in this way; they are used 1639:{\displaystyle m_{\mathrm {p} }=m\left({\frac {\pi }{2}}\right)\,} 335:

has a latitude of 90° North (written 90° N or +90°), and the

8960: 8007: 7772: 7364: 7327: 7290: 6386: 6267:

at a point on the surface. The true vertical, the direction of a

3051: 2128:

The distance in metres (correct to 0.01 metre) between latitudes

801: 652: 636: 628: 624: 580: 456: 328: 251: 92:, run east–west as circles parallel to the equator. Latitude and 81: 45: 6595:. National Imagery and Mapping Agency. p. 3-1. TR8350.2 6543: 6146:) construct an auxiliary ellipsoid which is confocal (same foci 6520:"ISO 19111 Geographic information — Referencing by coordinates" 6284: 4386:{\displaystyle R_{q}=a{\sqrt {\frac {q_{\mathrm {p} }}{2}}}\,.} 3050:, the distance above the equatorial plane, the equation of the 993:

Geodetic coordinates § Geodetic vs. geocentric coordinates

460: 431:. The time variation is discussed more fully in the article on 423:

and the latitude of the polar circles is its complement (90° -

360:

Besides the equator, four other parallels are of significance:

131: 6095:

is the angle between the radius vector and the polar axis and

5633: 635:. The other parameter is usually (1) the polar radius or 6218:. The relation of Cartesian and spherical polars is given in 6023:

Geocentric coordinate related to spherical polar coordinates

2253: 1650: 877: 592: 591:, it has become natural to use reference ellipsoids (such as 584: 122: 73: 69: 36:. The vertical lines from pole to pole are lines of constant 3138:

The Cartesian coordinates of the point are parameterized by

2840: 659:. These parameters are not independent: they are related by 6788:. Special Publication No. 67. US Coast and Geodetic Survey. 3335:

where the meridian distance from the equator to a latitude

1965:

When the latitude difference is 1 degree, corresponding to

1136: 459:

is it possible for the Sun to be directly overhead (at the

411:

The plane of the Earth's orbit about the Sun is called the

158:

as defined in the specification of the ISO 19111 standard.

5484:{\displaystyle y(\phi )={\frac {E}{2\pi }}\psi (\phi )\,.} 587:

over the limited area of a survey but, with the advent of

6714:. National Geospatial-Intelligence Agency. Archived from 6263:) is the angle between the equatorial plane and the true 6230: 3196:{\displaystyle p=a\cos \beta \,,\qquad z=b\sin \beta \,;} 588: 142:

on the reference surface. The latitude of a point on the

7091:

Karney, Charles F. F. (2023). "On auxiliary latitudes".

250:

together with its gravitational field is the science of

7140: 6567:

Newton, Isaac. "Book III Proposition XIX Problem III".

246:

or to determine GPS satellite orbits. The study of the

99:

On its own, the term "latitude" normally refers to the

7217:

Convert decimal degrees into degrees, minutes, seconds

7199:

Convert decimal degrees into degrees, minutes, seconds

7183:'s (NGA) database of foreign geographic feature names. 4396:

An example of the use of the authalic latitude is the

2471: 266:

A perspective view of the Earth showing how latitude (

7187:

Resources for determining your latitude and longitude

6948:

Karney, C. F. F. (2013). "Algorithms for geodesics".

5511: 5434: 5083: 4429: 4341: 4131: 3768: 3676: 3581: 3518: 3352: 3266: 3147: 3063: 2905: 2815: 2686: 2516: 2269: 2177: 2154: 2134: 1992: 1843: 1686: 1598: 1369: 1343: 1200: 1027: 1007: 668: 356:

The orientation of the Earth at the December solstice

292: 272: 109:) to the ellipsoidal surface from the point, and the 16:

Geographic coordinate specifying north–south position

7222:

Distance calculation based on latitude and longitude

6588:

National Imagery and Mapping Agency (23 June 2004).

6544:

The Corporation of Trinity House (10 January 2020).

2439: 986: 7267: 6106: 3615:{\displaystyle R={\frac {2m_{\mathrm {p} }}{\pi }}} 419:. The latitude of the tropical circles is equal to 8401: 6590:"Department of Defense World Geodetic System 1984" 5928: 5561: 5483: 5398: 4987: 4385: 4321: 4111: 3748: 3614: 3561: 3494: 3324: 3195: 3127: 3031: 2829: 2793: 2659: 2360: 2241: 2168: + 0.5 degrees on the WGS84 spheroid is 2160: 2140: 2117: 1849: 1824: 1638: 1554: 1304: 1167: 1160:The length of a degree of latitude depends on the 1033: 1013: 789: 298: 278: 6304:Astronomical latitude is not to be confused with 2252:The variation of this distance with latitude (on 983:illustrations usually exaggerate the flattening. 134:, but the geoid is more accurately modeled by an 9131: 2459: 602: 7134: 5706:Approximate difference from geodetic latitude ( 5417:(units of length or pixels) then the distance, 619:axis to form an oblate ellipsoid of revolution. 347: 7006:"Sur la Construction des Cartes Géographiques" 6014: 5678:, the auxiliary latitudes can be expressed as 5572: 446:when the Sun is overhead at some point of the 310: 8023: 7253: 5498:is closely related to the conformal latitude 200:lighthouse is at 50°39.734′ N 001°35.500′ W. 6867: 6771: 6769: 6657:Osborne, Peter (2013). "Chapters 5,6". 5999:are the latitude and longitude of the point 526: 6570:Philosophiæ Naturalis Principia Mathematica 5634:Numerical comparison of auxiliary latitudes 876:. For example, the defining values for the 549:Philosophiæ Naturalis Principia Mathematica 8030: 8016: 7260: 7246: 7141:Hofmann-Wellenhof, B.; Moritz, H. (2006). 6166:) of the auxiliary ellipsoid. Further let 6138:not on the reference ellipsoid (semi-axes 1125: 513: 499: 9115:Map projection of the tri-axial ellipsoid 8232: 7104: 7036: 6961: 6881: 6832: 6766: 6740: 6738: 6736: 6734: 6732: 6652: 6650: 6648: 6646: 6644: 6209: 5555: 5477: 4379: 4332:and the radius of the sphere is taken as 3555: 3488: 3476: 3189: 3166: 3121: 2841:Parametric latitude (or reduced latitude) 2653: 2357: 2199: 1815: 1717: 1635: 1589:distance from the equator to the pole is 1540: 1417: 783: 257: 7075: 7000: 6803: 6797: 6229: 6225: 6110: 6063:in which the coordinates of a point are 6018: 5951: 3210:because of the form of these equations. 3046:, the distance from the minor axis, and 2844: 2475: 2472:§ Geodetic and geocentric latitudes 2380: 1076: 996: 880:ellipsoid, used by all GPS devices, are 606: 351: 261: 25: 7181:National Geospatial-Intelligence Agency 7127:Holfmann-Wellenfor & Moritz (2006) 7012:(in French). Vol. IV. p. 667. 6656: 6618: 6616: 6614: 6612: 6610: 6573:. Translated by Motte, Andrew. p. 6353:Bowditch's American Practical Navigator 5941: 2849:Definition of the parametric latitude ( 2373:National Geospatial-Intelligence Agency 2256:) is shown in the table along with the 556:ellipsoid. (This article uses the term 9132: 7090: 7022: 6947: 6941: 6920: 6914: 6818: 6812: 6744: 6729: 6688:Rapp, Richard H. (1991). "Chapter 3". 6683: 6681: 6679: 6677: 6641: 5938:ellipsoidal coordinates respectively. 3220: 2465: 2397: 450:. The south polar latitudes below the 76:position of a point on the surface of 9083: 8978: 8880: 8496: 8072: 8011: 7241: 6775: 6622: 6560: 5578:There are two methods of proceeding. 5033: 4403: 2873:on the surrounding sphere (of radius 2480:The definition of geodetic latitude ( 2385:The definition of geodetic latitude ( 1001:The definition of geodetic latitude ( 6687: 6607: 6174:on the auxiliary ellipsoid. The set 3640: 1143: 230:Celestial navigation § Latitude 8881: 6674: 1983:radians, the arc distance is about 438:The figure shows the geometry of a 125:, a surface which approximates the 48:are lines of constant latitude, or 13: 8037: 6566: 6003:on the ellipsoid and the distance 5637: 4398:Albers equal-area conic projection 4367: 4142: 3734: 3600: 3525: 3314: 2271: 2179: 1994: 1605: 1344:Meridian distance on the ellipsoid 1296: 1293: 1290: 1287: 1284: 1281: 1278: 1260: 1257: 1254: 1251: 1248: 1245: 1242: 514: 500: 238:, latitude is determined with the 14: 9161: 7162: 6748:Map Projections: A Working Manual 5674:To first order in the flattening 5066:defined by the lines of constant 987:Geodetic and geocentric latitudes 579:have been used in the history of 9058:Quadrilateralized spherical cube 8738:Quadrilateralized spherical cube 7283: 6216:geographic coordinate conversion 6204:triaxial ellipsoidal coordinates 6192:ellipsoidal-harmonic coordinates 6107:Ellipsoidal-harmonic coordinates 4420:) transformation to the sphere. 2415:Parametric (or reduced) latitude 560:in preference to the older term 217: 7121: 7084: 7069: 7016: 6994: 6493: 5929:Latitude and coordinate systems 3170: 2830:{\displaystyle \phi {-}\theta } 2258:length of a degree of longitude 1168:Meridian distance on the sphere 1156:Length of a degree of longitude 732: 696: 161:Since there are many different 8647:Lambert cylindrical equal-area 8073: 6704: 6581: 6537: 6512: 6480: 6432:International Latitude Service 6170:be the parametric latitude of 5552: 5546: 5521: 5515: 5474: 5468: 5444: 5438: 5386: 5371: 5346: 5340: 5311: 5296: 5271: 5259: 5097: 5091: 5052:Transverse Mercator projection 5028:Transverse Mercator projection 4973: 4958: 4933: 4927: 4877: 4862: 4837: 4825: 4443: 4437: 4102: 4087: 3782: 3776: 3724: 3718: 3686: 3680: 3635:Transverse Mercator projection 3362: 3356: 3304: 3298: 3276: 3270: 3012: 3000: 2915: 2909: 2748: 2735: 2702: 2690: 2630: 2617: 2526: 2520: 1714: 1708: 1699: 1693: 1414: 1403: 1379: 1373: 1210: 1204: 754: 742: 476:Transverse Mercator projection 44:. The circles parallel to the 1: 9095:Interruption (map projection) 8497: 7131:, p.240, eq. (6-6) to (6-10). 7115:10.1080/00396265.2023.2217604 6712:"Length of degree calculator" 6468: 6437:List of countries by latitude 6202:); for a generalization, see 6200:oblate spheroidal coordinates 4416:, gives an angle-preserving ( 2881:on the ellipsoid at latitude 2503:) and the geodetic latitude ( 2148: − 0.5 degrees and 603:The geometry of the ellipsoid 531: 116: 9084: 8733:Lambert azimuthal equal-area 8529:Guyou hemisphere-in-a-square 8519:Adams hemisphere-in-a-square 7228:16th Century Latitude Survey 7076:Orihuela, Sebastián (2013). 6627:(3rd ed.). De Gruyter. 6506: 6473: 6457:Orders of magnitude (length) 6397:Geographic coordinate system 5421:, of a parallel of latitude 3631:equidistant conic projection 2436:geographic coordinate system 1356:to the equator is given by ( 348:Named latitudes on the Earth 156:geographic coordinate system 7: 6671:for LaTeX code and figures. 6546:"1/2020 Needles Lighthouse" 6488:Template:Circle of latitude 6335: 6220:spherical coordinate system 6061:spherical polar coordinates 6015:Spherical polar coordinates 5573:Inverse formulae and series 2484:) and geocentric latitude ( 2389:) and geocentric latitude ( 800:Many other parameters (see 311:The graticule on the sphere 10: 9166: 6277:gravitational acceleration 5945: 3644: 3627:equirectangular projection 3224: 3206:Cayley suggested the term 2469: 1153: 1147: 990: 643:; or (2) the (first) 535: 496: 482: 227: 221: 86:Lines of constant latitude 18: 9090: 9079: 9023: 9006: 8969: 8946: 8893: 8889: 8876: 8836: 8813: 8795: 8755: 8688: 8665: 8617: 8594: 8571: 8562: 8509: 8505: 8492: 8451: 8429: 8388: 8343: 8315: 8288: 8223: 8171: 8123: 8094: 8085: 8081: 8068: 8045: 7282: 7279: 7055:10.1007/s00190-011-0445-3 6980:10.1007/s00190-012-0578-z 6935:10.1080/14786447008640411 6690:Geometric Geodesy, Part I 6382:Degree Confluence Project 6046:is the complement of the 1677:meridian arc is given by 527:Latitude on the ellipsoid 490: 485: 210:For a brief history, see 167:Global Positioning System 21:Latitude (disambiguation) 7209:– info about decimal to 6900:10.1002/asna.18260041601 6808:. 1st semester: 130–161. 6745:Snyder, John P. (1987). 6659:The Mercator Projections 6281:centrifugal acceleration 6162:be the semi-minor axis ( 6115:Ellipsoidal coordinates 6042:The geocentric latitude 5698:takes on the values for 2454:, and the eccentricity, 1034:{\displaystyle \lambda } 963:(eccentricity squared): 493: 488: 483: 299:{\displaystyle \lambda } 8534:Lambert conformal conic 6196:ellipsoidal coordinates 5494:The isometric latitude 934:from which are derived 538:Ellipsoid of revolution 224:Longitude determination 205:planetographic latitude 136:ellipsoid of revolution 8667:Tobler hyperelliptical 8280:Tobler hyperelliptical 8206:Space-oblique Mercator 7078:"Funciones de Latitud" 7002:Lagrange, Joseph-Louis 6851:10.1002/asna.201011352 6318:equatorial coordinates 6275:(the resultant of the 6253: 6210:Coordinate conversions 6131: 6039: 5975:At an arbitrary point 5972: 5647:(positive latitudes), 5643: 5563: 5485: 5400: 4989: 4387: 4323: 4113: 3750: 3655:(after the Greek for " 3616: 3563: 3496: 3326: 3197: 3129: 3033: 2854: 2831: 2795: 2661: 2489: 2394: 2362: 2260:(east–west distance): 2243: 2162: 2142: 2119: 1851: 1826: 1640: 1556: 1306: 1042: 1035: 1015: 920:(inverse flattening): 791: 620: 519: 505: 357: 307: 300: 280: 258:Latitude on the sphere 53: 6548:. Notices to Mariners 6462:World Geodetic System 6417:Great-circle distance 6402:Geographical distance 6291:is accurately known. 6279:(mass-based) and the 6257:Astronomical latitude 6233: 6226:Astronomical latitude 6114: 6022: 5956:Geodetic coordinates 5955: 5641: 5584:fixed-point iteration 5564: 5486: 5401: 5008:Gudermannian function 4990: 4388: 4324: 4114: 3751: 3665:equal-area projection 3617: 3564: 3497: 3327: 3198: 3130: 3034: 2848: 2832: 2796: 2662: 2479: 2384: 2363: 2244: 2163: 2161:{\displaystyle \phi } 2143: 2141:{\displaystyle \phi } 2120: 1852: 1850:{\displaystyle \phi } 1827: 1641: 1557: 1307: 1126:astronomical latitude 1036: 1016: 1014:{\displaystyle \phi } 1000: 887:(equatorial radius): 792: 615:compressed along the 610: 597:datum transformations 518: 504: 355: 301: 281: 279:{\displaystyle \phi } 265: 228:Further information: 184:). It is measured in 29: 9043:Cahill–Keyes M-shape 8903:Chamberlin trimetric 7224:– JavaScript version 7169:GEONets Names Server 6929:(4th ser): 329–340. 6667:10.5281/zenodo.35392 6407:Geomagnetic latitude 6330:ecliptic coordinates 5948:Geodetic coordinates 5942:Geodetic coordinates 5509: 5432: 5425:from the equator is 5081: 4427: 4339: 4129: 3766: 3674: 3579: 3516: 3350: 3264: 3145: 3061: 2903: 2813: 2684: 2514: 2458:. (For inverses see 2267: 2175: 2152: 2132: 1990: 1841: 1684: 1596: 1367: 1198: 1113:. This article uses 1025: 1005: 666: 577:reference ellipsoids 491:Transverse Mercator 290: 270: 236:celestial navigation 163:reference ellipsoids 111:plane of the equator 9140:Circles of latitude 9110:Tissot's indicatrix 9011:Central cylindrical 8652:Smyth equal-surface 8554:Transverse Mercator 8403:General perspective 8158:Smyth equal-surface 8110:Transverse Mercator 7376:Tropic of Capricorn 7344:Tropic of Capricorn 7307:Tropic of Capricorn 7269:Circles of latitude 7047:2011JGeod..85..475K 6972:2013JGeod..87...43K 6892:1825AN......4..241B 6843:2010AN....331..852K 6422:History of latitude 6298:vertical deflection 6083:is the distance of 5711: 5694:where the constant 5048:Mercator projection 5012:Mercator projection 3411: 3233:rectifying latitude 3221:Rectifying latitude 3208:parametric latitude 2859:parametric latitude 2494:geocentric latitude 2466:Geocentric latitude 2418:Rectifying latitude 2411:Geocentric latitude 2404:auxiliary latitudes 2398:Auxiliary latitudes 2284: 2192: 2007: 1578:radius of curvature 1475: 1399: 1162:figure of the Earth 1122:Geographic latitude 1064:Geocentric latitude 850:Figure of the Earth 611:A sphere of radius 472:Mercator projection 448:Tropic of Capricorn 403:66° 34′ (66.57°) S 393:23° 26′ (23.43°) S 389:Tropic of Capricorn 383:23° 26′ (23.43°) N 373:66° 34′ (66.57°) N 248:figure of the Earth 212:History of latitude 190:minutes and seconds 68:that specifies the 9063:Waterman butterfly 8913:Miller cylindrical 8544:Peirce quincuncial 8439:Lambert equal-area 8191:Gall stereographic 7205:2012-11-07 at the 7192:2008-05-19 at the 7175:2008-03-09 at the 7025:Journal of Geodesy 6950:Journal of Geodesy 6806:Mém. Inst. Nat. Fr 6623:Torge, W. (2001). 6363:Circle of latitude 6358:Cardinal direction 6265:vertical direction 6254: 6132: 6040: 5979:consider the line 5973: 5705: 5644: 5604:Lagrange reversion 5559: 5481: 5396: 5394: 5040:isometric latitude 5034:Isometric latitude 4985: 4983: 4410:conformal latitude 4404:Conformal latitude 4383: 4319: 4317: 4109: 4107: 3746: 3612: 3559: 3492: 3397: 3322: 3193: 3125: 3029: 2855: 2853:) on the ellipsoid 2827: 2791: 2672:ellipsoidal height 2657: 2490: 2429:Isometric latitude 2426:Conformal latitude 2395: 2358: 2270: 2239: 2178: 2158: 2138: 2115: 1993: 1847: 1822: 1636: 1576:is the meridional 1552: 1461: 1385: 1302: 1069:spherical latitude 1043: 1031: 1011: 787: 651:; or (3) the 621: 520: 506: 358: 308: 296: 276: 54: 9127: 9126: 9123: 9122: 9075: 9074: 9071: 9070: 9019: 9018: 8872: 8871: 8868: 8867: 8751: 8750: 8488: 8487: 8484: 8483: 8447: 8446: 8335:Lambert conformal 8311: 8310: 8225:Pseudocylindrical 8219: 8218: 8005: 8004: 6447:Natural Area Code 6322:ecliptic latitude 6314:celestial equator 6308:, the coordinate 6273:gravity direction 5935:coordinate system 5926: 5925: 5463: 5227: 5173: 5150: 5137: 4765: 4746: 4732: 4678: 4665: 4610: 4596: 4571: 4518: 4377: 4376: 4294: 4249: 4213: 4166: 4069: 4041: 3952: 3898: 3865: 3740: 3653:authalic latitude 3641:Authalic latitude 3610: 3549: 3472: 3320: 3290: 3227:Rectifying radius 3113: 3086: 2960: 2775: 2422:Authalic latitude 2355: 2352: 2277: 2185: 2113: 2109: 2000: 1963: 1962: 1811: 1653:this distance is 1629: 1536: 1231: 1178:meridian distance 1144:Meridian distance 1052:Geodetic latitude 1021:) and longitude ( 781: 691: 524: 523: 407: 406: 286:) and longitude ( 240:meridian altitude 101:geodetic latitude 9157: 9081: 9080: 9038:Cahill Butterfly 8976: 8975: 8956:Goode homolosine 8891: 8890: 8878: 8877: 8843: 8842:(Mecca or Qibla) 8723:Goode homolosine 8569: 8568: 8507: 8506: 8494: 8493: 8399: 8398: 8394: 8265:Goode homolosine 8230: 8229: 8115:Oblique Mercator 8092: 8091: 8083: 8082: 8070: 8069: 8032: 8025: 8018: 8009: 8008: 7996: 7989: 7982: 7975: 7386:Antarctic Circle 7371:Tropic of Cancer 7359: 7358:Antarctic Circle 7352: 7345: 7338: 7337:Tropic of Cancer 7330: 7322: 7321:Antarctic Circle 7315: 7308: 7301: 7300:Tropic of Cancer 7293: 7287: 7262: 7255: 7248: 7239: 7238: 7179:. access to the 7157: 7156: 7145:(2nd ed.). 7143:Physical Geodesy 7138: 7132: 7129:Physical Geodesy 7125: 7119: 7118: 7108: 7099:(395): 165–180. 7088: 7082: 7081: 7073: 7067: 7066: 7040: 7020: 7014: 7013: 6998: 6992: 6991: 6965: 6945: 6939: 6938: 6918: 6912: 6911: 6885: 6862: 6836: 6816: 6810: 6809: 6801: 6795: 6789: 6787: 6773: 6764: 6763: 6761: 6760: 6742: 6727: 6726: 6724: 6723: 6708: 6702: 6701: 6685: 6672: 6670: 6654: 6639: 6638: 6620: 6605: 6604: 6602: 6600: 6594: 6585: 6579: 6578: 6564: 6558: 6557: 6555: 6553: 6541: 6535: 6534: 6532: 6531: 6516: 6500: 6497: 6491: 6484: 6377:celestial sphere 6287:and stars whose 6262: 6243:Local plumb line 6189: 6173: 6169: 6165: 6161: 6157: 6153: 6149: 6145: 6141: 6137: 6130: 6102: 6098: 6094: 6090: 6087:from the centre 6086: 6082: 6078: 6059:in conventional 6058: 6045: 6038: 6010: 6006: 6002: 5998: 5982: 5978: 5971: 5782: 5769: 5756: 5743: 5730: 5717: 5712: 5709: 5704: 5628: 5595:double precision 5568: 5566: 5565: 5560: 5539: 5538: 5501: 5497: 5490: 5488: 5487: 5482: 5464: 5462: 5451: 5424: 5420: 5416: 5405: 5403: 5402: 5397: 5395: 5367: 5366: 5336: 5335: 5317: 5292: 5291: 5255: 5254: 5236: 5232: 5228: 5226: 5206: 5186: 5174: 5166: 5161: 5157: 5156: 5152: 5151: 5143: 5138: 5130: 5073: 5069: 5061: 5057: 5045: 5005: 4994: 4992: 4991: 4986: 4984: 4980: 4976: 4954: 4953: 4923: 4922: 4893: 4889: 4885: 4884: 4880: 4858: 4857: 4821: 4820: 4789: 4788: 4770: 4766: 4758: 4753: 4749: 4748: 4747: 4739: 4737: 4733: 4731: 4711: 4691: 4684: 4680: 4679: 4671: 4666: 4658: 4637: 4636: 4615: 4611: 4603: 4598: 4597: 4589: 4587: 4583: 4582: 4581: 4576: 4572: 4570: 4550: 4530: 4523: 4519: 4517: 4500: 4483: 4468: 4467: 4415: 4392: 4390: 4389: 4384: 4378: 4372: 4371: 4370: 4360: 4359: 4351: 4350: 4328: 4326: 4325: 4320: 4318: 4308: 4307: 4295: 4290: 4289: 4288: 4272: 4258: 4254: 4250: 4248: 4237: 4226: 4214: 4212: 4204: 4203: 4202: 4186: 4171: 4167: 4159: 4147: 4146: 4145: 4118: 4116: 4115: 4110: 4108: 4083: 4082: 4070: 4065: 4064: 4063: 4047: 4042: 4040: 4033: 4032: 4023: 4022: 4006: 3996: 3992: 3991: 3990: 3969: 3961: 3957: 3953: 3951: 3931: 3911: 3899: 3897: 3889: 3888: 3887: 3871: 3866: 3864: 3857: 3856: 3847: 3846: 3830: 3820: 3816: 3815: 3814: 3793: 3755: 3753: 3752: 3747: 3745: 3741: 3739: 3738: 3737: 3727: 3713: 3704: 3703: 3662: 3621: 3619: 3618: 3613: 3611: 3606: 3605: 3604: 3603: 3589: 3568: 3566: 3565: 3560: 3554: 3550: 3542: 3530: 3529: 3528: 3501: 3499: 3498: 3493: 3487: 3475: 3474: 3473: 3465: 3459: 3455: 3454: 3443: 3442: 3433: 3432: 3410: 3405: 3396: 3392: 3391: 3390: 3338: 3331: 3329: 3328: 3323: 3321: 3319: 3318: 3317: 3307: 3293: 3291: 3283: 3256: 3254: 3253: 3250: 3247: 3246: 3238: 3202: 3200: 3199: 3194: 3134: 3132: 3131: 3126: 3114: 3112: 3111: 3102: 3101: 3092: 3087: 3085: 3084: 3075: 3074: 3065: 3049: 3045: 3038: 3036: 3035: 3030: 3028: 3024: 2991: 2990: 2975: 2971: 2961: 2959: 2958: 2943: 2933: 2932: 2895: 2884: 2880: 2876: 2872: 2868: 2863:reduced latitude 2852: 2836: 2834: 2833: 2828: 2823: 2807: 2800: 2798: 2797: 2792: 2790: 2786: 2776: 2774: 2763: 2756: 2755: 2730: 2720: 2719: 2666: 2664: 2663: 2658: 2652: 2648: 2638: 2637: 2608: 2607: 2592: 2588: 2578: 2574: 2573: 2572: 2544: 2543: 2506: 2502: 2487: 2483: 2457: 2453: 2392: 2388: 2367: 2365: 2364: 2359: 2356: 2354: 2353: 2345: 2344: 2335: 2334: 2319: 2317: 2316: 2306: 2289: 2283: 2278: 2275: 2248: 2246: 2245: 2240: 2191: 2186: 2183: 2167: 2165: 2164: 2159: 2147: 2145: 2144: 2139: 2124: 2122: 2121: 2116: 2114: 2112: 2111: 2110: 2102: 2100: 2096: 2089: 2088: 2079: 2078: 2057: 2056: 2046: 2045: 2041: 2040: 2039: 2012: 2006: 2001: 1998: 1982: 1980: 1979: 1976: 1973: 1972: 1904:107.550 km 1893:111.320 km 1882: 1881: 1880: 1869: 1868: 1867: 1856: 1854: 1853: 1848: 1835: 1834: 1831: 1829: 1828: 1823: 1814: 1813: 1812: 1804: 1798: 1794: 1787: 1786: 1777: 1776: 1755: 1751: 1750: 1749: 1664: 1662: 1658: 1645: 1643: 1642: 1637: 1634: 1630: 1622: 1610: 1609: 1608: 1586:quarter meridian 1575: 1561: 1559: 1558: 1553: 1551: 1539: 1538: 1537: 1529: 1523: 1519: 1518: 1507: 1506: 1497: 1496: 1474: 1469: 1460: 1456: 1455: 1454: 1428: 1413: 1398: 1393: 1359: 1355: 1334: 1330: 1326: 1318: 1311: 1309: 1308: 1303: 1301: 1300: 1299: 1265: 1264: 1263: 1232: 1230: 1229: 1217: 1190: 1175: 1116: 1112: 1103: 1094: 1090: 1086: 1082: 1058: 1040: 1038: 1037: 1032: 1020: 1018: 1017: 1012: 978: 977: 974: 971: 968: 962: 955: 953: 949: 946: 941:(polar radius): 940: 929: 928: 925: 919: 918: 916: 915: 910: 907: 897: 895: 892: 886: 875: 874: 872: 871: 866: 863: 847: 845: 844: 841: 838: 831: 827: 823: 819: 815: 811: 796: 794: 793: 788: 782: 780: 779: 764: 728: 727: 706: 705: 692: 687: 676: 658: 650: 642: 634: 486:Normal Mercator 481: 480: 452:Antarctic Circle 422: 418: 399:Antarctic Circle 379:Tropic of Cancer 365: 364: 305: 303: 302: 297: 285: 283: 282: 277: 183: 179: 9165: 9164: 9160: 9159: 9158: 9156: 9155: 9154: 9130: 9129: 9128: 9119: 9086: 9067: 9015: 9002: 8965: 8942: 8928:Van der Grinten 8885: 8883:By construction 8864: 8841: 8840: 8832: 8809: 8791: 8772:Equirectangular 8758: 8747: 8684: 8661: 8657:Trystan Edwards 8613: 8590: 8558: 8501: 8480: 8453:Pseudoazimuthal 8443: 8425: 8392: 8391: 8384: 8339: 8307: 8303:Winkel I and II 8284: 8215: 8196:Gall isographic 8186:Equirectangular 8167: 8163:Trystan Edwards 8119: 8077: 8064: 8041: 8036: 8006: 8001: 8000: 7999: 7998: 7994: 7991: 7987: 7984: 7980: 7977: 7973: 7970: 7965: 7960: 7955: 7950: 7943: 7938: 7933: 7928: 7918: 7913: 7908: 7903: 7898: 7891: 7886: 7881: 7876: 7866: 7861: 7856: 7851: 7846: 7841: 7836: 7831: 7826: 7821: 7816: 7811: 7806: 7801: 7796: 7791: 7786: 7781: 7776: 7769: 7764: 7759: 7754: 7749: 7744: 7739: 7734: 7729: 7724: 7719: 7714: 7709: 7704: 7699: 7694: 7689: 7684: 7679: 7674: 7669: 7664: 7659: 7654: 7649: 7644: 7639: 7634: 7629: 7624: 7619: 7614: 7609: 7604: 7599: 7594: 7589: 7584: 7579: 7574: 7569: 7564: 7559: 7554: 7549: 7544: 7539: 7534: 7529: 7524: 7519: 7514: 7509: 7504: 7499: 7494: 7489: 7484: 7479: 7474: 7469: 7462: 7457: 7452: 7445: 7440: 7435: 7428: 7423: 7418: 7411: 7406: 7401: 7388: 7383: 7378: 7373: 7368: 7361: 7357: 7354: 7350: 7347: 7343: 7340: 7336: 7333: 7328: 7324: 7320: 7317: 7313: 7310: 7306: 7303: 7299: 7296: 7291: 7275: 7266: 7207:Wayback Machine 7194:Wayback Machine 7177:Wayback Machine 7165: 7160: 7153: 7139: 7135: 7126: 7122: 7089: 7085: 7074: 7070: 7021: 7017: 6999: 6995: 6946: 6942: 6919: 6915: 6863: 6827:(86): 241–254. 6817: 6813: 6802: 6798: 6785: 6777:Adams, Oscar S. 6774: 6767: 6758: 6756: 6743: 6730: 6721: 6719: 6710: 6709: 6705: 6686: 6675: 6655: 6642: 6635: 6621: 6608: 6598: 6596: 6592: 6586: 6582: 6565: 6561: 6551: 6549: 6542: 6538: 6529: 6527: 6518: 6517: 6513: 6509: 6504: 6503: 6498: 6494: 6485: 6481: 6476: 6471: 6466: 6427:Horse latitudes 6338: 6260: 6252: 6228: 6212: 6175: 6171: 6167: 6163: 6159: 6155: 6151: 6147: 6143: 6139: 6135: 6116: 6109: 6100: 6096: 6092: 6088: 6084: 6080: 6064: 6056: 6043: 6024: 6017: 6008: 6004: 6000: 5984: 5980: 5976: 5957: 5950: 5944: 5931: 5774: 5773: 5761: 5760: 5748: 5747: 5735: 5734: 5722: 5721: 5715: 5707: 5636: 5607: 5575: 5531: 5527: 5510: 5507: 5506: 5499: 5495: 5455: 5450: 5433: 5430: 5429: 5422: 5418: 5414: 5393: 5392: 5359: 5355: 5328: 5324: 5315: 5314: 5284: 5280: 5247: 5243: 5234: 5233: 5207: 5187: 5185: 5181: 5165: 5142: 5129: 5128: 5124: 5117: 5113: 5100: 5084: 5082: 5079: 5078: 5071: 5067: 5059: 5055: 5043: 5036: 4999: 4982: 4981: 4946: 4942: 4915: 4911: 4910: 4906: 4891: 4890: 4850: 4846: 4813: 4809: 4808: 4804: 4797: 4793: 4781: 4777: 4768: 4767: 4757: 4738: 4712: 4692: 4690: 4686: 4685: 4670: 4657: 4656: 4652: 4645: 4641: 4629: 4625: 4613: 4612: 4602: 4588: 4577: 4551: 4531: 4529: 4525: 4524: 4501: 4484: 4482: 4478: 4477: 4473: 4472: 4460: 4456: 4446: 4430: 4428: 4425: 4424: 4413: 4406: 4366: 4365: 4361: 4358: 4346: 4342: 4340: 4337: 4336: 4316: 4315: 4300: 4296: 4284: 4280: 4273: 4271: 4256: 4255: 4238: 4227: 4225: 4221: 4205: 4198: 4194: 4187: 4185: 4172: 4158: 4154: 4141: 4140: 4136: 4132: 4130: 4127: 4126: 4106: 4105: 4075: 4071: 4059: 4055: 4048: 4046: 4028: 4024: 4018: 4014: 4007: 3986: 3982: 3975: 3971: 3970: 3968: 3959: 3958: 3932: 3912: 3910: 3906: 3890: 3883: 3879: 3872: 3870: 3852: 3848: 3842: 3838: 3831: 3810: 3806: 3799: 3795: 3794: 3792: 3785: 3769: 3767: 3764: 3763: 3733: 3732: 3728: 3714: 3712: 3708: 3696: 3692: 3675: 3672: 3671: 3660: 3649: 3647:Authalic radius 3643: 3599: 3598: 3594: 3590: 3588: 3580: 3577: 3576: 3541: 3537: 3524: 3523: 3519: 3517: 3514: 3513: 3480: 3464: 3460: 3447: 3438: 3434: 3428: 3424: 3417: 3413: 3412: 3406: 3401: 3386: 3382: 3375: 3371: 3351: 3348: 3347: 3336: 3313: 3312: 3308: 3294: 3292: 3282: 3265: 3262: 3261: 3251: 3248: 3244: 3243: 3242: 3240: 3236: 3229: 3223: 3146: 3143: 3142: 3107: 3103: 3097: 3093: 3091: 3080: 3076: 3070: 3066: 3064: 3062: 3059: 3058: 3047: 3043: 2999: 2995: 2983: 2979: 2954: 2950: 2942: 2941: 2937: 2925: 2921: 2904: 2901: 2900: 2886: 2882: 2878: 2874: 2870: 2866: 2850: 2843: 2819: 2814: 2811: 2810: 2805: 2764: 2751: 2747: 2731: 2729: 2728: 2724: 2712: 2708: 2685: 2682: 2681: 2633: 2629: 2616: 2612: 2600: 2596: 2568: 2564: 2557: 2553: 2552: 2548: 2536: 2532: 2515: 2512: 2511: 2504: 2500: 2485: 2481: 2474: 2468: 2455: 2451: 2400: 2390: 2386: 2340: 2336: 2330: 2326: 2318: 2312: 2308: 2307: 2290: 2288: 2279: 2274: 2268: 2265: 2264: 2187: 2182: 2176: 2173: 2172: 2153: 2150: 2149: 2133: 2130: 2129: 2101: 2084: 2080: 2074: 2070: 2063: 2059: 2058: 2052: 2048: 2047: 2035: 2031: 2024: 2020: 2013: 2011: 2002: 1997: 1991: 1988: 1987: 1977: 1974: 1970: 1969: 1968: 1966: 1956:111.694 km 1948:28.902 km 1945:111.618 km 1937:55.800 km 1934:111.412 km 1926:78.847 km 1923:111.132 km 1915:96.486 km 1912:110.852 km 1901:110.649 km 1890:110.574 km 1879: 1876: 1875: 1874: 1872: 1866: 1863: 1862: 1861: 1859: 1842: 1839: 1838: 1803: 1799: 1782: 1778: 1772: 1768: 1761: 1757: 1756: 1745: 1741: 1734: 1730: 1685: 1682: 1681: 1660: 1656: 1654: 1621: 1617: 1604: 1603: 1599: 1597: 1594: 1593: 1566: 1544: 1528: 1524: 1511: 1502: 1498: 1492: 1488: 1481: 1477: 1476: 1470: 1465: 1450: 1446: 1439: 1435: 1421: 1406: 1394: 1389: 1368: 1365: 1364: 1357: 1353: 1346: 1332: 1328: 1324: 1316: 1277: 1276: 1272: 1241: 1240: 1236: 1225: 1221: 1216: 1199: 1196: 1195: 1181: 1173: 1170: 1158: 1152: 1146: 1114: 1111: 1105: 1102: 1096: 1092: 1088: 1084: 1080: 1067:(also known as 1056: 1026: 1023: 1022: 1006: 1003: 1002: 995: 989: 975: 972: 969: 966: 964: 958: 951: 947: 944: 942: 938: 926: 923: 921: 911: 908: 905: 904: 902: 901: 893: 890: 888: 884: 867: 864: 861: 860: 858: 857: 842: 839: 836: 835: 833: 829: 825: 821: 817: 813: 809: 775: 771: 763: 723: 719: 701: 697: 677: 675: 667: 664: 663: 656: 648: 640: 637:semi-minor axis 632: 629:semi-major axis 605: 575:Many different 540: 534: 529: 468:map projections 420: 416: 350: 313: 291: 288: 287: 271: 268: 267: 260: 232: 226: 220: 194:decimal degrees 181: 177: 119: 24: 17: 12: 11: 5: 9163: 9153: 9152: 9147: 9142: 9125: 9124: 9121: 9120: 9118: 9117: 9112: 9107: 9102: 9097: 9091: 9088: 9087: 9077: 9076: 9073: 9072: 9069: 9068: 9066: 9065: 9060: 9055: 9050: 9045: 9040: 9035: 9029: 9027: 9021: 9020: 9017: 9016: 9014: 9013: 9007: 9004: 9003: 9001: 9000: 8995: 8990: 8984: 8982: 8973: 8967: 8966: 8964: 8963: 8958: 8952: 8950: 8944: 8943: 8941: 8940: 8935: 8930: 8925: 8920: 8915: 8910: 8908:Kavrayskiy VII 8905: 8899: 8897: 8887: 8886: 8874: 8873: 8870: 8869: 8866: 8865: 8863: 8862: 8857: 8852: 8846: 8844: 8838:Retroazimuthal 8834: 8833: 8831: 8830: 8825: 8819: 8817: 8811: 8810: 8808: 8807: 8801: 8799: 8793: 8792: 8790: 8789: 8784: 8779: 8774: 8769: 8763: 8761: 8757:Equidistant in 8753: 8752: 8749: 8748: 8746: 8745: 8740: 8735: 8730: 8725: 8720: 8715: 8710: 8705: 8700: 8695: 8689: 8686: 8685: 8683: 8682: 8677: 8671: 8669: 8663: 8662: 8660: 8659: 8654: 8649: 8644: 8639: 8634: 8629: 8623: 8621: 8615: 8614: 8612: 8611: 8606: 8600: 8598: 8592: 8591: 8589: 8588: 8583: 8577: 8575: 8566: 8560: 8559: 8557: 8556: 8551: 8546: 8541: 8536: 8531: 8526: 8521: 8515: 8513: 8503: 8502: 8490: 8489: 8486: 8485: 8482: 8481: 8479: 8478: 8473: 8468: 8463: 8457: 8455: 8449: 8448: 8445: 8444: 8442: 8441: 8436: 8430: 8427: 8426: 8424: 8423: 8418: 8413: 8407: 8405: 8396: 8386: 8385: 8383: 8382: 8377: 8376: 8375: 8370: 8360: 8355: 8349: 8347: 8341: 8340: 8338: 8337: 8332: 8327: 8321: 8319: 8313: 8312: 8309: 8308: 8306: 8305: 8300: 8295: 8293:Kavrayskiy VII 8289: 8286: 8285: 8283: 8282: 8277: 8272: 8267: 8262: 8257: 8252: 8247: 8242: 8236: 8234: 8227: 8221: 8220: 8217: 8216: 8214: 8213: 8208: 8203: 8198: 8193: 8188: 8183: 8178: 8172: 8169: 8168: 8166: 8165: 8160: 8155: 8150: 8145: 8140: 8135: 8129: 8127: 8121: 8120: 8118: 8117: 8112: 8107: 8101: 8099: 8089: 8079: 8078: 8066: 8065: 8063: 8062: 8057: 8052: 8046: 8043: 8042: 8039:Map projection 8035: 8034: 8027: 8020: 8012: 8003: 8002: 7992: 7985: 7978: 7971: 7966: 7961: 7956: 7951: 7944: 7939: 7934: 7929: 7919: 7914: 7909: 7904: 7899: 7892: 7887: 7882: 7877: 7867: 7862: 7857: 7852: 7847: 7842: 7837: 7832: 7827: 7822: 7817: 7812: 7807: 7802: 7797: 7792: 7787: 7782: 7777: 7770: 7765: 7760: 7755: 7750: 7745: 7740: 7735: 7730: 7725: 7720: 7715: 7710: 7705: 7700: 7695: 7690: 7685: 7680: 7675: 7670: 7665: 7660: 7655: 7650: 7645: 7640: 7635: 7630: 7625: 7620: 7615: 7610: 7605: 7600: 7595: 7590: 7585: 7580: 7575: 7570: 7565: 7560: 7555: 7550: 7545: 7540: 7535: 7530: 7525: 7520: 7515: 7510: 7505: 7500: 7495: 7490: 7485: 7480: 7475: 7470: 7463: 7458: 7453: 7446: 7441: 7436: 7429: 7424: 7419: 7412: 7407: 7402: 7389: 7384: 7379: 7374: 7369: 7362: 7355: 7348: 7341: 7334: 7325: 7318: 7311: 7304: 7297: 7288: 7281: 7280: 7277: 7276: 7265: 7264: 7257: 7250: 7242: 7236: 7235: 7230: 7225: 7219: 7214: 7196: 7184: 7164: 7163:External links 7161: 7159: 7158: 7151: 7133: 7120: 7083: 7068: 7031:(8): 475–485. 7015: 6993: 6940: 6913: 6876:(8): 852–861. 6811: 6796: 6765: 6728: 6703: 6673: 6640: 6633: 6606: 6580: 6559: 6536: 6510: 6508: 6505: 6502: 6501: 6492: 6478: 6477: 6475: 6472: 6470: 6467: 6465: 6464: 6459: 6454: 6449: 6444: 6439: 6434: 6429: 6424: 6419: 6414: 6409: 6404: 6399: 6394: 6392:Geodetic datum 6389: 6384: 6379: 6370: 6365: 6360: 6355: 6350: 6347:mean sea level 6339: 6337: 6334: 6271:, is also the 6251: 6250: 6247: 6244: 6241: 6238: 6234: 6227: 6224: 6211: 6208: 6108: 6105: 6016: 6013: 5946:Main article: 5943: 5940: 5930: 5927: 5924: 5923: 5920: 5917: 5914: 5911: 5908: 5904: 5903: 5900: 5897: 5894: 5891: 5888: 5884: 5883: 5880: 5877: 5874: 5871: 5868: 5864: 5863: 5860: 5857: 5854: 5851: 5848: 5844: 5843: 5840: 5837: 5834: 5831: 5828: 5824: 5823: 5820: 5817: 5814: 5811: 5808: 5804: 5803: 5800: 5797: 5794: 5791: 5788: 5784: 5783: 5770: 5757: 5744: 5731: 5718: 5635: 5632: 5631: 5630: 5600: 5599: 5598: 5590:root finding. 5588:Newton–Raphson 5574: 5571: 5570: 5569: 5558: 5554: 5551: 5548: 5545: 5542: 5537: 5534: 5530: 5526: 5523: 5520: 5517: 5514: 5492: 5491: 5480: 5476: 5473: 5470: 5467: 5461: 5458: 5454: 5449: 5446: 5443: 5440: 5437: 5407: 5406: 5391: 5388: 5385: 5382: 5379: 5376: 5373: 5370: 5365: 5362: 5358: 5354: 5351: 5348: 5345: 5342: 5339: 5334: 5331: 5327: 5323: 5320: 5318: 5316: 5313: 5310: 5307: 5304: 5301: 5298: 5295: 5290: 5287: 5283: 5279: 5276: 5273: 5270: 5267: 5264: 5261: 5258: 5253: 5250: 5246: 5242: 5239: 5237: 5235: 5231: 5225: 5222: 5219: 5216: 5213: 5210: 5205: 5202: 5199: 5196: 5193: 5190: 5184: 5180: 5177: 5172: 5169: 5164: 5160: 5155: 5149: 5146: 5141: 5136: 5133: 5127: 5123: 5120: 5116: 5112: 5109: 5106: 5103: 5101: 5099: 5096: 5093: 5090: 5087: 5086: 5058:and longitude 5035: 5032: 4996: 4995: 4979: 4975: 4972: 4969: 4966: 4963: 4960: 4957: 4952: 4949: 4945: 4941: 4938: 4935: 4932: 4929: 4926: 4921: 4918: 4914: 4909: 4905: 4902: 4899: 4896: 4894: 4892: 4888: 4883: 4879: 4876: 4873: 4870: 4867: 4864: 4861: 4856: 4853: 4849: 4845: 4842: 4839: 4836: 4833: 4830: 4827: 4824: 4819: 4816: 4812: 4807: 4803: 4800: 4796: 4792: 4787: 4784: 4780: 4776: 4773: 4771: 4769: 4764: 4761: 4756: 4752: 4745: 4742: 4736: 4730: 4727: 4724: 4721: 4718: 4715: 4710: 4707: 4704: 4701: 4698: 4695: 4689: 4683: 4677: 4674: 4669: 4664: 4661: 4655: 4651: 4648: 4644: 4640: 4635: 4632: 4628: 4624: 4621: 4618: 4616: 4614: 4609: 4606: 4601: 4595: 4592: 4586: 4580: 4575: 4569: 4566: 4563: 4560: 4557: 4554: 4549: 4546: 4543: 4540: 4537: 4534: 4528: 4522: 4516: 4513: 4510: 4507: 4504: 4499: 4496: 4493: 4490: 4487: 4481: 4476: 4471: 4466: 4463: 4459: 4455: 4452: 4449: 4447: 4445: 4442: 4439: 4436: 4433: 4432: 4405: 4402: 4394: 4393: 4382: 4375: 4369: 4364: 4357: 4354: 4349: 4345: 4330: 4329: 4314: 4311: 4306: 4303: 4299: 4293: 4287: 4283: 4279: 4276: 4270: 4267: 4264: 4261: 4259: 4257: 4253: 4247: 4244: 4241: 4236: 4233: 4230: 4224: 4220: 4217: 4211: 4208: 4201: 4197: 4193: 4190: 4184: 4181: 4178: 4175: 4173: 4170: 4165: 4162: 4157: 4153: 4150: 4144: 4139: 4135: 4134: 4120: 4119: 4104: 4101: 4098: 4095: 4092: 4089: 4086: 4081: 4078: 4074: 4068: 4062: 4058: 4054: 4051: 4045: 4039: 4036: 4031: 4027: 4021: 4017: 4013: 4010: 4005: 4002: 3999: 3995: 3989: 3985: 3981: 3978: 3974: 3967: 3964: 3962: 3960: 3956: 3950: 3947: 3944: 3941: 3938: 3935: 3930: 3927: 3924: 3921: 3918: 3915: 3909: 3905: 3902: 3896: 3893: 3886: 3882: 3878: 3875: 3869: 3863: 3860: 3855: 3851: 3845: 3841: 3837: 3834: 3829: 3826: 3823: 3819: 3813: 3809: 3805: 3802: 3798: 3791: 3788: 3786: 3784: 3781: 3778: 3775: 3772: 3771: 3757: 3756: 3744: 3736: 3731: 3726: 3723: 3720: 3717: 3711: 3707: 3702: 3699: 3695: 3691: 3688: 3685: 3682: 3679: 3642: 3639: 3623: 3622: 3609: 3602: 3597: 3593: 3587: 3584: 3570: 3569: 3558: 3553: 3548: 3545: 3540: 3536: 3533: 3527: 3522: 3507:polar distance 3503: 3502: 3491: 3486: 3483: 3479: 3471: 3468: 3463: 3458: 3453: 3450: 3446: 3441: 3437: 3431: 3427: 3423: 3420: 3416: 3409: 3404: 3400: 3395: 3389: 3385: 3381: 3378: 3374: 3370: 3367: 3364: 3361: 3358: 3355: 3333: 3332: 3316: 3311: 3306: 3303: 3300: 3297: 3289: 3286: 3281: 3278: 3275: 3272: 3269: 3222: 3219: 3204: 3203: 3192: 3188: 3185: 3182: 3179: 3176: 3173: 3169: 3165: 3162: 3159: 3156: 3153: 3150: 3136: 3135: 3124: 3120: 3117: 3110: 3106: 3100: 3096: 3090: 3083: 3079: 3073: 3069: 3040: 3039: 3027: 3023: 3020: 3017: 3014: 3011: 3008: 3005: 3002: 2998: 2994: 2989: 2986: 2982: 2978: 2974: 2970: 2967: 2964: 2957: 2953: 2949: 2946: 2940: 2936: 2931: 2928: 2924: 2920: 2917: 2914: 2911: 2908: 2842: 2839: 2826: 2822: 2818: 2802: 2801: 2789: 2785: 2782: 2779: 2773: 2770: 2767: 2762: 2759: 2754: 2750: 2746: 2743: 2740: 2737: 2734: 2727: 2723: 2718: 2715: 2711: 2707: 2704: 2701: 2698: 2695: 2692: 2689: 2668: 2667: 2656: 2651: 2647: 2644: 2641: 2636: 2632: 2628: 2625: 2622: 2619: 2615: 2611: 2606: 2603: 2599: 2595: 2591: 2587: 2584: 2581: 2577: 2571: 2567: 2563: 2560: 2556: 2551: 2547: 2542: 2539: 2535: 2531: 2528: 2525: 2522: 2519: 2467: 2464: 2431: 2430: 2427: 2424: 2419: 2416: 2413: 2402:There are six 2399: 2396: 2369: 2368: 2351: 2348: 2343: 2339: 2333: 2329: 2325: 2322: 2315: 2311: 2305: 2302: 2299: 2296: 2293: 2287: 2282: 2273: 2250: 2249: 2238: 2235: 2232: 2229: 2226: 2223: 2220: 2217: 2214: 2211: 2208: 2205: 2202: 2198: 2195: 2190: 2181: 2157: 2137: 2126: 2125: 2108: 2105: 2099: 2095: 2092: 2087: 2083: 2077: 2073: 2069: 2066: 2062: 2055: 2051: 2044: 2038: 2034: 2030: 2027: 2023: 2019: 2016: 2010: 2005: 1996: 1961: 1960: 1959:0.000 km 1957: 1954: 1950: 1949: 1946: 1943: 1939: 1938: 1935: 1932: 1928: 1927: 1924: 1921: 1917: 1916: 1913: 1910: 1906: 1905: 1902: 1899: 1895: 1894: 1891: 1888: 1884: 1883: 1877: 1870: 1864: 1857: 1846: 1833: 1832: 1821: 1818: 1810: 1807: 1802: 1797: 1793: 1790: 1785: 1781: 1775: 1771: 1767: 1764: 1760: 1754: 1748: 1744: 1740: 1737: 1733: 1729: 1726: 1723: 1720: 1716: 1713: 1710: 1707: 1704: 1701: 1698: 1695: 1692: 1689: 1647: 1646: 1633: 1628: 1625: 1620: 1616: 1613: 1607: 1602: 1563: 1562: 1550: 1547: 1543: 1535: 1532: 1527: 1522: 1517: 1514: 1510: 1505: 1501: 1495: 1491: 1487: 1484: 1480: 1473: 1468: 1464: 1459: 1453: 1449: 1445: 1442: 1438: 1434: 1431: 1427: 1424: 1420: 1416: 1412: 1409: 1405: 1402: 1397: 1392: 1388: 1384: 1381: 1378: 1375: 1372: 1345: 1342: 1323:of the Earth. 1313: 1312: 1298: 1295: 1292: 1289: 1286: 1283: 1280: 1275: 1271: 1268: 1262: 1259: 1256: 1253: 1250: 1247: 1244: 1239: 1235: 1228: 1224: 1220: 1215: 1212: 1209: 1206: 1203: 1180:is denoted by 1169: 1166: 1148:Main article: 1145: 1142: 1119: 1118: 1109: 1100: 1073:3D polar angle 1060: 1030: 1010: 988: 985: 980: 979: 956: 932: 931: 899: 798: 797: 786: 778: 774: 770: 767: 762: 759: 756: 753: 750: 747: 744: 741: 738: 735: 731: 726: 722: 718: 715: 712: 709: 704: 700: 695: 690: 686: 683: 680: 674: 671: 604: 601: 546:published the 536:Main article: 533: 530: 528: 525: 522: 521: 512: 507: 498: 495: 494: 492: 489: 487: 484: 409: 408: 405: 404: 401: 395: 394: 391: 385: 384: 381: 375: 374: 371: 349: 346: 325:Prime Meridian 312: 309: 295: 275: 259: 256: 219: 216: 127:mean sea level 118: 115: 15: 9: 6: 4: 3: 2: 9162: 9151: 9148: 9146: 9143: 9141: 9138: 9137: 9135: 9116: 9113: 9111: 9108: 9106: 9103: 9101: 9098: 9096: 9093: 9092: 9089: 9082: 9078: 9064: 9061: 9059: 9056: 9054: 9051: 9049: 9046: 9044: 9041: 9039: 9036: 9034: 9031: 9030: 9028: 9026: 9022: 9012: 9009: 9008: 9005: 8999: 8998:Stereographic 8996: 8994: 8991: 8989: 8986: 8985: 8983: 8981: 8977: 8974: 8972: 8968: 8962: 8959: 8957: 8954: 8953: 8951: 8949: 8945: 8939: 8938:Winkel tripel 8936: 8934: 8931: 8929: 8926: 8924: 8921: 8919: 8918:Natural Earth 8916: 8914: 8911: 8909: 8906: 8904: 8901: 8900: 8898: 8896: 8892: 8888: 8884: 8879: 8875: 8861: 8858: 8856: 8853: 8851: 8848: 8847: 8845: 8839: 8835: 8829: 8826: 8824: 8821: 8820: 8818: 8816: 8812: 8806: 8803: 8802: 8800: 8798: 8794: 8788: 8785: 8783: 8780: 8778: 8775: 8773: 8770: 8768: 8765: 8764: 8762: 8760: 8754: 8744: 8741: 8739: 8736: 8734: 8731: 8729: 8726: 8724: 8721: 8719: 8716: 8714: 8711: 8709: 8706: 8704: 8701: 8699: 8698:Briesemeister 8696: 8694: 8691: 8690: 8687: 8681: 8678: 8676: 8673: 8672: 8670: 8668: 8664: 8658: 8655: 8653: 8650: 8648: 8645: 8643: 8640: 8638: 8635: 8633: 8630: 8628: 8625: 8624: 8622: 8620: 8616: 8610: 8607: 8605: 8602: 8601: 8599: 8597: 8593: 8587: 8584: 8582: 8579: 8578: 8576: 8574: 8570: 8567: 8565: 8561: 8555: 8552: 8550: 8549:Stereographic 8547: 8545: 8542: 8540: 8537: 8535: 8532: 8530: 8527: 8525: 8522: 8520: 8517: 8516: 8514: 8512: 8508: 8504: 8500: 8495: 8491: 8477: 8476:Winkel tripel 8474: 8472: 8469: 8467: 8464: 8462: 8459: 8458: 8456: 8454: 8450: 8440: 8437: 8435: 8432: 8431: 8428: 8422: 8421:Stereographic 8419: 8417: 8414: 8412: 8409: 8408: 8406: 8404: 8400: 8397: 8395: 8387: 8381: 8378: 8374: 8371: 8369: 8366: 8365: 8364: 8361: 8359: 8356: 8354: 8351: 8350: 8348: 8346: 8345:Pseudoconical 8342: 8336: 8333: 8331: 8328: 8326: 8323: 8322: 8320: 8318: 8314: 8304: 8301: 8299: 8296: 8294: 8291: 8290: 8287: 8281: 8278: 8276: 8273: 8271: 8268: 8266: 8263: 8261: 8258: 8256: 8253: 8251: 8248: 8246: 8243: 8241: 8238: 8237: 8235: 8231: 8228: 8226: 8222: 8212: 8209: 8207: 8204: 8202: 8199: 8197: 8194: 8192: 8189: 8187: 8184: 8182: 8179: 8177: 8174: 8173: 8170: 8164: 8161: 8159: 8156: 8154: 8151: 8149: 8146: 8144: 8141: 8139: 8136: 8134: 8131: 8130: 8128: 8126: 8122: 8116: 8113: 8111: 8108: 8106: 8103: 8102: 8100: 8097: 8093: 8090: 8088: 8084: 8080: 8076: 8071: 8067: 8061: 8058: 8056: 8053: 8051: 8048: 8047: 8044: 8040: 8033: 8028: 8026: 8021: 8019: 8014: 8013: 8010: 7997: 7990: 7983: 7976: 7969: 7964: 7959: 7954: 7949: 7948: 7942: 7937: 7932: 7927: 7926: 7922: 7917: 7912: 7907: 7902: 7897: 7896: 7890: 7885: 7880: 7875: 7874: 7870: 7865: 7860: 7855: 7850: 7845: 7840: 7835: 7830: 7825: 7820: 7815: 7810: 7805: 7800: 7795: 7790: 7785: 7780: 7775: 7774: 7768: 7763: 7758: 7753: 7748: 7743: 7738: 7733: 7728: 7723: 7718: 7713: 7708: 7703: 7698: 7693: 7688: 7683: 7678: 7673: 7668: 7663: 7658: 7653: 7648: 7643: 7638: 7633: 7628: 7623: 7618: 7613: 7608: 7603: 7598: 7593: 7588: 7583: 7578: 7573: 7568: 7563: 7558: 7553: 7548: 7543: 7538: 7533: 7528: 7523: 7518: 7513: 7508: 7503: 7498: 7493: 7488: 7483: 7478: 7473: 7468: 7467: 7461: 7456: 7451: 7450: 7444: 7439: 7434: 7433: 7427: 7422: 7417: 7416: 7410: 7405: 7400: 7399: 7396: 7393: 7387: 7382: 7381:Arctic Circle 7377: 7372: 7367: 7366: 7360: 7353: 7351:Arctic Circle 7346: 7339: 7332: 7331: 7323: 7316: 7314:Arctic Circle 7309: 7302: 7295: 7294: 7286: 7278: 7274: 7270: 7263: 7258: 7256: 7251: 7249: 7244: 7243: 7240: 7234: 7231: 7229: 7226: 7223: 7220: 7218: 7215: 7212: 7208: 7204: 7200: 7197: 7195: 7191: 7188: 7185: 7182: 7178: 7174: 7170: 7167: 7166: 7154: 7152:3-211-33544-7 7148: 7144: 7137: 7130: 7124: 7116: 7112: 7107: 7102: 7098: 7094: 7093:Survey Review 7087: 7079: 7072: 7064: 7060: 7056: 7052: 7048: 7044: 7039: 7034: 7030: 7026: 7019: 7011: 7007: 7003: 6997: 6989: 6985: 6981: 6977: 6973: 6969: 6964: 6959: 6955: 6951: 6944: 6936: 6932: 6928: 6924: 6917: 6909: 6905: 6901: 6897: 6893: 6889: 6884: 6879: 6875: 6871: 6870:Astron. Nachr 6866: 6860: 6856: 6852: 6848: 6844: 6840: 6835: 6830: 6826: 6822: 6821:Astron. Nachr 6815: 6807: 6800: 6793: 6784: 6783: 6778: 6772: 6770: 6755:on 2008-05-16 6754: 6750: 6749: 6741: 6739: 6737: 6735: 6733: 6718:on 2012-12-11 6717: 6713: 6707: 6699: 6695: 6691: 6684: 6682: 6680: 6678: 6668: 6664: 6660: 6653: 6651: 6649: 6647: 6645: 6636: 6634:3-11-017072-8 6630: 6626: 6619: 6617: 6615: 6613: 6611: 6591: 6584: 6576: 6572: 6571: 6563: 6547: 6540: 6525: 6521: 6515: 6511: 6496: 6489: 6483: 6479: 6463: 6460: 6458: 6455: 6453: 6450: 6448: 6445: 6443: 6440: 6438: 6435: 6433: 6430: 6428: 6425: 6423: 6420: 6418: 6415: 6413: 6410: 6408: 6405: 6403: 6400: 6398: 6395: 6393: 6390: 6388: 6385: 6383: 6380: 6378: 6374: 6371: 6369: 6366: 6364: 6361: 6359: 6356: 6354: 6351: 6348: 6344: 6341: 6340: 6333: 6331: 6327: 6323: 6319: 6315: 6311: 6307: 6302: 6300: 6299: 6292: 6290: 6286: 6282: 6278: 6274: 6270: 6266: 6258: 6248: 6245: 6242: 6239: 6236: 6235: 6232: 6223: 6221: 6217: 6207: 6205: 6201: 6197: 6193: 6187: 6183: 6179: 6128: 6124: 6120: 6113: 6104: 6076: 6072: 6068: 6062: 6055: 6054: 6049: 6036: 6032: 6028: 6021: 6012: 5996: 5992: 5988: 5969: 5965: 5961: 5954: 5949: 5939: 5936: 5921: 5918: 5915: 5912: 5909: 5906: 5905: 5901: 5898: 5895: 5892: 5889: 5886: 5885: 5881: 5878: 5875: 5872: 5869: 5866: 5865: 5861: 5858: 5855: 5852: 5849: 5846: 5845: 5841: 5838: 5835: 5832: 5829: 5826: 5825: 5821: 5818: 5815: 5812: 5809: 5806: 5805: 5801: 5798: 5795: 5792: 5789: 5786: 5785: 5781: 5777: 5771: 5768: 5764: 5758: 5755: 5751: 5745: 5742: 5738: 5732: 5729: 5725: 5719: 5714: 5713: 5703: 5701: 5697: 5693: 5689: 5685: 5681: 5677: 5672: 5670: 5666: 5662: 5658: 5654: 5650: 5640: 5626: 5622: 5618: 5614: 5610: 5605: 5601: 5596: 5592: 5591: 5589: 5585: 5581: 5580: 5579: 5556: 5549: 5543: 5540: 5535: 5532: 5528: 5524: 5518: 5512: 5505: 5504: 5503: 5478: 5471: 5465: 5459: 5456: 5452: 5447: 5441: 5435: 5428: 5427: 5426: 5412: 5389: 5383: 5380: 5377: 5374: 5368: 5363: 5360: 5356: 5352: 5349: 5343: 5337: 5332: 5329: 5325: 5321: 5319: 5308: 5305: 5302: 5299: 5293: 5288: 5285: 5281: 5277: 5274: 5268: 5265: 5262: 5256: 5251: 5248: 5244: 5240: 5238: 5229: 5223: 5220: 5217: 5214: 5211: 5208: 5203: 5200: 5197: 5194: 5191: 5188: 5182: 5178: 5175: 5170: 5167: 5162: 5158: 5153: 5147: 5144: 5139: 5134: 5131: 5125: 5121: 5118: 5114: 5110: 5107: 5104: 5102: 5094: 5088: 5077: 5076: 5075: 5070:and constant 5065: 5053: 5049: 5041: 5031: 5029: 5024: 5020: 5015: 5013: 5009: 5003: 4977: 4970: 4967: 4964: 4961: 4955: 4950: 4947: 4943: 4939: 4936: 4930: 4924: 4919: 4916: 4912: 4907: 4903: 4900: 4897: 4895: 4886: 4881: 4874: 4871: 4868: 4865: 4859: 4854: 4851: 4847: 4843: 4840: 4834: 4831: 4828: 4822: 4817: 4814: 4810: 4805: 4801: 4798: 4794: 4790: 4785: 4782: 4778: 4774: 4772: 4762: 4759: 4754: 4750: 4743: 4740: 4734: 4728: 4725: 4722: 4719: 4716: 4713: 4708: 4705: 4702: 4699: 4696: 4693: 4687: 4681: 4675: 4672: 4667: 4662: 4659: 4653: 4649: 4646: 4642: 4638: 4633: 4630: 4626: 4622: 4619: 4617: 4607: 4604: 4599: 4593: 4590: 4584: 4578: 4573: 4567: 4564: 4561: 4558: 4555: 4552: 4547: 4544: 4541: 4538: 4535: 4532: 4526: 4520: 4514: 4511: 4508: 4505: 4502: 4497: 4494: 4491: 4488: 4485: 4479: 4474: 4469: 4464: 4461: 4457: 4453: 4450: 4448: 4440: 4434: 4423: 4422: 4421: 4419: 4411: 4401: 4399: 4380: 4373: 4362: 4355: 4352: 4347: 4343: 4335: 4334: 4333: 4312: 4309: 4304: 4301: 4297: 4291: 4285: 4281: 4277: 4274: 4268: 4265: 4262: 4260: 4251: 4245: 4242: 4239: 4234: 4231: 4228: 4222: 4218: 4215: 4209: 4206: 4199: 4195: 4191: 4188: 4182: 4179: 4176: 4174: 4168: 4163: 4160: 4155: 4151: 4148: 4137: 4125: 4124: 4123: 4099: 4096: 4093: 4090: 4084: 4079: 4076: 4072: 4066: 4060: 4056: 4052: 4049: 4043: 4037: 4034: 4029: 4025: 4019: 4015: 4011: 4008: 4003: 4000: 3997: 3993: 3987: 3983: 3979: 3976: 3972: 3965: 3963: 3954: 3948: 3945: 3942: 3939: 3936: 3933: 3928: 3925: 3922: 3919: 3916: 3913: 3907: 3903: 3900: 3894: 3891: 3884: 3880: 3876: 3873: 3867: 3861: 3858: 3853: 3849: 3843: 3839: 3835: 3832: 3827: 3824: 3821: 3817: 3811: 3807: 3803: 3800: 3796: 3789: 3787: 3779: 3773: 3762: 3761: 3760: 3742: 3729: 3721: 3715: 3709: 3705: 3700: 3697: 3693: 3689: 3683: 3677: 3670: 3669: 3668: 3667:to a sphere. 3666: 3658: 3654: 3648: 3638: 3636: 3632: 3628: 3607: 3595: 3591: 3585: 3582: 3575: 3574: 3573: 3556: 3551: 3546: 3543: 3538: 3534: 3531: 3520: 3512: 3511: 3510: 3508: 3489: 3484: 3481: 3477: 3469: 3466: 3461: 3456: 3451: 3448: 3444: 3439: 3435: 3429: 3425: 3421: 3418: 3414: 3407: 3402: 3398: 3393: 3387: 3383: 3379: 3376: 3372: 3368: 3365: 3359: 3353: 3346: 3345: 3344: 3342: 3309: 3301: 3295: 3287: 3284: 3279: 3273: 3267: 3260: 3259: 3258: 3234: 3228: 3218: 3216: 3211: 3209: 3190: 3186: 3183: 3180: 3177: 3174: 3171: 3167: 3163: 3160: 3157: 3154: 3151: 3148: 3141: 3140: 3139: 3122: 3118: 3115: 3108: 3104: 3098: 3094: 3088: 3081: 3077: 3071: 3067: 3057: 3056: 3055: 3053: 3025: 3021: 3018: 3015: 3009: 3006: 3003: 2996: 2992: 2987: 2984: 2980: 2976: 2972: 2968: 2965: 2962: 2955: 2951: 2947: 2944: 2938: 2934: 2929: 2926: 2922: 2918: 2912: 2906: 2899: 2898: 2897: 2893: 2889: 2864: 2860: 2847: 2838: 2824: 2820: 2816: 2787: 2783: 2780: 2777: 2771: 2768: 2765: 2760: 2757: 2752: 2744: 2741: 2738: 2732: 2725: 2721: 2716: 2713: 2709: 2705: 2699: 2696: 2693: 2687: 2680: 2679: 2678: 2676: 2673: 2654: 2649: 2645: 2642: 2639: 2634: 2626: 2623: 2620: 2613: 2609: 2604: 2601: 2597: 2593: 2589: 2585: 2582: 2579: 2575: 2569: 2565: 2561: 2558: 2554: 2549: 2545: 2540: 2537: 2533: 2529: 2523: 2517: 2510: 2509: 2508: 2497: 2495: 2478: 2473: 2463: 2461: 2448: 2445: 2441: 2438:as discussed 2437: 2428: 2425: 2423: 2420: 2417: 2414: 2412: 2409: 2408: 2407: 2405: 2383: 2379: 2376: 2374: 2349: 2346: 2341: 2337: 2331: 2327: 2323: 2320: 2313: 2309: 2303: 2300: 2297: 2294: 2291: 2285: 2280: 2263: 2262: 2261: 2259: 2255: 2236: 2233: 2230: 2227: 2224: 2221: 2218: 2215: 2212: 2209: 2206: 2203: 2200: 2196: 2193: 2188: 2171: 2170: 2169: 2155: 2135: 2106: 2103: 2097: 2093: 2090: 2085: 2081: 2075: 2071: 2067: 2064: 2060: 2053: 2049: 2042: 2036: 2032: 2028: 2025: 2021: 2017: 2014: 2008: 2003: 1986: 1985: 1984: 1958: 1955: 1952: 1951: 1947: 1944: 1941: 1940: 1936: 1933: 1930: 1929: 1925: 1922: 1919: 1918: 1914: 1911: 1908: 1907: 1903: 1900: 1897: 1896: 1892: 1889: 1886: 1885: 1871: 1858: 1844: 1837: 1836: 1819: 1816: 1808: 1805: 1800: 1795: 1791: 1788: 1783: 1779: 1773: 1769: 1765: 1762: 1758: 1752: 1746: 1742: 1738: 1735: 1731: 1727: 1724: 1721: 1718: 1711: 1705: 1702: 1696: 1690: 1687: 1680: 1679: 1678: 1676: 1672: 1666: 1652: 1631: 1626: 1623: 1618: 1614: 1611: 1600: 1592: 1591: 1590: 1588: 1587: 1581: 1579: 1573: 1569: 1548: 1545: 1541: 1533: 1530: 1525: 1520: 1515: 1512: 1508: 1503: 1499: 1493: 1489: 1485: 1482: 1478: 1471: 1466: 1462: 1457: 1451: 1447: 1443: 1440: 1436: 1432: 1429: 1425: 1422: 1418: 1410: 1407: 1400: 1395: 1390: 1386: 1382: 1376: 1370: 1363: 1362: 1361: 1351: 1341: 1339: 1338:nautical mile 1322: 1273: 1269: 1266: 1237: 1233: 1226: 1222: 1218: 1213: 1207: 1201: 1194: 1193: 1192: 1188: 1184: 1179: 1165: 1163: 1157: 1151: 1141: 1138: 1134: 1129: 1127: 1123: 1108: 1099: 1078: 1074: 1070: 1066: 1065: 1061: 1054: 1053: 1049: 1048: 1047: 1028: 1008: 999: 994: 984: 961: 957: 937: 936: 935: 914: 900: 883: 882: 881: 879: 870: 855: 851: 807: 803: 784: 776: 772: 768: 765: 760: 757: 751: 748: 745: 739: 736: 733: 729: 724: 720: 716: 713: 710: 707: 702: 698: 693: 688: 684: 681: 678: 672: 669: 662: 661: 660: 654: 646: 638: 630: 626: 618: 614: 609: 600: 598: 594: 590: 586: 582: 578: 573: 571: 567: 563: 559: 555: 551: 550: 545: 539: 517: 511: 508: 503: 497: 479: 477: 473: 469: 464: 462: 458: 453: 449: 445: 441: 440:cross-section 436: 434: 430: 426: 414: 402: 400: 397: 396: 392: 390: 387: 386: 382: 380: 377: 376: 372: 370: 369:Arctic Circle 367: 366: 363: 362: 361: 354: 345: 341: 338: 334: 330: 326: 322: 318: 293: 273: 264: 255: 253: 249: 245: 241: 237: 231: 225: 218:Determination 215: 213: 208: 206: 201: 199: 195: 191: 187: 175: 170: 168: 164: 159: 157: 154:constitute a 153: 149: 145: 141: 137: 133: 128: 124: 114: 112: 108: 107: 102: 97: 95: 91: 87: 83: 79: 75: 71: 67: 63: 59: 51: 47: 43: 39: 35: 34: 28: 22: 9099: 8993:Orthographic 8524:Gauss–Krüger 8416:Orthographic 8211:Web Mercator 8105:Gauss–Krüger 7946: 7923: 7894: 7871: 7771: 7465: 7448: 7431: 7414: 7390: 7363: 7326: 7289: 7142: 7136: 7128: 7123: 7096: 7092: 7086: 7071: 7028: 7024: 7018: 7009: 6996: 6956:(1): 43–55. 6953: 6949: 6943: 6926: 6922: 6916: 6873: 6869: 6865:Translation: 6864: 6824: 6820: 6814: 6805: 6799: 6791: 6781: 6757:. 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(See also 5001: 4997: 4409: 4407: 4395: 4331: 4121: 3758: 3652: 3650: 3624: 3571: 3504: 3341:Meridian arc 3334: 3232: 3230: 3212: 3207: 3205: 3137: 3041: 2891: 2887: 2862: 2858: 2856: 2803: 2674: 2669: 2498: 2493: 2491: 2449: 2443: 2432: 2403: 2401: 2377: 2370: 2251: 2127: 1964: 1674: 1671:Meridian arc 1667: 1648: 1584: 1582: 1571: 1567: 1564: 1360:in radians) 1350:Meridian arc 1347: 1319:denotes the 1314: 1186: 1182: 1171: 1159: 1150:Meridian arc 1133:Eiffel Tower 1130: 1121: 1120: 1106: 1097: 1071:, after the 1068: 1062: 1050: 1044: 981: 959: 933: 912: 868: 856:flattening, 853: 799: 653:eccentricity 622: 616: 612: 574: 566:Meridian arc 561: 557: 547: 544:Isaac Newton 541: 509: 465: 437: 424: 410: 359: 342: 314: 233: 209: 202: 171: 160: 143: 120: 104: 100: 98: 89: 61: 55: 49: 41: 32: 8971:Perspective 8759:some aspect 8743:Strebe 1995 8718:Equal Earth 8637:Gall–Peters 8619:Cylindrical 8434:Equidistant 8330:Equidistant 8260:Equal Earth 8143:Gall–Peters 8087:Cylindrical 7213:conversion. 7211:sexagesimal 6373:Declination 6310:astronomers 6306:declination 6289:declination 6190:define the 6048:polar angle 3663:, gives an 3217:, Karney). 1321:mean radius 244:theodolites 198:The Needles 9150:Navigation 9134:Categories 9033:AuthaGraph 9025:Polyhedral 8895:Compromise 8823:Loximuthal 8815:Loxodromic 8777:Sinusoidal 8627:Balthasart 8604:Sinusoidal 8581:Sinusoidal 8564:Equal-area 8275:Sinusoidal 8233:Equal-area 8133:Balthasart 8125:Equal-area 8098:-conformal 8075:By surface 7106:2212.05818 6759:2017-09-02 6722:2011-02-08 6698:1811/24333 6530:2022-01-16 6469:References 6452:Navigation 6412:Geotagging 6368:Colatitude 6269:plumb line 6194:or simply 6053:colatitude 5772:Geocentric 5746:Rectifying 5720:Parametric 3645:See also: 3225:See also: 2470:See also: 1154:See also: 991:See also: 645:flattening 532:Ellipsoids 433:axial tilt 337:South Pole 333:North Pole 222:See also: 117:Background 66:coordinate 9105:Longitude 8933:Wagner VI 8782:Two-point 8713:Eckert VI 8708:Eckert IV 8703:Eckert II 8680:Mollweide 8675:Collignon 8642:Hobo–Dyer 8596:Bottomley 8511:Conformal 8499:By metric 8390:Azimuthal 8363:Polyconic 8358:Bottomley 8298:Wagner VI 8270:Mollweide 8255:Eckert VI 8250:Eckert IV 8245:Eckert II 8240:Collignon 8148:Hobo–Dyer 7273:meridians 7063:118619524 7038:1002.1417 6988:119310141 6963:1109.4448 6923:Phil. Mag 6908:118630614 6883:0908.1824 6859:118760590 6834:0908.1824 6507:Citations 6474:Footnotes 6442:Longitude 6246:Continent 6240:Ellipsoid 5759:Conformal 5597:accuracy. 5550:ϕ 5544:χ 5541:⁡ 5533:− 5519:ϕ 5513:ψ 5472:ϕ 5466:ψ 5460:π 5442:ϕ 5384:ϕ 5381:⁡ 5369:⁡ 5361:− 5350:− 5344:ϕ 5338:⁡ 5330:− 5309:ϕ 5306:⁡ 5294:⁡ 5286:− 5275:− 5269:ϕ 5266:⁡ 5257:⁡ 5249:− 5224:ϕ 5221:⁡ 5204:ϕ 5201:⁡ 5192:− 5179:⁡ 5145:ϕ 5132:π 5122:⁡ 5111:⁡ 5095:ϕ 5089:ψ 5064:graticule 5019:arbitrary 4971:ϕ 4968:⁡ 4956:⁡ 4948:− 4937:− 4931:ϕ 4925:⁡ 4917:− 4904:⁡ 4875:ϕ 4872:⁡ 4860:⁡ 4852:− 4841:− 4835:ϕ 4832:⁡ 4823:⁡ 4815:− 4802:⁡ 4791:⁡ 4783:− 4760:π 4755:− 4729:ϕ 4726:⁡ 4709:ϕ 4706:⁡ 4697:− 4673:π 4660:ϕ 4650:⁡ 4639:⁡ 4631:− 4605:π 4600:− 4568:ϕ 4565:⁡ 4548:ϕ 4545:⁡ 4536:− 4515:ϕ 4512:⁡ 4506:− 4498:ϕ 4495:⁡ 4470:⁡ 4462:− 4441:ϕ 4435:χ 4418:conformal 4310:⁡ 4302:− 4278:− 4232:− 4219:⁡ 4192:− 4183:− 4161:π 4100:ϕ 4097:⁡ 4085:⁡ 4077:− 4053:− 4038:ϕ 4035:⁡ 4012:− 4004:ϕ 4001:⁡ 3980:− 3949:ϕ 3946:⁡ 3929:ϕ 3926:⁡ 3917:− 3904:⁡ 3877:− 3868:− 3862:ϕ 3859:⁡ 3836:− 3828:ϕ 3825:⁡ 3804:− 3780:ϕ 3722:ϕ 3706:⁡ 3698:− 3684:ϕ 3678:ξ 3657:same area 3608:π 3544:π 3482:ϕ 3462:− 3449:ϕ 3445:⁡ 3422:− 3408:ϕ 3399:∫ 3380:− 3360:ϕ 3302:ϕ 3285:π 3274:ϕ 3268:μ 3257:radians: 3187:β 3184:⁡ 3164:β 3161:⁡ 3022:ϕ 3019:⁡ 3007:− 2993:⁡ 2985:− 2969:ϕ 2966:⁡ 2948:− 2935:⁡ 2927:− 2913:ϕ 2907:β 2825:θ 2821:− 2817:ϕ 2784:ϕ 2781:⁡ 2742:− 2722:⁡ 2714:− 2694:ϕ 2688:θ 2646:ϕ 2643:⁡ 2624:− 2610:⁡ 2602:− 2586:ϕ 2583:⁡ 2562:− 2546:⁡ 2538:− 2524:ϕ 2518:θ 2350:ϕ 2347:⁡ 2324:− 2314:∘ 2304:ϕ 2301:⁡ 2292:π 2272:Δ 2237:ϕ 2231:⁡ 2219:ϕ 2213:⁡ 2204:− 2180:Δ 2156:ϕ 2136:ϕ 2094:ϕ 2091:⁡ 2068:− 2054:∘ 2029:− 2015:π 1995:Δ 1845:ϕ 1820:ϕ 1817:δ 1801:− 1792:ϕ 1789:⁡ 1766:− 1739:− 1722:ϕ 1719:δ 1712:ϕ 1697:ϕ 1688:δ 1624:π 1546:ϕ 1526:− 1513:ϕ 1509:⁡ 1486:− 1472:ϕ 1463:∫ 1444:− 1423:ϕ 1408:ϕ 1396:ϕ 1387:∫ 1377:ϕ 1274:ϕ 1238:ϕ 1227:∘ 1219:π 1208:ϕ 1164:assumed. 1029:λ 1009:ϕ 896:.0 m 806:ellipsoid 769:− 749:− 717:− 682:− 558:ellipsoid 321:meridians 294:λ 274:ϕ 140:graticule 94:longitude 90:parallels 78:the Earth 58:geography 50:parallels 42:meridians 38:longitude 33:graticule 9100:Latitude 9085:See also 9048:Dymaxion 8988:Gnomonic 8923:Robinson 8828:Mercator 8805:Gnomonic 8797:Gnomonic 8632:Behrmann 8539:Mercator 8411:Gnomonic 8393:(planar) 8368:American 8138:Behrmann 8096:Mercator 7203:Archived 7190:Archived 7173:Archived 7004:(1779). 6779:(1921). 6599:25 April 6343:Altitude 6336:See also 6326:ecliptic 5882:−10.02′ 5862:−11.55′ 5733:Authalic 5409:For the 5050:and the 3485:′ 3452:′ 3339:is (see 3215:Vincenty 1663: km 1549:′ 1516:′ 1426:′ 1411:′ 570:triaxial 562:spheroid 542:In 1687 474:and the 444:solstice 413:ecliptic 62:latitude 30:Earth's 9145:Geodesy 8961:HEALPix 8860:Littrow 8471:Wiechel 8373:Chinese 8317:Conical 8181:Central 8176:Cassini 8153:Lambert 8050:History 7365:Equator 7329:Equator 7292:Equator 7043:Bibcode 6968:Bibcode 6888:Bibcode 6839:Bibcode 6625:Geodesy 6387:Geodesy 5902:−5.79′ 5879:−10.01′ 5859:−11.54′ 5842:−9.98′ 5822:−5.76′ 5006:is the 3255:⁠ 3241:⁠ 3052:ellipse 2375:(NGA). 2207:559.822 2201:132.954 1981:⁠ 1967:⁠ 954: m 930:exactly 922:298.257 917:⁠ 903:⁠ 898:exactly 873:⁠ 859:⁠ 854:inverse 846:⁠ 834:⁠ 824:. Both 802:ellipse 625:ellipse 581:geodesy 457:tropics 329:Equator 252:geodesy 186:degrees 82:Equator 46:equator 8980:Planar 8948:Hybrid 8855:Hammer 8787:Werner 8728:Hammer 8693:Albers 8609:Werner 8586:Werner 8466:Hammer 8461:Aitoff 8380:Werner 8325:Albers 8201:Miller 8060:Portal 7149: 7061: 7010:Oevres 6986: 6906: 6857: 6631: 6552:24 May 6285:zenith 6079:where 5922:0.00′ 5899:−5.78′ 5896:−4.34′ 5893:−3.86′ 5890:−2.89′ 5876:−7.51′ 5873:−6.67′ 5870:−5.00′ 5856:−8.66′ 5853:−7.70′ 5850:−5.77′ 5839:−9.98′ 5836:−7.49′ 5833:−6.66′ 5830:−5.00′ 5819:−5.76′ 5816:−4.32′ 5813:−3.84′ 5810:−2.88′ 5802:0.00′ 5642:inline 5411:normal 4998:where 3759:where 2804:where 2507:) is: 1565:where 1315:where 554:oblate 461:zenith 152:height 148:normal 144:actual 132:sphere 106:normal 8850:Craig 8767:Conic 8573:Bonne 8353:Bonne 7995:45x90 7988:45x90 7981:45x90 7974:45x90 7101:arXiv 7059:S2CID 7033:arXiv 6984:S2CID 6958:arXiv 6904:S2CID 6878:arXiv 6855:S2CID 6829:arXiv 6786:(PDF) 6593:(PDF) 6328:(see 6316:(see 6249:Geoid 6237:Ocean 5919:0.00′ 5916:0.00′ 5913:0.00′ 5910:0.00′ 5799:0.00′ 5796:0.00′ 5793:0.00′ 5790:0.00′ 5690:sin 2 5023:small 3509:) is 2460:below 2440:below 2254:WGS84 2225:1.175 1675:small 1651:WGS84 1191:then 1077:below 965:0.006 878:WGS84 593:WGS84 585:geoid 429:epoch 317:poles 123:geoid 88:, or 74:south 70:north 64:is a 40:, or 9053:ISEA 8055:List 7767:170° 7762:160° 7757:140° 7752:130° 7747:110° 7742:100° 7707:170° 7702:160° 7697:140° 7692:130° 7687:110° 7682:100° 7647:175° 7642:165° 7637:155° 7632:145° 7627:135° 7622:125° 7617:115° 7612:105° 7557:175° 7552:165° 7547:155° 7542:145° 7537:135° 7532:125° 7527:115° 7522:105° 7466:180° 7460:150° 7455:120° 7432:180° 7426:150° 7421:120° 7147:ISBN 6792:Note 6629:ISBN 6601:2020 6554:2020 6142:and 5702:= . 5586:and 5357:tanh 5282:tanh 5245:sinh 5038:The 4944:tanh 4848:tanh 4811:sinh 4799:sinh 4408:The 4298:tanh 4122:and 4073:tanh 3659:"), 3651:The 3231:The 3054:is: 2857:The 2492:The 2444:only 2276:long 1878:long 1659:.965 1649:For 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6101:P' 6093:θ′ 6091:, 6073:′, 6065:P( 6057:θ′ 6033:′, 6025:P( 6009:PN 6005:PN 5985:P( 5981:PN 5958:P( 5787:0° 5778:− 5765:− 5752:− 5739:− 5726:− 5710:) 5688:Cf 5686:− 5682:= 5667:≤ 5663:≤ 5659:≤ 5655:≤ 5651:≤ 5623:+ 5615:- 5529:gd 5502:: 5326:gd 5176:ln 5108:ln 5042:, 4913:gd 4901:gd 4412:, 4400:. 4216:ln 3901:ln 3637:. 3343:) 3235:, 2865:, 2677:: 2393:). 1887:0° 1665:. 1655:10 1580:. 1104:, 1095:, 1093:ϕ′ 1091:, 1087:, 1083:, 976:14 952:25 816:, 812:, 804:, 655:, 647:, 639:, 631:, 510:\ 435:. 254:. 214:. 188:, 113:. 84:. 60:, 8031:e 8024:t 8017:v 7925:S 7873:N 7398:E 7392:W 7261:e 7254:t 7247:v 7155:. 7117:. 7113:: 7103:: 7080:. 7065:. 7053:: 7045:: 7035:: 6990:. 6978:: 6970:: 6960:: 6937:. 6933:: 6910:. 6898:: 6890:: 6880:: 6861:. 6849:: 6841:: 6831:: 6825:4 6790:( 6762:. 6725:. 6700:. 6696:: 6669:. 6665:: 6637:. 6603:. 6577:. 6556:. 6533:. 6490:. 6349:) 6345:( 6261:Φ 6259:( 6188:) 6186:λ 6184:, 6182:β 6180:, 6178:u 6176:( 6172:P 6168:β 6160:u 6148:F 6136:P 6129:) 6127:λ 6125:, 6123:β 6121:, 6119:u 6097:λ 6089:O 6085:P 6081:r 6077:) 6075:λ 6071:θ 6069:, 6067:r 6044:θ 6037:) 6035:λ 6031:θ 6029:, 6027:r 6001:N 5997:) 5995:h 5993:, 5991:λ 5989:, 5987:ɸ 5977:P 5970:) 5968:h 5966:, 5964:λ 5962:, 5960:ɸ 5780:ϕ 5776:θ 5767:ϕ 5763:χ 5754:ϕ 5750:μ 5741:ϕ 5737:ξ 5728:ϕ 5724:β 5716:ϕ 5708:ϕ 5700:ζ 5696:C 5692:ϕ 5684:ϕ 5680:ζ 5676:f 5669:ϕ 5665:β 5661:ξ 5657:μ 5653:χ 5649:θ 5627:) 5625:b 5621:a 5617:b 5613:a 5609:n 5557:. 5553:) 5547:( 5536:1 5525:= 5522:) 5516:( 5500:χ 5496:ψ 5479:. 5475:) 5469:( 5457:2 5453:E 5448:= 5445:) 5439:( 5436:y 5423:ϕ 5419:y 5415:E 5390:. 5387:) 5375:e 5372:( 5364:1 5353:e 5347:) 5341:( 5333:1 5322:= 5312:) 5300:e 5297:( 5289:1 5278:e 5272:) 5260:( 5252:1 5241:= 5230:] 5215:e 5212:+ 5209:1 5195:e 5189:1 5183:[ 5171:2 5168:e 5163:+ 5159:] 5154:) 5148:2 5140:+ 5135:4 5126:( 5115:[ 5105:= 5098:) 5092:( 5072:λ 5068:ψ 5060:λ 5056:ψ 5044:ψ 5004:) 5002:x 4978:] 4974:) 4962:e 4959:( 4951:1 4940:e 4934:) 4928:( 4920:1 4908:[ 4898:= 4887:] 4882:) 4878:) 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3710:( 3701:1 3690:= 3687:) 3681:( 3661:ξ 3601:p 3596:m 3592:2 3586:= 3583:R 3557:. 3552:) 3547:2 3539:( 3535:m 3532:= 3526:p 3521:m 3490:, 3478:d 3470:2 3467:3 3457:) 3440:2 3430:2 3426:e 3419:1 3415:( 3403:0 3394:) 3388:2 3384:e 3377:1 3373:( 3369:a 3366:= 3363:) 3357:( 3354:m 3337:ϕ 3315:p 3310:m 3305:) 3299:( 3296:m 3288:2 3280:= 3277:) 3271:( 3252:2 3249:/ 3245:π 3237:μ 3191:; 3178:b 3175:= 3172:z 3168:, 3155:a 3152:= 3149:p 3123:. 3119:1 3116:= 3109:2 3105:b 3099:2 3095:z 3089:+ 3082:2 3078:a 3072:2 3068:p 3048:z 3044:p 3026:) 3013:) 3010:f 3004:1 3001:( 2997:( 2988:1 2977:= 2973:) 2956:2 2952:e 2945:1 2939:( 2930:1 2919:= 2916:) 2910:( 2894:) 2892:ϕ 2890:( 2888:u 2883:ϕ 2879:P 2875:a 2871:Q 2867:β 2851:β 2806:N 2788:) 2772:h 2769:+ 2766:N 2761:h 2758:+ 2753:2 2749:) 2745:f 2739:1 2736:( 2733:N 2726:( 2717:1 2706:= 2703:) 2700:h 2697:, 2691:( 2675:h 2655:. 2650:) 2635:2 2631:) 2627:f 2621:1 2618:( 2614:( 2605:1 2594:= 2590:) 2576:) 2570:2 2566:e 2559:1 2555:( 2550:( 2541:1 2530:= 2527:) 2521:( 2505:ϕ 2501:θ 2488:) 2486:θ 2482:ϕ 2456:e 2452:a 2391:θ 2387:ϕ 2342:2 2332:2 2328:e 2321:1 2295:a 2286:= 2281:1 2234:4 2222:+ 2216:2 2194:= 2189:1 2107:2 2104:3 2098:) 2086:2 2076:2 2072:e 2065:1 2061:( 2043:) 2037:2 2033:e 2026:1 2022:( 2018:a 2009:= 2004:1 1975:/ 1971:π 1873:Δ 1860:Δ 1809:2 1806:3 1796:) 1784:2 1774:2 1770:e 1763:1 1759:( 1753:) 1747:2 1743:e 1736:1 1732:( 1728:a 1725:= 1715:) 1709:( 1706:M 1703:= 1700:) 1694:( 1691:m 1632:) 1627:2 1619:( 1615:m 1612:= 1606:p 1601:m 1574:) 1572:ϕ 1570:( 1568:M 1542:d 1534:2 1531:3 1521:) 1504:2 1494:2 1490:e 1483:1 1479:( 1467:0 1458:) 1452:2 1448:e 1441:1 1437:( 1433:a 1430:= 1419:d 1415:) 1404:( 1401:M 1391:0 1383:= 1380:) 1374:( 1371:m 1358:ϕ 1354:ϕ 1333:R 1329:R 1325:R 1317:R 1297:s 1294:n 1291:a 1288:i 1285:d 1282:a 1279:r 1270:R 1267:= 1261:s 1258:e 1255:e 1252:r 1249:g 1246:e 1243:d 1234:R 1214:= 1211:) 1205:( 1202:m 1189:) 1187:ϕ 1185:( 1183:m 1174:ϕ 1117:. 1115:θ 1110:g 1107:ϕ 1101:c 1098:ϕ 1089:q 1085:ψ 1081:θ 1057:ϕ 960:e 943:6 939:b 913:f 909:/ 906:1 889:6 885:a 869:f 865:/ 862:1 840:/ 837:1 830:e 826:f 822:e 818:f 814:b 810:a 785:. 777:2 773:e 766:1 761:a 758:= 755:) 752:f 746:1 743:( 740:a 737:= 734:b 730:, 725:2 721:f 714:f 711:2 708:= 703:2 699:e 694:, 689:a 685:b 679:a 673:= 670:f 657:e 649:f 641:b 633:a 617:z 613:a 425:i 421:i 417:i 182:φ 178:ϕ 176:( 72:– 23:.

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