Latitude - Knowledge (XXG)

606:) 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 38: 1009: 274: 2857: 2393: 5650: 619: 5682:; 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. 364: 4128: 6123: 6031: 5964: 2488: 527: 513: 3776: 5415: 5004: 176:, 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 4338: 6242: 1571: 4123:{\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}}} 2819:

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

5588:

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.

1056:

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

993:

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

6305:

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

5657:

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

5948:

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

579:.) 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 5091: 1841: 2134: 3511: 4437: 1346:

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

5640:. 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. 2676: 5036:

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

4139: 2473:.) 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. 426:, 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 2457:

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.

5085:, 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): 3048: 2377: 1321: 350:

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.

5617:. 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, 806: 2810: 6209:(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 2258: 1377: 3765: 3341: 465:

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

5410:{\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}}} 489:. 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. 2896:. 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, 354:

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.

4442: 1694: 5578: 4999:{\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}}} 3578: 1679:

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

1150:

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.

3144: 2000: 1655: 4402: 3781: 2444:

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

140:

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

5500: 338:) 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 4333:{\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}}} 3212: 2524: 3360: 5096: 4144: 3631: 326:

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

2913: 6022:

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.

6225:

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

3640:

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

2277: 6114:

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.

8383: 1208: 2846: 149:. 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 7270: 3636:

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

1146:

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

1050: 315: 1684:

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

2177: 2157: 1866: 1030: 676: 295: 2694: 559: 6306:

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

1566:{\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 '} 5613:

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

1070:. This is the definition assumed when the word latitude is used without qualification. The definition must be accompanied with a specification of the ellipsoid. 342:. 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 2185: 63:. 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. 5032:

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

3684: 481:

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

6363: 6018:. 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 3274: 9125: 8657: 8163: 8040: 3053:

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

5593:

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

7296: 6722: 8743: 8539: 8529: 8449: 7191: 2383: 1142:

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

7263: 3224:

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

8534: 8115: 1836:{\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 } 330:

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

132:

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

5424:

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

161:

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

2129:{\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}}}}} 5944:

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

5519: 5041:

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

91:

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

6793:

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

3526: 819:) 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 3071: 1606: 8544: 8345: 1588: 4349: 8677: 8667: 8662: 8637: 8629: 8290: 8216: 8173: 8168: 8143: 8135: 7256: 7213: 7161: 6805:: Adams uses the nomenclature isometric latitude for the conformal latitude of this article (and throughout the modern literature).) 6643: 638:

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

214:

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

7200: 2671:{\displaystyle \theta (\phi )=\tan ^{-1}\left(\left(1-e^{2}\right)\tan \phi \right)=\tan ^{-1}\left((1-f)^{2}\tan \phi \right)\,.} 37: 9068: 8865: 8792: 8748: 8444: 6226: 6214: 253:

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

5442: 3506:{\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 '\,,} 1135:

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

8913: 8860: 3155: 2268: 1166: 6530: 1187:) is therefore equal to the angle subtended at the centre by the meridian arc from the equator to the point concerned. If the 9021: 8990: 8564: 8413: 8191: 8120: 6442: 5062: 5038: 3645: 486: 207:, north or south of the equator. For navigational purposes positions are given in degrees and decimal minutes. For instance, 6110:

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

6510:

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

9105: 9073: 8923: 8554: 8378: 8211: 8201: 8033: 6762:. U.S. Geological Survey Professional Paper 1395. Washington, DC: United States Government Printing Office. Archived from 6447: 6210: 1008: 3589: 9150: 9063: 8777: 8431: 8340: 6467: 6407: 5074: 3641: 2446: 166: 150: 76: 9053: 3043:{\displaystyle \beta (\phi )=\tan ^{-1}\left({\sqrt {1-e^{2}}}\tan \phi \right)=\tan ^{-1}\left((1-f)\tan \phi \right)} 9003: 8966: 8733: 8426: 8275: 8125: 6498: 6230: 6071: 1052:) on an ellipsoid. The normal to the surface does not pass through the centre, except at the equator and at the poles. 8647: 8153: 114:

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

6757: 2389:

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

273: 8938: 8782: 8373: 8206: 8196: 6287: 3637: 2372:{\displaystyle \Delta _{\text{long}}^{1}={\frac {\pi a\cos \phi }{180^{\circ }{\sqrt {1-e^{2}\sin ^{2}\phi }}}}\,} 8918: 8303: 7183: 6830:

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

6392: 5649: 177: 31: 8652: 8158: 6879:

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

180:(GPS), but in common usage, where high accuracy is not required, the reference ellipsoid is not usually stated. 9008: 8948: 8928: 8559: 8521: 8486: 8026: 6291: 450: 6556: 8221: 8065: 7405: 1316:{\displaystyle m(\phi )={\frac {\pi }{180^{\circ }}}R\phi _{\mathrm {degrees} }=R\phi _{\mathrm {radians} }} 1086:): the angle between the radius (from centre to the point on the surface) and the equatorial plane. (Figure 548: 453:

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

234: 215: 146: 42: 2461:

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

1057:

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

9120: 8753: 8728: 8270: 8060: 6328: 3225: 2907:, is also used in the current literature. The parametric latitude is related to the geodetic latitude by: 1066:: the angle between the normal and the equatorial plane. The standard notation in English publications is 9043: 8833: 8787: 8614: 8591: 8574: 8285: 7012: 6726: 6472: 6427: 6412: 5594: 5018: 3675: 157:

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

3644:. (Snyder, Section 16). The rectifying latitude is also of great importance in the construction of the 2856: 2392: 183:

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

6791: 6600: 9048: 8943: 8723: 8718: 8713: 8690: 8685: 8606: 8368: 8308: 8280: 8265: 8260: 8255: 8250: 7283: 7053: 6978: 6898: 6849: 6417: 6340: 5958: 2823: 2417:

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

1003: 563:, in which he proved that a rotating self-gravitating fluid body in equilibrium takes the form of an 331: 246: 240: 438:). The axis of rotation varies slowly over time and the values given here are those for the current 9160: 8998: 8933: 8838: 8815: 8642: 8549: 8421: 8148: 8106: 7973: 7968: 7963: 7956: 7951: 7946: 7941: 7935: 7926: 7921: 7916: 7911: 7904: 7899: 7894: 7889: 7883: 7869: 7864: 7859: 7854: 7849: 7844: 7839: 7834: 7824: 7819: 7814: 7809: 7804: 7799: 7794: 7789: 7777: 7772: 7767: 7762: 7757: 7752: 7717: 7712: 7707: 7702: 7697: 7692: 7657: 7652: 7647: 7642: 7637: 7632: 7627: 7622: 7567: 7562: 7557: 7552: 7547: 7542: 7537: 7532: 7470: 7465: 7436: 7431: 7386: 7353: 7316: 6432: 6308: 6145:

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

5065:. The name "isometric" arises from the fact that at any point on the ellipsoid equal increments of 5058: 5022: 2421: 1172: 1074: 860: 618: 587: 482: 458: 399: 258: 222: 173: 96: 6497:

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

6335:, the coordinate that astronomers use to specify the angular position of stars north–south of the 2848:

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

8870: 8481: 8186: 7931: 7879: 7747: 7742: 7737: 7732: 7727: 7722: 7687: 7682: 7677: 7672: 7667: 7662: 7617: 7612: 7607: 7602: 7597: 7592: 7587: 7582: 7577: 7527: 7522: 7517: 7512: 7507: 7502: 7497: 7492: 7487: 7458: 7453: 7448: 7424: 7419: 7414: 7408: 7402: 7279: 7238: 7111: 7069: 7043: 7034:

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

6994: 6968: 6914: 6888: 6865: 6839: 6373: 6368: 6275: 5614: 2682: 2510:

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

2507:

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

580: 5604:

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

5598: 8797: 8738: 8708: 8703: 8619: 8596: 8476: 8471: 8390: 8335: 8313: 7572: 7482: 7157: 7016: 6639: 6457: 6332: 6324: 6283: 5945: 5073:

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

4408: 3237: 2432: 1062: 1035: 801:{\displaystyle f={\frac {a-b}{a}},\qquad e^{2}=2f-f^{2},\qquad b=a(1-f)=a{\sqrt {1-e^{2}}}\,.} 439: 327: 300: 250: 158: 116: 2805:{\displaystyle \theta (\phi ,h)=\tan ^{-1}\left({\frac {N(1-f)^{2}+h}{N+h}}\tan \phi \right)} 9155: 8583: 8363: 7396: 7381: 7367: 7346: 7330: 7309: 7121: 7061: 6986: 6941: 6906: 6857: 6704: 6673: 6387: 5605: 1596: 462: 409: 389: 363: 121: 3250:, is the meridian distance scaled so that its value at the poles is equal to 90 degrees or 2681:

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

2162: 2142: 1851: 1015: 280: 7227: 7217: 7204: 7187: 6437: 3657: 607: 564: 478: 204: 107:

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

8004: 7997: 7990: 7983: 7232: 7209: 2253:{\displaystyle \Delta _{\text{lat}}^{1}=111\,132.954-559.822\cos 2\phi +1.175\cos 4\phi } 7197: 7057: 6982: 6902: 6853: 6122: 9035: 8981: 8958: 8905: 8893: 8848: 8825: 8807: 8767: 8509: 8463: 8400: 8355: 8327: 8235: 8097: 8085: 8049: 7475: 7441: 6787: 6402: 6294:) at that latitude. Astronomic latitude is calculated from angles measured between the 3760:{\displaystyle \xi (\phi )=\sin ^{-1}\left({\frac {q(\phi )}{q_{\mathrm {p} }}}\right)} 1083: 843:

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

335: 196: 6815:

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

9144: 7391: 7360: 7323: 7073: 6998: 6918: 6869: 4428: 1348: 379: 7295: 6030: 5963: 2487: 575:.) Newton's result was confirmed by geodetic measurements in the 18th century. (See 526: 512: 9058: 3517: 3351: 1681: 1360: 1342:

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

1331: 1188: 1160: 1143: 576: 554: 184: 7248: 7125: 5028:

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

3667: 6585: 6323:

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

3336:{\displaystyle \mu (\phi )={\frac {\pi }{2}}{\frac {m(\phi )}{m_{\mathrm {p} }}}} 8070: 7221: 6383: 6316: 6299: 2880:, is defined by the radius drawn from the centre of the ellipsoid to that point 1363:

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

1338:

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

208: 6233:. The relation of Cartesian and ellipsoidal coordinates is discussed in Torge. 334:; and the angle between any one meridian plane and that through Greenwich (the 7978: 7874: 7829: 7065: 6990: 6945: 6462: 6422: 6378: 6320: 6279: 6063: 655: 443: 347: 343: 254: 7244:

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

6910: 30:

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

9115: 6452: 6357: 6165:) with the reference ellipsoid: the necessary condition is that the product 816: 610:

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

200: 137: 104: 88: 68: 48: 6861: 1139:. "Latitude" (unqualified) should normally refer to the geodetic latitude. 17: 6763: 6703:. Columbus, OH: Dept. of Geodetic Science and Surveying, Ohio State Univ. 3516:

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

863:. Reference ellipsoids are usually defined by the semi-major axis and the 634:

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

6677: 6353: 6336: 6169:

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

3583:

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

454: 423: 162: 6579: 859:

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

317:) are defined on a spherical model. The graticule spacing is 10 degrees. 8971: 7783: 7375: 7338: 7301: 6397: 5057:, is used in the development of the ellipsoidal versions of the normal 3062: 812: 663: 647: 639: 635: 591: 467: 339: 262: 92: 56: 7179: 6708: 1090:). There is no standard notation: examples from various texts include 6295: 5994:

which is normal to the reference ellipsoid. The geodetic coordinates

1183:

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

471: 142: 6312:

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

5573:{\displaystyle \psi (\phi )=\operatorname {gd} ^{-1}\chi (\phi )\,.} 594:. In pre-satellite days they were devised to give a good fit to the 7243: 7116: 7088: 3573:{\displaystyle m_{\mathrm {p} }=m\left({\frac {\pi }{2}}\right)\,.} 2382:

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

7048: 6973: 6932:

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

6893: 6844: 6241: 6240: 6121: 6029: 5962: 3139:{\displaystyle {\frac {p^{2}}{a^{2}}}+{\frac {z^{2}}{b^{2}}}=1\,.} 2888:) which is the projection parallel to the Earth's axis of a point 2855: 2486: 2453:. The remaining latitudes are not used in this way; they are used 2391: 2264: 1661: 1650:{\displaystyle m_{\mathrm {p} }=m\left({\frac {\pi }{2}}\right)\,} 1007: 888: 617: 603: 595: 362: 346:

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

272: 133: 84: 80: 36: 8018: 6278:

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

2139:

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

103:, run east–west as circles parallel to the equator. Latitude and 6606:. National Imagery and Mapping Agency. p. 3-1. TR8350.2 6157:) construct an auxiliary ellipsoid which is confocal (same foci 1147: 9094: 8891: 8507: 8083: 8022: 7252: 6531:"ISO 19111 Geographic information — Referencing by coordinates" 4397:{\displaystyle R_{q}=a{\sqrt {\frac {q_{\mathrm {p} }}{2}}}\,.} 3061:, the distance above the equatorial plane, the equation of the 1004:

Geodetic coordinates § Geodetic vs. geocentric coordinates

442:. The time variation is discussed more fully in the article on 434:

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

371:

Besides the equator, four other parallels are of significance:

6106:

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

646:. The other parameter is usually (1) the polar radius or 599: 6229:. The relation of Cartesian and spherical polars is given in 6034:

Geocentric coordinate related to spherical polar coordinates

602:, it has become natural to use reference ellipsoids (such as 47:. The vertical lines from pole to pole are lines of constant 5648: 3149:

The Cartesian coordinates of the point are parameterized by

670:. These parameters are not independent: they are related by 525: 511: 6799:. Special Publication No. 67. US Coast and Geodetic Survey. 3346:

where the meridian distance from the equator to a latitude

1976:

When the latitude difference is 1 degree, corresponding to

470:

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

422:

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

169:

as defined in the specification of the ISO 19111 standard.

5495:{\displaystyle y(\phi )={\frac {E}{2\pi }}\psi (\phi )\,.} 598:

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

6725:. National Geospatial-Intelligence Agency. Archived from 6274:) is the angle between the equatorial plane and the true 3207:{\displaystyle p=a\cos \beta \,,\qquad z=b\sin \beta \,;} 153:

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

7102:

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

261:

together with its gravitational field is the science of

6578:

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

257:

or to determine GPS satellite orbits. The study of the

110:

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

7228:

Convert decimal degrees into degrees, minutes, seconds

7210:

Convert decimal degrees into degrees, minutes, seconds

7194:'s (NGA) database of foreign geographic feature names. 4407:

An example of the use of the authalic latitude is the

2482: 277:

A perspective view of the Earth showing how latitude (

7198:

Resources for determining your latitude and longitude

6959:

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

5522: 5445: 5094: 4440: 4352: 4142: 3779: 3687: 3592: 3529: 3363: 3277: 3158: 3074: 2916: 2826: 2697: 2527: 2280: 2188: 2165: 2145: 2003: 1854: 1697: 1609: 1380: 1211: 1038: 1018: 679: 367:

The orientation of the Earth at the December solstice

303: 283: 120:) to the ellipsoidal surface from the point, and the 27:

Geographic coordinate specifying north–south position

7233:

Distance calculation based on latitude and longitude

6599:

National Imagery and Mapping Agency (23 June 2004).

6555:

The Corporation of Trinity House (10 January 2020).

2450: 9034: 8989: 8980: 8957: 8904: 8847: 8824: 8806: 8766: 8676: 8628: 8605: 8582: 8573: 8520: 8462: 8412: 8399: 8354: 8326: 8243: 8234: 8134: 8105: 8096: 3626:{\displaystyle R={\frac {2m_{\mathrm {p} }}{\pi }}} 430:. The latitude of the tropical circles is equal to 6601:"Department of Defense World Geodetic System 1984" 5572: 5494: 5409: 4998: 4396: 4332: 4122: 3759: 3625: 3572: 3505: 3335: 3206: 3138: 3042: 2840: 2804: 2670: 2371: 2252: 2179: + 0.5 degrees on the WGS84 spheroid is 2171: 2151: 2128: 1860: 1835: 1649: 1565: 1315: 1171:The length of a degree of latitude depends on the 1044: 1024: 800: 309: 289: 6315:Astronomical latitude is not to be confused with 2263:The variation of this distance with latitude (on 994:illustrations usually exaggerate the flattening. 145:, but the geoid is more accurately modeled by an 2470: 5717:Approximate difference from geodetic latitude ( 5428:(units of length or pixels) then the distance, 630:axis to form an oblate ellipsoid of revolution. 7017:"Sur la Construction des Cartes Géographiques" 5689:, the auxiliary latitudes can be expressed as 457:when the Sun is overhead at some point of the 8034: 7264: 5509:is closely related to the conformal latitude 211:lighthouse is at 50°39.734′ N 001°35.500′ W. 8: 6782: 6780: 6668:Osborne, Peter (2013). "Chapters 5,6". 6010:are the latitude and longitude of the point 6581:Philosophiæ Naturalis Principia Mathematica 5645:Numerical comparison of auxiliary latitudes 887:. For example, the defining values for the 560:Philosophiæ Naturalis Principia Mathematica 9091: 8986: 8901: 8888: 8579: 8517: 8504: 8409: 8240: 8102: 8093: 8080: 8041: 8027: 8019: 7271: 7257: 7249: 7152:Hofmann-Wellenhof, B.; Moritz, H. (2006). 6177:) of the auxiliary ellipsoid. Further let 6149:not on the reference ellipsoid (semi-axes 1136: 524: 510: 9126:Map projection of the tri-axial ellipsoid 7115: 7047: 6972: 6892: 6843: 6751: 6749: 6747: 6745: 6743: 6663: 6661: 6659: 6657: 6655: 5566: 5542: 5521: 5488: 5461: 5444: 5370: 5339: 5295: 5258: 5196: 5176: 5153: 5140: 5095: 5093: 4957: 4926: 4861: 4824: 4792: 4768: 4749: 4701: 4681: 4668: 4640: 4613: 4599: 4588: 4540: 4493: 4471: 4441: 4439: 4390: 4377: 4376: 4369: 4357: 4351: 4343:and the radius of the sphere is taken as 4311: 4295: 4282: 4236: 4209: 4196: 4169: 4152: 4151: 4143: 4141: 4086: 4070: 4057: 4039: 4029: 3997: 3979: 3921: 3894: 3881: 3863: 3853: 3821: 3803: 3780: 3778: 3744: 3743: 3723: 3707: 3686: 3610: 3609: 3599: 3591: 3566: 3552: 3535: 3534: 3528: 3499: 3487: 3475: 3471: 3449: 3439: 3417: 3412: 3397: 3362: 3324: 3323: 3303: 3293: 3276: 3200: 3177: 3157: 3132: 3118: 3108: 3102: 3091: 3081: 3075: 3073: 2994: 2965: 2953: 2936: 2915: 2852:Parametric latitude (or reduced latitude) 2830: 2825: 2762: 2740: 2723: 2696: 2664: 2644: 2611: 2579: 2547: 2526: 2368: 2351: 2341: 2329: 2323: 2299: 2290: 2285: 2279: 2210: 2198: 2193: 2187: 2164: 2144: 2112: 2095: 2085: 2063: 2046: 2022: 2013: 2008: 2002: 1853: 1826: 1814: 1810: 1793: 1783: 1756: 1728: 1696: 1646: 1632: 1615: 1614: 1608: 1600:distance from the equator to the pole is 1551: 1539: 1535: 1513: 1503: 1481: 1476: 1461: 1428: 1405: 1400: 1379: 1288: 1287: 1252: 1251: 1236: 1227: 1210: 1037: 1017: 794: 786: 774: 734: 712: 686: 678: 302: 282: 6074:in which the coordinates of a point are 5715: 3221:because of the form of these equations. 3057:, the distance from the minor axis, and 2483:§ Geodetic and geocentric latitudes 1845: 1087: 891:ellipsoid, used by all GPS devices, are 375: 7192:National Geospatial-Intelligence Agency 7138:Holfmann-Wellenfor & Moritz (2006) 7023:(in French). Vol. IV. p. 667. 6629: 6627: 6625: 6623: 6621: 6584:. Translated by Motte, Andrew. p. 6522: 6490: 6364:Bowditch's American Practical Navigator 2860:Definition of the parametric latitude ( 2384:National Geospatial-Intelligence Agency 2267:) is shown in the table along with the 567:ellipsoid. (This article uses the term 6699:Rapp, Richard H. (1991). "Chapter 3". 6694: 6692: 6690: 6688: 5949:ellipsoidal coordinates respectively. 461:. The south polar latitudes below the 87:position of a point on the surface of 5589:There are two methods of proceeding. 2884:on the surrounding sphere (of radius 2491:The definition of geodetic latitude ( 2396:The definition of geodetic latitude ( 1012:The definition of geodetic latitude ( 7: 6185:on the auxiliary ellipsoid. The set 241:Celestial navigation § Latitude 1994:radians, the arc distance is about 449:The figure shows the geometry of a 136:, a surface which approximates the 59:are lines of constant latitude, or 6014:on the ellipsoid and the distance 4409:Albers equal-area conic projection 4378: 4153: 3745: 3611: 3536: 3325: 2282: 2190: 2005: 1616: 1355:Meridian distance on the ellipsoid 1307: 1304: 1301: 1298: 1295: 1292: 1289: 1271: 1268: 1265: 1262: 1259: 1256: 1253: 249:, latitude is determined with the 25: 6759:Map Projections: A Working Manual 5685:To first order in the flattening 5077:defined by the lines of constant 998:Geodetic and geocentric latitudes 590:have been used in the history of 9069:Quadrilateralized spherical cube 8749:Quadrilateralized spherical cube 7294: 6227:geographic coordinate conversion 6215:triaxial ellipsoidal coordinates 6203:ellipsoidal-harmonic coordinates 6118:Ellipsoidal-harmonic coordinates 4431:) transformation to the sphere. 2426:Parametric (or reduced) latitude 571:in preference to the older term 5940:Latitude and coordinate systems 3181: 2841:{\displaystyle \phi {-}\theta } 2269:length of a degree of longitude 1179:Meridian distance on the sphere 1167:Length of a degree of longitude 743: 707: 172:Since there are many different 8658:Lambert cylindrical equal-area 6443:International Latitude Service 6181:be the parametric latitude of 5563: 5557: 5532: 5526: 5485: 5479: 5455: 5449: 5397: 5382: 5357: 5351: 5322: 5307: 5282: 5270: 5108: 5102: 5063:Transverse Mercator projection 5039:Transverse Mercator projection 4984: 4969: 4944: 4938: 4888: 4873: 4848: 4836: 4454: 4448: 4113: 4098: 3793: 3787: 3735: 3729: 3697: 3691: 3646:Transverse Mercator projection 3373: 3367: 3315: 3309: 3287: 3281: 3023: 3011: 2926: 2920: 2759: 2746: 2713: 2701: 2641: 2628: 2537: 2531: 1725: 1719: 1710: 1704: 1425: 1414: 1390: 1384: 1221: 1215: 765: 753: 487:Transverse Mercator projection 55:. The circles parallel to the 1: 9106:Interruption (map projection) 7142:, p.240, eq. (6-6) to (6-10). 7126:10.1080/00396265.2023.2217604 6723:"Length of degree calculator" 6448:List of countries by latitude 6213:); for a generalization, see 6211:oblate spheroidal coordinates 4427:, gives an angle-preserving ( 2892:on the ellipsoid at latitude 2514:) and the geodetic latitude ( 2159: − 0.5 degrees and 614:The geometry of the ellipsoid 8744:Lambert azimuthal equal-area 8540:Guyou hemisphere-in-a-square 8530:Adams hemisphere-in-a-square 7239:16th Century Latitude Survey 7087:Orihuela, Sebastián (2013). 6638:(3rd ed.). De Gruyter. 6468:Orders of magnitude (length) 6408:Geographic coordinate system 5432:, of a parallel of latitude 3642:equidistant conic projection 2447:geographic coordinate system 1367:to the equator is given by ( 359:Named latitudes on the Earth 167:geographic coordinate system 6682:for LaTeX code and figures. 6557:"1/2020 Needles Lighthouse" 6499:Template:Circle of latitude 6231:spherical coordinate system 6072:spherical polar coordinates 6026:Spherical polar coordinates 5584:Inverse formulae and series 2495:) and geocentric latitude ( 2400:) and geocentric latitude ( 811:Many other parameters (see 322:The graticule on the sphere 9177: 6288:gravitational acceleration 5956: 3655: 3638:equirectangular projection 3235: 3217:Cayley suggested the term 2480: 1164: 1158: 1001: 654:; or (2) the (first) 546: 507: 493: 238: 232: 97:Lines of constant latitude 29: 9101: 9090: 9017: 8900: 8887: 8699: 8516: 8503: 8440: 8299: 8182: 8092: 8079: 8056: 7293: 7290: 7066:10.1007/s00190-011-0445-3 6991:10.1007/s00190-012-0578-z 6946:10.1080/14786447008640411 6701:Geometric Geodesy, Part I 6393:Degree Confluence Project 6057:is the complement of the 1688:meridian arc is given by 538:Latitude on the ellipsoid 501: 496: 221:For a brief history, see 178:Global Positioning System 32:Latitude (disambiguation) 7220:– info about decimal to 6911:10.1002/asna.18260041601 6819:. 1st semester: 130–161. 6756:Snyder, John P. (1987). 6670:The Mercator Projections 6292:centrifugal acceleration 6173:be the semi-minor axis ( 6126:Ellipsoidal coordinates 6053:The geocentric latitude 5709:takes on the values for 2465:, and the eccentricity, 1045:{\displaystyle \lambda } 974:(eccentricity squared): 504: 499: 494: 310:{\displaystyle \lambda } 8545:Lambert conformal conic 6207:ellipsoidal coordinates 5505:The isometric latitude 945:from which are derived 549:Ellipsoid of revolution 235:Longitude determination 216:planetographic latitude 147:ellipsoid of revolution 8678:Tobler hyperelliptical 8291:Tobler hyperelliptical 8217:Space-oblique Mercator 7089:"Funciones de Latitud" 7013:Lagrange, Joseph-Louis 6862:10.1002/asna.201011352 6329:equatorial coordinates 6286:(the resultant of the 6264: 6221:Coordinate conversions 6142: 6050: 5986:At an arbitrary point 5983: 5658:(positive latitudes), 5654: 5574: 5496: 5411: 5000: 4398: 4334: 4124: 3761: 3666:(after the Greek for " 3627: 3574: 3507: 3337: 3208: 3140: 3044: 2865: 2842: 2806: 2672: 2500: 2405: 2373: 2271:(east–west distance): 2254: 2173: 2153: 2130: 1862: 1837: 1651: 1567: 1317: 1053: 1046: 1026: 931:(inverse flattening): 802: 631: 530: 516: 368: 318: 311: 291: 269:Latitude on the sphere 64: 6559:. Notices to Mariners 6473:World Geodetic System 6428:Great-circle distance 6413:Geographical distance 6302:is accurately known. 6290:(mass-based) and the 6268:Astronomical latitude 6244: 6237:Astronomical latitude 6125: 6033: 5967:Geodetic coordinates 5966: 5652: 5595:fixed-point iteration 5575: 5497: 5412: 5019:Gudermannian function 5001: 4399: 4335: 4125: 3762: 3676:equal-area projection 3628: 3575: 3508: 3338: 3209: 3141: 3045: 2859: 2843: 2807: 2673: 2490: 2395: 2374: 2255: 2174: 2172:{\displaystyle \phi } 2154: 2152:{\displaystyle \phi } 2131: 1863: 1861:{\displaystyle \phi } 1838: 1652: 1568: 1318: 1137:astronomical latitude 1047: 1027: 1025:{\displaystyle \phi } 1011: 898:(equatorial radius): 803: 626:compressed along the 621: 608:datum transformations 529: 515: 366: 312: 292: 290:{\displaystyle \phi } 276: 239:Further information: 195:). It is measured in 40: 9054:Cahill–Keyes M-shape 8914:Chamberlin trimetric 7235:– JavaScript version 7180:GEONets Names Server 6940:(4th ser): 329–340. 6678:10.5281/zenodo.35392 6418:Geomagnetic latitude 6341:ecliptic coordinates 5959:Geodetic coordinates 5953:Geodetic coordinates 5520: 5443: 5436:from the equator is 5092: 4438: 4350: 4140: 3777: 3685: 3590: 3527: 3361: 3275: 3156: 3072: 2914: 2824: 2695: 2525: 2469:. (For inverses see 2278: 2186: 2163: 2143: 2001: 1852: 1695: 1607: 1378: 1209: 1124:. This article uses 1036: 1016: 677: 588:reference ellipsoids 502:Transverse Mercator 301: 281: 247:celestial navigation 174:reference ellipsoids 122:plane of the equator 9151:Circles of latitude 9121:Tissot's indicatrix 9022:Central cylindrical 8663:Smyth equal-surface 8565:Transverse Mercator 8414:General perspective 8169:Smyth equal-surface 8121:Transverse Mercator 7387:Tropic of Capricorn 7355:Tropic of Capricorn 7318:Tropic of Capricorn 7280:Circles of latitude 7058:2011JGeod..85..475K 6983:2013JGeod..87...43K 6903:1825AN......4..241B 6854:2010AN....331..852K 6433:History of latitude 6309:vertical deflection 6094:is the distance of 5722: 5705:where the constant 5059:Mercator projection 5023:Mercator projection 3422: 3244:rectifying latitude 3232:Rectifying latitude 3219:parametric latitude 2870:parametric latitude 2505:geocentric latitude 2477:Geocentric latitude 2429:Rectifying latitude 2422:Geocentric latitude 2415:auxiliary latitudes 2409:Auxiliary latitudes 2295: 2203: 2018: 1589:radius of curvature 1486: 1410: 1173:figure of the Earth 1133:Geographic latitude 1075:Geocentric latitude 861:Figure of the Earth 622:A sphere of radius 483:Mercator projection 459:Tropic of Capricorn 414:66° 34′ (66.57°) S 404:23° 26′ (23.43°) S 400:Tropic of Capricorn 394:23° 26′ (23.43°) N 384:66° 34′ (66.57°) N 259:figure of the Earth 223:History of latitude 201:minutes and seconds 79:that specifies the 9074:Waterman butterfly 8924:Miller cylindrical 8555:Peirce quincuncial 8450:Lambert equal-area 8202:Gall stereographic 7216:2012-11-07 at the 7203:2008-05-19 at the 7186:2008-03-09 at the 7036:Journal of Geodesy 6961:Journal of Geodesy 6817:Mém. Inst. Nat. Fr 6634:Torge, W. (2001). 6374:Circle of latitude 6369:Cardinal direction 6276:vertical direction 6265: 6143: 6051: 5990:consider the line 5984: 5716: 5655: 5615:Lagrange reversion 5570: 5492: 5407: 5405: 5051:isometric latitude 5045:Isometric latitude 4996: 4994: 4421:conformal latitude 4415:Conformal latitude 4394: 4330: 4328: 4120: 4118: 3757: 3623: 3570: 3503: 3408: 3333: 3204: 3136: 3040: 2866: 2864:) on the ellipsoid 2838: 2802: 2683:ellipsoidal height 2668: 2501: 2440:Isometric latitude 2437:Conformal latitude 2406: 2369: 2281: 2250: 2189: 2169: 2149: 2126: 2004: 1858: 1833: 1647: 1587:is the meridional 1563: 1472: 1396: 1313: 1080:spherical latitude 1054: 1042: 1022: 798: 662:; or (3) the 632: 531: 517: 369: 319: 307: 287: 65: 9138: 9137: 9134: 9133: 9086: 9085: 9082: 9081: 9030: 9029: 8883: 8882: 8879: 8878: 8762: 8761: 8499: 8498: 8495: 8494: 8458: 8457: 8346:Lambert conformal 8322: 8321: 8236:Pseudocylindrical 8230: 8229: 8016: 8015: 6458:Natural Area Code 6333:ecliptic latitude 6325:celestial equator 6319:, the coordinate 6284:gravity direction 5946:coordinate system 5937: 5936: 5474: 5238: 5184: 5161: 5148: 4776: 4757: 4743: 4689: 4676: 4621: 4607: 4582: 4529: 4388: 4387: 4305: 4260: 4224: 4177: 4080: 4052: 3963: 3909: 3876: 3751: 3664:authalic latitude 3652:Authalic latitude 3621: 3560: 3483: 3331: 3301: 3238:Rectifying radius 3124: 3097: 2971: 2786: 2433:Authalic latitude 2366: 2363: 2288: 2196: 2124: 2120: 2011: 1974: 1973: 1822: 1664:this distance is 1640: 1547: 1242: 1189:meridian distance 1155:Meridian distance 1063:Geodetic latitude 1032:) and longitude ( 792: 702: 535: 534: 418: 417: 297:) and longitude ( 251:meridian altitude 112:geodetic latitude 16:(Redirected from 9168: 9092: 9049:Cahill Butterfly 8987: 8967:Goode homolosine 8902: 8889: 8854: 8853:(Mecca or Qibla) 8734:Goode homolosine 8580: 8518: 8505: 8410: 8405: 8276:Goode homolosine 8241: 8126:Oblique Mercator 8103: 8094: 8081: 8043: 8036: 8029: 8020: 8007: 8000: 7993: 7986: 7397:Antarctic Circle 7382:Tropic of Cancer 7370: 7369:Antarctic Circle 7363: 7356: 7349: 7348:Tropic of Cancer 7341: 7333: 7332:Antarctic Circle 7326: 7319: 7312: 7311:Tropic of Cancer 7304: 7298: 7273: 7266: 7259: 7250: 7190:. access to the 7168: 7167: 7156:(2nd ed.). 7154:Physical Geodesy 7149: 7143: 7140:Physical Geodesy 7136: 7130: 7129: 7119: 7110:(395): 165–180. 7099: 7093: 7092: 7084: 7078: 7077: 7051: 7031: 7025: 7024: 7009: 7003: 7002: 6976: 6956: 6950: 6949: 6929: 6923: 6922: 6896: 6873: 6847: 6827: 6821: 6820: 6812: 6806: 6800: 6798: 6784: 6775: 6774: 6772: 6771: 6753: 6738: 6737: 6735: 6734: 6719: 6713: 6712: 6696: 6683: 6681: 6665: 6650: 6649: 6631: 6616: 6615: 6613: 6611: 6605: 6596: 6590: 6589: 6575: 6569: 6568: 6566: 6564: 6552: 6546: 6545: 6543: 6542: 6527: 6511: 6508: 6502: 6495: 6388:celestial sphere 6298:and stars whose 6273: 6254:Local plumb line 6200: 6184: 6180: 6176: 6172: 6168: 6164: 6160: 6156: 6152: 6148: 6141: 6113: 6109: 6105: 6101: 6098:from the centre 6097: 6093: 6089: 6070:in conventional 6069: 6056: 6049: 6021: 6017: 6013: 6009: 5993: 5989: 5982: 5793: 5780: 5767: 5754: 5741: 5728: 5723: 5720: 5639: 5606:double precision 5579: 5577: 5576: 5571: 5550: 5549: 5512: 5508: 5501: 5499: 5498: 5493: 5475: 5473: 5462: 5435: 5431: 5427: 5416: 5414: 5413: 5408: 5406: 5378: 5377: 5347: 5346: 5328: 5303: 5302: 5266: 5265: 5247: 5243: 5239: 5237: 5217: 5197: 5185: 5177: 5172: 5168: 5167: 5163: 5162: 5154: 5149: 5141: 5084: 5080: 5072: 5068: 5056: 5016: 5005: 5003: 5002: 4997: 4995: 4991: 4987: 4965: 4964: 4934: 4933: 4904: 4900: 4896: 4895: 4891: 4869: 4868: 4832: 4831: 4800: 4799: 4781: 4777: 4769: 4764: 4760: 4759: 4758: 4750: 4748: 4744: 4742: 4722: 4702: 4695: 4691: 4690: 4682: 4677: 4669: 4648: 4647: 4626: 4622: 4614: 4609: 4608: 4600: 4598: 4594: 4593: 4592: 4587: 4583: 4581: 4561: 4541: 4534: 4530: 4528: 4511: 4494: 4479: 4478: 4426: 4403: 4401: 4400: 4395: 4389: 4383: 4382: 4381: 4371: 4370: 4362: 4361: 4339: 4337: 4336: 4331: 4329: 4319: 4318: 4306: 4301: 4300: 4299: 4283: 4269: 4265: 4261: 4259: 4248: 4237: 4225: 4223: 4215: 4214: 4213: 4197: 4182: 4178: 4170: 4158: 4157: 4156: 4129: 4127: 4126: 4121: 4119: 4094: 4093: 4081: 4076: 4075: 4074: 4058: 4053: 4051: 4044: 4043: 4034: 4033: 4017: 4007: 4003: 4002: 4001: 3980: 3972: 3968: 3964: 3962: 3942: 3922: 3910: 3908: 3900: 3899: 3898: 3882: 3877: 3875: 3868: 3867: 3858: 3857: 3841: 3831: 3827: 3826: 3825: 3804: 3766: 3764: 3763: 3758: 3756: 3752: 3750: 3749: 3748: 3738: 3724: 3715: 3714: 3673: 3632: 3630: 3629: 3624: 3622: 3617: 3616: 3615: 3614: 3600: 3579: 3577: 3576: 3571: 3565: 3561: 3553: 3541: 3540: 3539: 3512: 3510: 3509: 3504: 3498: 3486: 3485: 3484: 3476: 3470: 3466: 3465: 3454: 3453: 3444: 3443: 3421: 3416: 3407: 3403: 3402: 3401: 3349: 3342: 3340: 3339: 3334: 3332: 3330: 3329: 3328: 3318: 3304: 3302: 3294: 3267: 3265: 3264: 3261: 3258: 3257: 3249: 3213: 3211: 3210: 3205: 3145: 3143: 3142: 3137: 3125: 3123: 3122: 3113: 3112: 3103: 3098: 3096: 3095: 3086: 3085: 3076: 3060: 3056: 3049: 3047: 3046: 3041: 3039: 3035: 3002: 3001: 2986: 2982: 2972: 2970: 2969: 2954: 2944: 2943: 2906: 2895: 2891: 2887: 2883: 2879: 2874:reduced latitude 2863: 2847: 2845: 2844: 2839: 2834: 2818: 2811: 2809: 2808: 2803: 2801: 2797: 2787: 2785: 2774: 2767: 2766: 2741: 2731: 2730: 2677: 2675: 2674: 2669: 2663: 2659: 2649: 2648: 2619: 2618: 2603: 2599: 2589: 2585: 2584: 2583: 2555: 2554: 2517: 2513: 2498: 2494: 2468: 2464: 2403: 2399: 2378: 2376: 2375: 2370: 2367: 2365: 2364: 2356: 2355: 2346: 2345: 2330: 2328: 2327: 2317: 2300: 2294: 2289: 2286: 2259: 2257: 2256: 2251: 2202: 2197: 2194: 2178: 2176: 2175: 2170: 2158: 2156: 2155: 2150: 2135: 2133: 2132: 2127: 2125: 2123: 2122: 2121: 2113: 2111: 2107: 2100: 2099: 2090: 2089: 2068: 2067: 2057: 2056: 2052: 2051: 2050: 2023: 2017: 2012: 2009: 1993: 1991: 1990: 1987: 1984: 1983: 1915:107.550 km 1904:111.320 km 1893: 1892: 1891: 1880: 1879: 1878: 1867: 1865: 1864: 1859: 1846: 1842: 1840: 1839: 1834: 1825: 1824: 1823: 1815: 1809: 1805: 1798: 1797: 1788: 1787: 1766: 1762: 1761: 1760: 1675: 1673: 1669: 1656: 1654: 1653: 1648: 1645: 1641: 1633: 1621: 1620: 1619: 1597:quarter meridian 1586: 1572: 1570: 1569: 1564: 1562: 1550: 1549: 1548: 1540: 1534: 1530: 1529: 1518: 1517: 1508: 1507: 1485: 1480: 1471: 1467: 1466: 1465: 1439: 1424: 1409: 1404: 1370: 1366: 1345: 1341: 1337: 1329: 1322: 1320: 1319: 1314: 1312: 1311: 1310: 1276: 1275: 1274: 1243: 1241: 1240: 1228: 1201: 1186: 1127: 1123: 1114: 1105: 1101: 1097: 1093: 1069: 1051: 1049: 1048: 1043: 1031: 1029: 1028: 1023: 989: 988: 985: 982: 979: 973: 966: 964: 960: 957: 952:(polar radius): 951: 940: 939: 936: 930: 929: 927: 926: 921: 918: 908: 906: 903: 897: 886: 885: 883: 882: 877: 874: 858: 856: 855: 852: 849: 842: 838: 834: 830: 826: 822: 807: 805: 804: 799: 793: 791: 790: 775: 739: 738: 717: 716: 703: 698: 687: 669: 661: 653: 645: 497:Normal Mercator 492: 491: 463:Antarctic Circle 433: 429: 410:Antarctic Circle 390:Tropic of Cancer 376: 316: 314: 313: 308: 296: 294: 293: 288: 194: 190: 21: 9176: 9175: 9171: 9170: 9169: 9167: 9166: 9165: 9141: 9140: 9139: 9130: 9097: 9078: 9026: 9013: 8976: 8953: 8939:Van der Grinten 8896: 8894:By construction 8875: 8852: 8851: 8843: 8820: 8802: 8783:Equirectangular 8769: 8758: 8695: 8672: 8668:Trystan Edwards 8624: 8601: 8569: 8512: 8491: 8464:Pseudoazimuthal 8454: 8436: 8403: 8402: 8395: 8350: 8318: 8314:Winkel I and II 8295: 8226: 8207:Gall isographic 8197:Equirectangular 8178: 8174:Trystan Edwards 8130: 8088: 8075: 8052: 8047: 8017: 8012: 8011: 8010: 8009: 8005: 8002: 7998: 7995: 7991: 7988: 7984: 7981: 7976: 7971: 7966: 7961: 7954: 7949: 7944: 7939: 7929: 7924: 7919: 7914: 7909: 7902: 7897: 7892: 7887: 7877: 7872: 7867: 7862: 7857: 7852: 7847: 7842: 7837: 7832: 7827: 7822: 7817: 7812: 7807: 7802: 7797: 7792: 7787: 7780: 7775: 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: 7473: 7468: 7463: 7456: 7451: 7446: 7439: 7434: 7429: 7422: 7417: 7412: 7399: 7394: 7389: 7384: 7379: 7372: 7368: 7365: 7361: 7358: 7354: 7351: 7347: 7344: 7339: 7335: 7331: 7328: 7324: 7321: 7317: 7314: 7310: 7307: 7302: 7286: 7277: 7218:Wayback Machine 7205:Wayback Machine 7188:Wayback Machine 7176: 7171: 7164: 7151: 7150: 7146: 7137: 7133: 7101: 7100: 7096: 7086: 7085: 7081: 7033: 7032: 7028: 7011: 7010: 7006: 6958: 6957: 6953: 6931: 6930: 6926: 6878: 6874: 6838:(86): 241–254. 6829: 6828: 6824: 6814: 6813: 6809: 6796: 6788:Adams, Oscar S. 6786: 6785: 6778: 6769: 6767: 6755: 6754: 6741: 6732: 6730: 6721: 6720: 6716: 6698: 6697: 6686: 6667: 6666: 6653: 6646: 6633: 6632: 6619: 6609: 6607: 6603: 6598: 6597: 6593: 6577: 6576: 6572: 6562: 6560: 6554: 6553: 6549: 6540: 6538: 6529: 6528: 6524: 6520: 6515: 6514: 6509: 6505: 6496: 6492: 6487: 6482: 6477: 6438:Horse latitudes 6349: 6271: 6263: 6239: 6223: 6186: 6182: 6178: 6174: 6170: 6166: 6162: 6158: 6154: 6150: 6146: 6127: 6120: 6111: 6107: 6103: 6099: 6095: 6091: 6075: 6067: 6054: 6035: 6028: 6019: 6015: 6011: 5995: 5991: 5987: 5968: 5961: 5955: 5942: 5785: 5784: 5772: 5771: 5759: 5758: 5746: 5745: 5733: 5732: 5726: 5718: 5647: 5618: 5586: 5538: 5518: 5517: 5510: 5506: 5466: 5441: 5440: 5433: 5429: 5425: 5404: 5403: 5366: 5335: 5326: 5325: 5291: 5254: 5245: 5244: 5218: 5198: 5192: 5139: 5135: 5128: 5124: 5111: 5090: 5089: 5082: 5078: 5070: 5066: 5054: 5047: 5010: 4993: 4992: 4953: 4922: 4921: 4917: 4902: 4901: 4857: 4820: 4819: 4815: 4808: 4804: 4788: 4779: 4778: 4723: 4703: 4697: 4696: 4667: 4663: 4656: 4652: 4636: 4624: 4623: 4562: 4542: 4536: 4535: 4512: 4495: 4489: 4488: 4484: 4483: 4467: 4457: 4436: 4435: 4424: 4417: 4372: 4353: 4348: 4347: 4327: 4326: 4307: 4291: 4284: 4267: 4266: 4249: 4238: 4232: 4216: 4205: 4198: 4183: 4165: 4147: 4138: 4137: 4117: 4116: 4082: 4066: 4059: 4035: 4025: 4018: 3993: 3986: 3982: 3981: 3970: 3969: 3943: 3923: 3917: 3901: 3890: 3883: 3859: 3849: 3842: 3817: 3810: 3806: 3805: 3796: 3775: 3774: 3739: 3725: 3719: 3703: 3683: 3682: 3671: 3660: 3658:Authalic radius 3654: 3605: 3601: 3588: 3587: 3548: 3530: 3525: 3524: 3491: 3458: 3445: 3435: 3428: 3424: 3423: 3393: 3386: 3382: 3359: 3358: 3347: 3319: 3305: 3273: 3272: 3262: 3259: 3255: 3254: 3253: 3251: 3247: 3240: 3234: 3154: 3153: 3114: 3104: 3087: 3077: 3070: 3069: 3058: 3054: 3010: 3006: 2990: 2961: 2952: 2948: 2932: 2912: 2911: 2897: 2893: 2889: 2885: 2881: 2877: 2861: 2854: 2822: 2821: 2816: 2775: 2758: 2742: 2739: 2735: 2719: 2693: 2692: 2640: 2627: 2623: 2607: 2575: 2568: 2564: 2563: 2559: 2543: 2523: 2522: 2515: 2511: 2496: 2492: 2485: 2479: 2466: 2462: 2411: 2401: 2397: 2347: 2337: 2319: 2318: 2301: 2276: 2275: 2184: 2183: 2161: 2160: 2141: 2140: 2091: 2081: 2074: 2070: 2069: 2059: 2058: 2042: 2035: 2031: 2024: 1999: 1998: 1988: 1985: 1981: 1980: 1979: 1977: 1967:111.694 km 1959:28.902 km 1956:111.618 km 1948:55.800 km 1945:111.412 km 1937:78.847 km 1934:111.132 km 1926:96.486 km 1923:110.852 km 1912:110.649 km 1901:110.574 km 1890: 1887: 1886: 1885: 1883: 1877: 1874: 1873: 1872: 1870: 1850: 1849: 1789: 1779: 1772: 1768: 1767: 1752: 1745: 1741: 1693: 1692: 1671: 1667: 1665: 1628: 1610: 1605: 1604: 1577: 1555: 1522: 1509: 1499: 1492: 1488: 1487: 1457: 1450: 1446: 1432: 1417: 1376: 1375: 1368: 1364: 1357: 1343: 1339: 1335: 1327: 1283: 1247: 1232: 1207: 1206: 1192: 1184: 1181: 1169: 1163: 1157: 1125: 1122: 1116: 1113: 1107: 1103: 1099: 1095: 1091: 1078:(also known as 1067: 1034: 1033: 1014: 1013: 1006: 1000: 986: 983: 980: 977: 975: 969: 962: 958: 955: 953: 949: 937: 934: 932: 922: 919: 916: 915: 913: 912: 904: 901: 899: 895: 878: 875: 872: 871: 869: 868: 853: 850: 847: 846: 844: 840: 836: 832: 828: 824: 820: 782: 730: 708: 688: 675: 674: 667: 659: 651: 648:semi-minor axis 643: 640:semi-major axis 616: 586:Many different 551: 545: 540: 479:map projections 431: 427: 361: 324: 299: 298: 279: 278: 271: 243: 237: 231: 205:decimal degrees 192: 188: 130: 35: 28: 23: 22: 15: 12: 11: 5: 9174: 9172: 9164: 9163: 9158: 9153: 9143: 9142: 9136: 9135: 9132: 9131: 9129: 9128: 9123: 9118: 9113: 9108: 9102: 9099: 9098: 9095: 9088: 9087: 9084: 9083: 9080: 9079: 9077: 9076: 9071: 9066: 9061: 9056: 9051: 9046: 9040: 9038: 9032: 9031: 9028: 9027: 9025: 9024: 9018: 9015: 9014: 9012: 9011: 9006: 9001: 8995: 8993: 8984: 8978: 8977: 8975: 8974: 8969: 8963: 8961: 8955: 8954: 8952: 8951: 8946: 8941: 8936: 8931: 8926: 8921: 8919:Kavrayskiy VII 8916: 8910: 8908: 8898: 8897: 8892: 8885: 8884: 8881: 8880: 8877: 8876: 8874: 8873: 8868: 8863: 8857: 8855: 8849:Retroazimuthal 8845: 8844: 8842: 8841: 8836: 8830: 8828: 8822: 8821: 8819: 8818: 8812: 8810: 8804: 8803: 8801: 8800: 8795: 8790: 8785: 8780: 8774: 8772: 8768:Equidistant in 8764: 8763: 8760: 8759: 8757: 8756: 8751: 8746: 8741: 8736: 8731: 8726: 8721: 8716: 8711: 8706: 8700: 8697: 8696: 8694: 8693: 8688: 8682: 8680: 8674: 8673: 8671: 8670: 8665: 8660: 8655: 8650: 8645: 8640: 8634: 8632: 8626: 8625: 8623: 8622: 8617: 8611: 8609: 8603: 8602: 8600: 8599: 8594: 8588: 8586: 8577: 8571: 8570: 8568: 8567: 8562: 8557: 8552: 8547: 8542: 8537: 8532: 8526: 8524: 8514: 8513: 8508: 8501: 8500: 8497: 8496: 8493: 8492: 8490: 8489: 8484: 8479: 8474: 8468: 8466: 8460: 8459: 8456: 8455: 8453: 8452: 8447: 8441: 8438: 8437: 8435: 8434: 8429: 8424: 8418: 8416: 8407: 8397: 8396: 8394: 8393: 8388: 8387: 8386: 8381: 8371: 8366: 8360: 8358: 8352: 8351: 8349: 8348: 8343: 8338: 8332: 8330: 8324: 8323: 8320: 8319: 8317: 8316: 8311: 8306: 8304:Kavrayskiy VII 8300: 8297: 8296: 8294: 8293: 8288: 8283: 8278: 8273: 8268: 8263: 8258: 8253: 8247: 8245: 8238: 8232: 8231: 8228: 8227: 8225: 8224: 8219: 8214: 8209: 8204: 8199: 8194: 8189: 8183: 8180: 8179: 8177: 8176: 8171: 8166: 8161: 8156: 8151: 8146: 8140: 8138: 8132: 8131: 8129: 8128: 8123: 8118: 8112: 8110: 8100: 8090: 8089: 8084: 8077: 8076: 8074: 8073: 8068: 8063: 8057: 8054: 8053: 8050:Map projection 8048: 8046: 8045: 8038: 8031: 8023: 8014: 8013: 8003: 7996: 7989: 7982: 7977: 7972: 7967: 7962: 7955: 7950: 7945: 7940: 7930: 7925: 7920: 7915: 7910: 7903: 7898: 7893: 7888: 7878: 7873: 7868: 7863: 7858: 7853: 7848: 7843: 7838: 7833: 7828: 7823: 7818: 7813: 7808: 7803: 7798: 7793: 7788: 7781: 7776: 7771: 7766: 7761: 7756: 7751: 7746: 7741: 7736: 7731: 7726: 7721: 7716: 7711: 7706: 7701: 7696: 7691: 7686: 7681: 7676: 7671: 7666: 7661: 7656: 7651: 7646: 7641: 7636: 7631: 7626: 7621: 7616: 7611: 7606: 7601: 7596: 7591: 7586: 7581: 7576: 7571: 7566: 7561: 7556: 7551: 7546: 7541: 7536: 7531: 7526: 7521: 7516: 7511: 7506: 7501: 7496: 7491: 7486: 7481: 7474: 7469: 7464: 7457: 7452: 7447: 7440: 7435: 7430: 7423: 7418: 7413: 7400: 7395: 7390: 7385: 7380: 7373: 7366: 7359: 7352: 7345: 7336: 7329: 7322: 7315: 7308: 7299: 7292: 7291: 7288: 7287: 7278: 7276: 7275: 7268: 7261: 7253: 7247: 7246: 7241: 7236: 7230: 7225: 7207: 7195: 7175: 7174:External links 7172: 7170: 7169: 7162: 7144: 7131: 7094: 7079: 7042:(8): 475–485. 7026: 7004: 6951: 6924: 6887:(8): 852–861. 6822: 6807: 6776: 6739: 6714: 6684: 6651: 6644: 6617: 6591: 6570: 6547: 6521: 6519: 6516: 6513: 6512: 6503: 6489: 6488: 6486: 6483: 6481: 6478: 6476: 6475: 6470: 6465: 6460: 6455: 6450: 6445: 6440: 6435: 6430: 6425: 6420: 6415: 6410: 6405: 6403:Geodetic datum 6400: 6395: 6390: 6381: 6376: 6371: 6366: 6361: 6358:mean sea level 6350: 6348: 6345: 6282:, is also the 6262: 6261: 6258: 6255: 6252: 6249: 6245: 6238: 6235: 6222: 6219: 6119: 6116: 6027: 6024: 5957:Main article: 5954: 5951: 5941: 5938: 5935: 5934: 5931: 5928: 5925: 5922: 5919: 5915: 5914: 5911: 5908: 5905: 5902: 5899: 5895: 5894: 5891: 5888: 5885: 5882: 5879: 5875: 5874: 5871: 5868: 5865: 5862: 5859: 5855: 5854: 5851: 5848: 5845: 5842: 5839: 5835: 5834: 5831: 5828: 5825: 5822: 5819: 5815: 5814: 5811: 5808: 5805: 5802: 5799: 5795: 5794: 5781: 5768: 5755: 5742: 5729: 5646: 5643: 5642: 5641: 5611: 5610: 5609: 5601:root finding. 5599:Newton–Raphson 5585: 5582: 5581: 5580: 5569: 5565: 5562: 5559: 5556: 5553: 5548: 5545: 5541: 5537: 5534: 5531: 5528: 5525: 5503: 5502: 5491: 5487: 5484: 5481: 5478: 5472: 5469: 5465: 5460: 5457: 5454: 5451: 5448: 5418: 5417: 5402: 5399: 5396: 5393: 5390: 5387: 5384: 5381: 5376: 5373: 5369: 5365: 5362: 5359: 5356: 5353: 5350: 5345: 5342: 5338: 5334: 5331: 5329: 5327: 5324: 5321: 5318: 5315: 5312: 5309: 5306: 5301: 5298: 5294: 5290: 5287: 5284: 5281: 5278: 5275: 5272: 5269: 5264: 5261: 5257: 5253: 5250: 5248: 5246: 5242: 5236: 5233: 5230: 5227: 5224: 5221: 5216: 5213: 5210: 5207: 5204: 5201: 5195: 5191: 5188: 5183: 5180: 5175: 5171: 5166: 5160: 5157: 5152: 5147: 5144: 5138: 5134: 5131: 5127: 5123: 5120: 5117: 5114: 5112: 5110: 5107: 5104: 5101: 5098: 5097: 5069:and longitude 5046: 5043: 5007: 5006: 4990: 4986: 4983: 4980: 4977: 4974: 4971: 4968: 4963: 4960: 4956: 4952: 4949: 4946: 4943: 4940: 4937: 4932: 4929: 4925: 4920: 4916: 4913: 4910: 4907: 4905: 4903: 4899: 4894: 4890: 4887: 4884: 4881: 4878: 4875: 4872: 4867: 4864: 4860: 4856: 4853: 4850: 4847: 4844: 4841: 4838: 4835: 4830: 4827: 4823: 4818: 4814: 4811: 4807: 4803: 4798: 4795: 4791: 4787: 4784: 4782: 4780: 4775: 4772: 4767: 4763: 4756: 4753: 4747: 4741: 4738: 4735: 4732: 4729: 4726: 4721: 4718: 4715: 4712: 4709: 4706: 4700: 4694: 4688: 4685: 4680: 4675: 4672: 4666: 4662: 4659: 4655: 4651: 4646: 4643: 4639: 4635: 4632: 4629: 4627: 4625: 4620: 4617: 4612: 4606: 4603: 4597: 4591: 4586: 4580: 4577: 4574: 4571: 4568: 4565: 4560: 4557: 4554: 4551: 4548: 4545: 4539: 4533: 4527: 4524: 4521: 4518: 4515: 4510: 4507: 4504: 4501: 4498: 4492: 4487: 4482: 4477: 4474: 4470: 4466: 4463: 4460: 4458: 4456: 4453: 4450: 4447: 4444: 4443: 4416: 4413: 4405: 4404: 4393: 4386: 4380: 4375: 4368: 4365: 4360: 4356: 4341: 4340: 4325: 4322: 4317: 4314: 4310: 4304: 4298: 4294: 4290: 4287: 4281: 4278: 4275: 4272: 4270: 4268: 4264: 4258: 4255: 4252: 4247: 4244: 4241: 4235: 4231: 4228: 4222: 4219: 4212: 4208: 4204: 4201: 4195: 4192: 4189: 4186: 4184: 4181: 4176: 4173: 4168: 4164: 4161: 4155: 4150: 4146: 4145: 4131: 4130: 4115: 4112: 4109: 4106: 4103: 4100: 4097: 4092: 4089: 4085: 4079: 4073: 4069: 4065: 4062: 4056: 4050: 4047: 4042: 4038: 4032: 4028: 4024: 4021: 4016: 4013: 4010: 4006: 4000: 3996: 3992: 3989: 3985: 3978: 3975: 3973: 3971: 3967: 3961: 3958: 3955: 3952: 3949: 3946: 3941: 3938: 3935: 3932: 3929: 3926: 3920: 3916: 3913: 3907: 3904: 3897: 3893: 3889: 3886: 3880: 3874: 3871: 3866: 3862: 3856: 3852: 3848: 3845: 3840: 3837: 3834: 3830: 3824: 3820: 3816: 3813: 3809: 3802: 3799: 3797: 3795: 3792: 3789: 3786: 3783: 3782: 3768: 3767: 3755: 3747: 3742: 3737: 3734: 3731: 3728: 3722: 3718: 3713: 3710: 3706: 3702: 3699: 3696: 3693: 3690: 3653: 3650: 3634: 3633: 3620: 3613: 3608: 3604: 3598: 3595: 3581: 3580: 3569: 3564: 3559: 3556: 3551: 3547: 3544: 3538: 3533: 3518:polar distance 3514: 3513: 3502: 3497: 3494: 3490: 3482: 3479: 3474: 3469: 3464: 3461: 3457: 3452: 3448: 3442: 3438: 3434: 3431: 3427: 3420: 3415: 3411: 3406: 3400: 3396: 3392: 3389: 3385: 3381: 3378: 3375: 3372: 3369: 3366: 3344: 3343: 3327: 3322: 3317: 3314: 3311: 3308: 3300: 3297: 3292: 3289: 3286: 3283: 3280: 3233: 3230: 3215: 3214: 3203: 3199: 3196: 3193: 3190: 3187: 3184: 3180: 3176: 3173: 3170: 3167: 3164: 3161: 3147: 3146: 3135: 3131: 3128: 3121: 3117: 3111: 3107: 3101: 3094: 3090: 3084: 3080: 3051: 3050: 3038: 3034: 3031: 3028: 3025: 3022: 3019: 3016: 3013: 3009: 3005: 3000: 2997: 2993: 2989: 2985: 2981: 2978: 2975: 2968: 2964: 2960: 2957: 2951: 2947: 2942: 2939: 2935: 2931: 2928: 2925: 2922: 2919: 2853: 2850: 2837: 2833: 2829: 2813: 2812: 2800: 2796: 2793: 2790: 2784: 2781: 2778: 2773: 2770: 2765: 2761: 2757: 2754: 2751: 2748: 2745: 2738: 2734: 2729: 2726: 2722: 2718: 2715: 2712: 2709: 2706: 2703: 2700: 2679: 2678: 2667: 2662: 2658: 2655: 2652: 2647: 2643: 2639: 2636: 2633: 2630: 2626: 2622: 2617: 2614: 2610: 2606: 2602: 2598: 2595: 2592: 2588: 2582: 2578: 2574: 2571: 2567: 2562: 2558: 2553: 2550: 2546: 2542: 2539: 2536: 2533: 2530: 2478: 2475: 2442: 2441: 2438: 2435: 2430: 2427: 2424: 2413:There are six 2410: 2407: 2380: 2379: 2362: 2359: 2354: 2350: 2344: 2340: 2336: 2333: 2326: 2322: 2316: 2313: 2310: 2307: 2304: 2298: 2293: 2284: 2261: 2260: 2249: 2246: 2243: 2240: 2237: 2234: 2231: 2228: 2225: 2222: 2219: 2216: 2213: 2209: 2206: 2201: 2192: 2168: 2148: 2137: 2136: 2119: 2116: 2110: 2106: 2103: 2098: 2094: 2088: 2084: 2080: 2077: 2073: 2066: 2062: 2055: 2049: 2045: 2041: 2038: 2034: 2030: 2027: 2021: 2016: 2007: 1972: 1971: 1970:0.000 km 1968: 1965: 1961: 1960: 1957: 1954: 1950: 1949: 1946: 1943: 1939: 1938: 1935: 1932: 1928: 1927: 1924: 1921: 1917: 1916: 1913: 1910: 1906: 1905: 1902: 1899: 1895: 1894: 1888: 1881: 1875: 1868: 1857: 1844: 1843: 1832: 1829: 1821: 1818: 1813: 1808: 1804: 1801: 1796: 1792: 1786: 1782: 1778: 1775: 1771: 1765: 1759: 1755: 1751: 1748: 1744: 1740: 1737: 1734: 1731: 1727: 1724: 1721: 1718: 1715: 1712: 1709: 1706: 1703: 1700: 1658: 1657: 1644: 1639: 1636: 1631: 1627: 1624: 1618: 1613: 1574: 1573: 1561: 1558: 1554: 1546: 1543: 1538: 1533: 1528: 1525: 1521: 1516: 1512: 1506: 1502: 1498: 1495: 1491: 1484: 1479: 1475: 1470: 1464: 1460: 1456: 1453: 1449: 1445: 1442: 1438: 1435: 1431: 1427: 1423: 1420: 1416: 1413: 1408: 1403: 1399: 1395: 1392: 1389: 1386: 1383: 1356: 1353: 1334:of the Earth. 1324: 1323: 1309: 1306: 1303: 1300: 1297: 1294: 1291: 1286: 1282: 1279: 1273: 1270: 1267: 1264: 1261: 1258: 1255: 1250: 1246: 1239: 1235: 1231: 1226: 1223: 1220: 1217: 1214: 1191:is denoted by 1180: 1177: 1159:Main article: 1156: 1153: 1130: 1129: 1120: 1111: 1084:3D polar angle 1071: 1041: 1021: 999: 996: 991: 990: 967: 943: 942: 910: 809: 808: 797: 789: 785: 781: 778: 773: 770: 767: 764: 761: 758: 755: 752: 749: 746: 742: 737: 733: 729: 726: 723: 720: 715: 711: 706: 701: 697: 694: 691: 685: 682: 615: 612: 557:published the 547:Main article: 544: 541: 539: 536: 533: 532: 523: 518: 509: 506: 505: 503: 500: 498: 495: 420: 419: 416: 415: 412: 406: 405: 402: 396: 395: 392: 386: 385: 382: 360: 357: 336:Prime Meridian 323: 320: 306: 286: 270: 267: 230: 227: 138:mean sea level 129: 126: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 9173: 9162: 9159: 9157: 9154: 9152: 9149: 9148: 9146: 9127: 9124: 9122: 9119: 9117: 9114: 9112: 9109: 9107: 9104: 9103: 9100: 9093: 9089: 9075: 9072: 9070: 9067: 9065: 9062: 9060: 9057: 9055: 9052: 9050: 9047: 9045: 9042: 9041: 9039: 9037: 9033: 9023: 9020: 9019: 9016: 9010: 9009:Stereographic 9007: 9005: 9002: 9000: 8997: 8996: 8994: 8992: 8988: 8985: 8983: 8979: 8973: 8970: 8968: 8965: 8964: 8962: 8960: 8956: 8950: 8949:Winkel tripel 8947: 8945: 8942: 8940: 8937: 8935: 8932: 8930: 8929:Natural Earth 8927: 8925: 8922: 8920: 8917: 8915: 8912: 8911: 8909: 8907: 8903: 8899: 8895: 8890: 8886: 8872: 8869: 8867: 8864: 8862: 8859: 8858: 8856: 8850: 8846: 8840: 8837: 8835: 8832: 8831: 8829: 8827: 8823: 8817: 8814: 8813: 8811: 8809: 8805: 8799: 8796: 8794: 8791: 8789: 8786: 8784: 8781: 8779: 8776: 8775: 8773: 8771: 8765: 8755: 8752: 8750: 8747: 8745: 8742: 8740: 8737: 8735: 8732: 8730: 8727: 8725: 8722: 8720: 8717: 8715: 8712: 8710: 8709:Briesemeister 8707: 8705: 8702: 8701: 8698: 8692: 8689: 8687: 8684: 8683: 8681: 8679: 8675: 8669: 8666: 8664: 8661: 8659: 8656: 8654: 8651: 8649: 8646: 8644: 8641: 8639: 8636: 8635: 8633: 8631: 8627: 8621: 8618: 8616: 8613: 8612: 8610: 8608: 8604: 8598: 8595: 8593: 8590: 8589: 8587: 8585: 8581: 8578: 8576: 8572: 8566: 8563: 8561: 8560:Stereographic 8558: 8556: 8553: 8551: 8548: 8546: 8543: 8541: 8538: 8536: 8533: 8531: 8528: 8527: 8525: 8523: 8519: 8515: 8511: 8506: 8502: 8488: 8487:Winkel tripel 8485: 8483: 8480: 8478: 8475: 8473: 8470: 8469: 8467: 8465: 8461: 8451: 8448: 8446: 8443: 8442: 8439: 8433: 8432:Stereographic 8430: 8428: 8425: 8423: 8420: 8419: 8417: 8415: 8411: 8408: 8406: 8398: 8392: 8389: 8385: 8382: 8380: 8377: 8376: 8375: 8372: 8370: 8367: 8365: 8362: 8361: 8359: 8357: 8356:Pseudoconical 8353: 8347: 8344: 8342: 8339: 8337: 8334: 8333: 8331: 8329: 8325: 8315: 8312: 8310: 8307: 8305: 8302: 8301: 8298: 8292: 8289: 8287: 8284: 8282: 8279: 8277: 8274: 8272: 8269: 8267: 8264: 8262: 8259: 8257: 8254: 8252: 8249: 8248: 8246: 8242: 8239: 8237: 8233: 8223: 8220: 8218: 8215: 8213: 8210: 8208: 8205: 8203: 8200: 8198: 8195: 8193: 8190: 8188: 8185: 8184: 8181: 8175: 8172: 8170: 8167: 8165: 8162: 8160: 8157: 8155: 8152: 8150: 8147: 8145: 8142: 8141: 8139: 8137: 8133: 8127: 8124: 8122: 8119: 8117: 8114: 8113: 8111: 8108: 8104: 8101: 8099: 8095: 8091: 8087: 8082: 8078: 8072: 8069: 8067: 8064: 8062: 8059: 8058: 8055: 8051: 8044: 8039: 8037: 8032: 8030: 8025: 8024: 8021: 8008: 8001: 7994: 7987: 7980: 7975: 7970: 7965: 7960: 7959: 7953: 7948: 7943: 7938: 7937: 7933: 7928: 7923: 7918: 7913: 7908: 7907: 7901: 7896: 7891: 7886: 7885: 7881: 7876: 7871: 7866: 7861: 7856: 7851: 7846: 7841: 7836: 7831: 7826: 7821: 7816: 7811: 7806: 7801: 7796: 7791: 7786: 7785: 7779: 7774: 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: 7478: 7472: 7467: 7462: 7461: 7455: 7450: 7445: 7444: 7438: 7433: 7428: 7427: 7421: 7416: 7411: 7410: 7407: 7404: 7398: 7393: 7392:Arctic Circle 7388: 7383: 7378: 7377: 7371: 7364: 7362:Arctic Circle 7357: 7350: 7343: 7342: 7334: 7327: 7325:Arctic Circle 7320: 7313: 7306: 7305: 7297: 7289: 7285: 7281: 7274: 7269: 7267: 7262: 7260: 7255: 7254: 7251: 7245: 7242: 7240: 7237: 7234: 7231: 7229: 7226: 7223: 7219: 7215: 7211: 7208: 7206: 7202: 7199: 7196: 7193: 7189: 7185: 7181: 7178: 7177: 7173: 7165: 7163:3-211-33544-7 7159: 7155: 7148: 7145: 7141: 7135: 7132: 7127: 7123: 7118: 7113: 7109: 7105: 7104:Survey Review 7098: 7095: 7090: 7083: 7080: 7075: 7071: 7067: 7063: 7059: 7055: 7050: 7045: 7041: 7037: 7030: 7027: 7022: 7018: 7014: 7008: 7005: 7000: 6996: 6992: 6988: 6984: 6980: 6975: 6970: 6966: 6962: 6955: 6952: 6947: 6943: 6939: 6935: 6928: 6925: 6920: 6916: 6912: 6908: 6904: 6900: 6895: 6890: 6886: 6882: 6881:Astron. Nachr 6877: 6871: 6867: 6863: 6859: 6855: 6851: 6846: 6841: 6837: 6833: 6832:Astron. Nachr 6826: 6823: 6818: 6811: 6808: 6804: 6795: 6794: 6789: 6783: 6781: 6777: 6766:on 2008-05-16 6765: 6761: 6760: 6752: 6750: 6748: 6746: 6744: 6740: 6729:on 2012-12-11 6728: 6724: 6718: 6715: 6710: 6706: 6702: 6695: 6693: 6691: 6689: 6685: 6679: 6675: 6671: 6664: 6662: 6660: 6658: 6656: 6652: 6647: 6645:3-11-017072-8 6641: 6637: 6630: 6628: 6626: 6624: 6622: 6618: 6602: 6595: 6592: 6587: 6583: 6582: 6574: 6571: 6558: 6551: 6548: 6536: 6532: 6526: 6523: 6517: 6507: 6504: 6500: 6494: 6491: 6484: 6479: 6474: 6471: 6469: 6466: 6464: 6461: 6459: 6456: 6454: 6451: 6449: 6446: 6444: 6441: 6439: 6436: 6434: 6431: 6429: 6426: 6424: 6421: 6419: 6416: 6414: 6411: 6409: 6406: 6404: 6401: 6399: 6396: 6394: 6391: 6389: 6385: 6382: 6380: 6377: 6375: 6372: 6370: 6367: 6365: 6362: 6359: 6355: 6352: 6351: 6346: 6344: 6342: 6338: 6334: 6330: 6326: 6322: 6318: 6313: 6311: 6310: 6303: 6301: 6297: 6293: 6289: 6285: 6281: 6277: 6269: 6259: 6256: 6253: 6250: 6247: 6246: 6243: 6236: 6234: 6232: 6228: 6220: 6218: 6216: 6212: 6208: 6204: 6198: 6194: 6190: 6139: 6135: 6131: 6124: 6117: 6115: 6087: 6083: 6079: 6073: 6066: 6065: 6060: 6047: 6043: 6039: 6032: 6025: 6023: 6007: 6003: 5999: 5980: 5976: 5972: 5965: 5960: 5952: 5950: 5947: 5939: 5932: 5929: 5926: 5923: 5920: 5917: 5916: 5912: 5909: 5906: 5903: 5900: 5897: 5896: 5892: 5889: 5886: 5883: 5880: 5877: 5876: 5872: 5869: 5866: 5863: 5860: 5857: 5856: 5852: 5849: 5846: 5843: 5840: 5837: 5836: 5832: 5829: 5826: 5823: 5820: 5817: 5816: 5812: 5809: 5806: 5803: 5800: 5797: 5796: 5792: 5788: 5782: 5779: 5775: 5769: 5766: 5762: 5756: 5753: 5749: 5743: 5740: 5736: 5730: 5725: 5724: 5714: 5712: 5708: 5704: 5700: 5696: 5692: 5688: 5683: 5681: 5677: 5673: 5669: 5665: 5661: 5651: 5644: 5637: 5633: 5629: 5625: 5621: 5616: 5612: 5607: 5603: 5602: 5600: 5596: 5592: 5591: 5590: 5583: 5567: 5560: 5554: 5551: 5546: 5543: 5539: 5535: 5529: 5523: 5516: 5515: 5514: 5489: 5482: 5476: 5470: 5467: 5463: 5458: 5452: 5446: 5439: 5438: 5437: 5423: 5400: 5394: 5391: 5388: 5385: 5379: 5374: 5371: 5367: 5363: 5360: 5354: 5348: 5343: 5340: 5336: 5332: 5330: 5319: 5316: 5313: 5310: 5304: 5299: 5296: 5292: 5288: 5285: 5279: 5276: 5273: 5267: 5262: 5259: 5255: 5251: 5249: 5240: 5234: 5231: 5228: 5225: 5222: 5219: 5214: 5211: 5208: 5205: 5202: 5199: 5193: 5189: 5186: 5181: 5178: 5173: 5169: 5164: 5158: 5155: 5150: 5145: 5142: 5136: 5132: 5129: 5125: 5121: 5118: 5115: 5113: 5105: 5099: 5088: 5087: 5086: 5081:and constant 5076: 5064: 5060: 5052: 5044: 5042: 5040: 5035: 5031: 5026: 5024: 5020: 5014: 4988: 4981: 4978: 4975: 4972: 4966: 4961: 4958: 4954: 4950: 4947: 4941: 4935: 4930: 4927: 4923: 4918: 4914: 4911: 4908: 4906: 4897: 4892: 4885: 4882: 4879: 4876: 4870: 4865: 4862: 4858: 4854: 4851: 4845: 4842: 4839: 4833: 4828: 4825: 4821: 4816: 4812: 4809: 4805: 4801: 4796: 4793: 4789: 4785: 4783: 4773: 4770: 4765: 4761: 4754: 4751: 4745: 4739: 4736: 4733: 4730: 4727: 4724: 4719: 4716: 4713: 4710: 4707: 4704: 4698: 4692: 4686: 4683: 4678: 4673: 4670: 4664: 4660: 4657: 4653: 4649: 4644: 4641: 4637: 4633: 4630: 4628: 4618: 4615: 4610: 4604: 4601: 4595: 4589: 4584: 4578: 4575: 4572: 4569: 4566: 4563: 4558: 4555: 4552: 4549: 4546: 4543: 4537: 4531: 4525: 4522: 4519: 4516: 4513: 4508: 4505: 4502: 4499: 4496: 4490: 4485: 4480: 4475: 4472: 4468: 4464: 4461: 4459: 4451: 4445: 4434: 4433: 4432: 4430: 4422: 4414: 4412: 4410: 4391: 4384: 4373: 4366: 4363: 4358: 4354: 4346: 4345: 4344: 4323: 4320: 4315: 4312: 4308: 4302: 4296: 4292: 4288: 4285: 4279: 4276: 4273: 4271: 4262: 4256: 4253: 4250: 4245: 4242: 4239: 4233: 4229: 4226: 4220: 4217: 4210: 4206: 4202: 4199: 4193: 4190: 4187: 4185: 4179: 4174: 4171: 4166: 4162: 4159: 4148: 4136: 4135: 4134: 4110: 4107: 4104: 4101: 4095: 4090: 4087: 4083: 4077: 4071: 4067: 4063: 4060: 4054: 4048: 4045: 4040: 4036: 4030: 4026: 4022: 4019: 4014: 4011: 4008: 4004: 3998: 3994: 3990: 3987: 3983: 3976: 3974: 3965: 3959: 3956: 3953: 3950: 3947: 3944: 3939: 3936: 3933: 3930: 3927: 3924: 3918: 3914: 3911: 3905: 3902: 3895: 3891: 3887: 3884: 3878: 3872: 3869: 3864: 3860: 3854: 3850: 3846: 3843: 3838: 3835: 3832: 3828: 3822: 3818: 3814: 3811: 3807: 3800: 3798: 3790: 3784: 3773: 3772: 3771: 3753: 3740: 3732: 3726: 3720: 3716: 3711: 3708: 3704: 3700: 3694: 3688: 3681: 3680: 3679: 3678:to a sphere. 3677: 3669: 3665: 3659: 3651: 3649: 3647: 3643: 3639: 3618: 3606: 3602: 3596: 3593: 3586: 3585: 3584: 3567: 3562: 3557: 3554: 3549: 3545: 3542: 3531: 3523: 3522: 3521: 3519: 3500: 3495: 3492: 3488: 3480: 3477: 3472: 3467: 3462: 3459: 3455: 3450: 3446: 3440: 3436: 3432: 3429: 3425: 3418: 3413: 3409: 3404: 3398: 3394: 3390: 3387: 3383: 3379: 3376: 3370: 3364: 3357: 3356: 3355: 3353: 3320: 3312: 3306: 3298: 3295: 3290: 3284: 3278: 3271: 3270: 3269: 3245: 3239: 3231: 3229: 3227: 3222: 3220: 3201: 3197: 3194: 3191: 3188: 3185: 3182: 3178: 3174: 3171: 3168: 3165: 3162: 3159: 3152: 3151: 3150: 3133: 3129: 3126: 3119: 3115: 3109: 3105: 3099: 3092: 3088: 3082: 3078: 3068: 3067: 3066: 3064: 3036: 3032: 3029: 3026: 3020: 3017: 3014: 3007: 3003: 2998: 2995: 2991: 2987: 2983: 2979: 2976: 2973: 2966: 2962: 2958: 2955: 2949: 2945: 2940: 2937: 2933: 2929: 2923: 2917: 2910: 2909: 2908: 2904: 2900: 2875: 2871: 2858: 2851: 2849: 2835: 2831: 2827: 2798: 2794: 2791: 2788: 2782: 2779: 2776: 2771: 2768: 2763: 2755: 2752: 2749: 2743: 2736: 2732: 2727: 2724: 2720: 2716: 2710: 2707: 2704: 2698: 2691: 2690: 2689: 2687: 2684: 2665: 2660: 2656: 2653: 2650: 2645: 2637: 2634: 2631: 2624: 2620: 2615: 2612: 2608: 2604: 2600: 2596: 2593: 2590: 2586: 2580: 2576: 2572: 2569: 2565: 2560: 2556: 2551: 2548: 2544: 2540: 2534: 2528: 2521: 2520: 2519: 2508: 2506: 2489: 2484: 2476: 2474: 2472: 2459: 2456: 2452: 2449:as discussed 2448: 2439: 2436: 2434: 2431: 2428: 2425: 2423: 2420: 2419: 2418: 2416: 2408: 2394: 2390: 2387: 2385: 2360: 2357: 2352: 2348: 2342: 2338: 2334: 2331: 2324: 2320: 2314: 2311: 2308: 2305: 2302: 2296: 2291: 2274: 2273: 2272: 2270: 2266: 2247: 2244: 2241: 2238: 2235: 2232: 2229: 2226: 2223: 2220: 2217: 2214: 2211: 2207: 2204: 2199: 2182: 2181: 2180: 2166: 2146: 2117: 2114: 2108: 2104: 2101: 2096: 2092: 2086: 2082: 2078: 2075: 2071: 2064: 2060: 2053: 2047: 2043: 2039: 2036: 2032: 2028: 2025: 2019: 2014: 1997: 1996: 1995: 1969: 1966: 1963: 1962: 1958: 1955: 1952: 1951: 1947: 1944: 1941: 1940: 1936: 1933: 1930: 1929: 1925: 1922: 1919: 1918: 1914: 1911: 1908: 1907: 1903: 1900: 1897: 1896: 1882: 1869: 1855: 1848: 1847: 1830: 1827: 1819: 1816: 1811: 1806: 1802: 1799: 1794: 1790: 1784: 1780: 1776: 1773: 1769: 1763: 1757: 1753: 1749: 1746: 1742: 1738: 1735: 1732: 1729: 1722: 1716: 1713: 1707: 1701: 1698: 1691: 1690: 1689: 1687: 1683: 1677: 1663: 1642: 1637: 1634: 1629: 1625: 1622: 1611: 1603: 1602: 1601: 1599: 1598: 1592: 1590: 1584: 1580: 1559: 1556: 1552: 1544: 1541: 1536: 1531: 1526: 1523: 1519: 1514: 1510: 1504: 1500: 1496: 1493: 1489: 1482: 1477: 1473: 1468: 1462: 1458: 1454: 1451: 1447: 1443: 1440: 1436: 1433: 1429: 1421: 1418: 1411: 1406: 1401: 1397: 1393: 1387: 1381: 1374: 1373: 1372: 1362: 1354: 1352: 1350: 1349:nautical mile 1333: 1284: 1280: 1277: 1248: 1244: 1237: 1233: 1229: 1224: 1218: 1212: 1205: 1204: 1203: 1199: 1195: 1190: 1178: 1176: 1174: 1168: 1162: 1154: 1152: 1149: 1145: 1140: 1138: 1134: 1119: 1110: 1089: 1085: 1081: 1077: 1076: 1072: 1065: 1064: 1060: 1059: 1058: 1039: 1019: 1010: 1005: 997: 995: 972: 968: 948: 947: 946: 925: 911: 894: 893: 892: 890: 881: 866: 862: 818: 814: 795: 787: 783: 779: 776: 771: 768: 762: 759: 756: 750: 747: 744: 740: 735: 731: 727: 724: 721: 718: 713: 709: 704: 699: 695: 692: 689: 683: 680: 673: 672: 671: 665: 657: 649: 641: 637: 629: 625: 620: 613: 611: 609: 605: 601: 597: 593: 589: 584: 582: 578: 574: 570: 566: 562: 561: 556: 550: 542: 537: 528: 522: 519: 514: 508: 490: 488: 484: 480: 475: 473: 469: 464: 460: 456: 452: 451:cross-section 447: 445: 441: 437: 425: 413: 411: 408: 407: 403: 401: 398: 397: 393: 391: 388: 387: 383: 381: 380:Arctic Circle 378: 377: 374: 373: 372: 365: 358: 356: 352: 349: 345: 341: 337: 333: 329: 321: 304: 284: 275: 268: 266: 264: 260: 256: 252: 248: 242: 236: 229:Determination 228: 226: 224: 219: 217: 212: 210: 206: 202: 198: 186: 181: 179: 175: 170: 168: 165:constitute a 164: 160: 156: 152: 148: 144: 139: 135: 127: 125: 123: 119: 118: 113: 108: 106: 102: 98: 94: 90: 86: 82: 78: 74: 70: 62: 58: 54: 50: 46: 45: 39: 33: 19: 9110: 9004:Orthographic 8535:Gauss–Krüger 8427:Orthographic 8222:Web Mercator 8116:Gauss–Krüger 7957: 7934: 7905: 7882: 7782: 7476: 7459: 7442: 7425: 7401: 7374: 7337: 7300: 7153: 7147: 7139: 7134: 7107: 7103: 7097: 7082: 7039: 7035: 7029: 7020: 7007: 6967:(1): 43–55. 6964: 6960: 6954: 6937: 6933: 6927: 6884: 6880: 6876:Translation: 6875: 6835: 6831: 6825: 6816: 6810: 6802: 6792: 6768:. Retrieved 6764:the original 6758: 6731:. Retrieved 6727:the original 6717: 6700: 6669: 6635: 6608:. Retrieved 6594: 6580: 6573: 6561:. Retrieved 6550: 6539:. Retrieved 6537:. 2021-06-01 6534: 6525: 6506: 6493: 6331:), nor with 6314: 6307: 6304: 6267: 6266: 6224: 6206: 6202: 6196: 6192: 6188: 6144: 6137: 6133: 6129: 6085: 6081: 6077: 6062: 6058: 6052: 6045: 6041: 6037: 6005: 6001: 5997: 5985: 5978: 5974: 5970: 5943: 5790: 5786: 5777: 5773: 5764: 5760: 5751: 5747: 5738: 5734: 5710: 5706: 5702: 5698: 5694: 5690: 5686: 5684: 5679: 5675: 5671: 5667: 5663: 5659: 5656: 5635: 5631: 5627: 5623: 5619: 5587: 5504: 5421: 5419: 5050: 5048: 5033: 5029: 5027: 5021:. (See also 5012: 5008: 4420: 4418: 4406: 4342: 4132: 3769: 3663: 3661: 3635: 3582: 3515: 3352:Meridian arc 3345: 3243: 3241: 3223: 3218: 3216: 3148: 3052: 2902: 2898: 2873: 2869: 2867: 2814: 2685: 2680: 2509: 2504: 2502: 2460: 2454: 2443: 2414: 2412: 2388: 2381: 2262: 2138: 1975: 1685: 1682:Meridian arc 1678: 1659: 1595: 1593: 1582: 1578: 1575: 1371:in radians) 1361:Meridian arc 1358: 1330:denotes the 1325: 1197: 1193: 1182: 1170: 1161:Meridian arc 1144:Eiffel Tower 1141: 1132: 1131: 1117: 1108: 1082:, after the 1079: 1073: 1061: 1055: 992: 970: 944: 923: 879: 867:flattening, 864: 810: 664:eccentricity 633: 627: 623: 585: 577:Meridian arc 572: 568: 558: 555:Isaac Newton 552: 520: 476: 448: 435: 421: 370: 353: 325: 244: 220: 213: 182: 171: 154: 131: 115: 111: 109: 100: 72: 66: 60: 52: 43: 8982:Perspective 8770:some aspect 8754:Strebe 1995 8729:Equal Earth 8648:Gall–Peters 8630:Cylindrical 8445:Equidistant 8341:Equidistant 8271:Equal Earth 8154:Gall–Peters 8098:Cylindrical 7224:conversion. 7222:sexagesimal 6384:Declination 6321:astronomers 6317:declination 6300:declination 6201:define the 6059:polar angle 3674:, gives an 3228:, Karney). 1332:mean radius 255:theodolites 209:The Needles 18:Latitudinal 9161:Navigation 9145:Categories 9044:AuthaGraph 9036:Polyhedral 8906:Compromise 8834:Loximuthal 8826:Loxodromic 8788:Sinusoidal 8638:Balthasart 8615:Sinusoidal 8592:Sinusoidal 8575:Equal-area 8286:Sinusoidal 8244:Equal-area 8144:Balthasart 8136:Equal-area 8109:-conformal 8086:By surface 7117:2212.05818 6770:2017-09-02 6733:2011-02-08 6709:1811/24333 6541:2022-01-16 6480:References 6463:Navigation 6423:Geotagging 6379:Colatitude 6280:plumb line 6205:or simply 6064:colatitude 5783:Geocentric 5757:Rectifying 5731:Parametric 3656:See also: 3236:See also: 2481:See also: 1165:See also: 1002:See also: 656:flattening 543:Ellipsoids 444:axial tilt 348:South Pole 344:North Pole 233:See also: 128:Background 77:coordinate 9116:Longitude 8944:Wagner VI 8793:Two-point 8724:Eckert VI 8719:Eckert IV 8714:Eckert II 8691:Mollweide 8686:Collignon 8653:Hobo–Dyer 8607:Bottomley 8522:Conformal 8510:By metric 8401:Azimuthal 8374:Polyconic 8369:Bottomley 8309:Wagner VI 8281:Mollweide 8266:Eckert VI 8261:Eckert IV 8256:Eckert II 8251:Collignon 8159:Hobo–Dyer 7284:meridians 7074:118619524 7049:1002.1417 6999:119310141 6974:1109.4448 6934:Phil. Mag 6919:118630614 6894:0908.1824 6870:118760590 6845:0908.1824 6518:Citations 6485:Footnotes 6453:Longitude 6257:Continent 6251:Ellipsoid 5770:Conformal 5608:accuracy. 5561:ϕ 5555:χ 5552:⁡ 5544:− 5530:ϕ 5524:ψ 5483:ϕ 5477:ψ 5471:π 5453:ϕ 5395:ϕ 5392:⁡ 5380:⁡ 5372:− 5361:− 5355:ϕ 5349:⁡ 5341:− 5320:ϕ 5317:⁡ 5305:⁡ 5297:− 5286:− 5280:ϕ 5277:⁡ 5268:⁡ 5260:− 5235:ϕ 5232:⁡ 5215:ϕ 5212:⁡ 5203:− 5190:⁡ 5156:ϕ 5143:π 5133:⁡ 5122:⁡ 5106:ϕ 5100:ψ 5075:graticule 5030:arbitrary 4982:ϕ 4979:⁡ 4967:⁡ 4959:− 4948:− 4942:ϕ 4936:⁡ 4928:− 4915:⁡ 4886:ϕ 4883:⁡ 4871:⁡ 4863:− 4852:− 4846:ϕ 4843:⁡ 4834:⁡ 4826:− 4813:⁡ 4802:⁡ 4794:− 4771:π 4766:− 4740:ϕ 4737:⁡ 4720:ϕ 4717:⁡ 4708:− 4684:π 4671:ϕ 4661:⁡ 4650:⁡ 4642:− 4616:π 4611:− 4579:ϕ 4576:⁡ 4559:ϕ 4556:⁡ 4547:− 4526:ϕ 4523:⁡ 4517:− 4509:ϕ 4506:⁡ 4481:⁡ 4473:− 4452:ϕ 4446:χ 4429:conformal 4321:⁡ 4313:− 4289:− 4243:− 4230:⁡ 4203:− 4194:− 4172:π 4111:ϕ 4108:⁡ 4096:⁡ 4088:− 4064:− 4049:ϕ 4046:⁡ 4023:− 4015:ϕ 4012:⁡ 3991:− 3960:ϕ 3957:⁡ 3940:ϕ 3937:⁡ 3928:− 3915:⁡ 3888:− 3879:− 3873:ϕ 3870:⁡ 3847:− 3839:ϕ 3836:⁡ 3815:− 3791:ϕ 3733:ϕ 3717:⁡ 3709:− 3695:ϕ 3689:ξ 3668:same area 3619:π 3555:π 3493:ϕ 3473:− 3460:ϕ 3456:⁡ 3433:− 3419:ϕ 3410:∫ 3391:− 3371:ϕ 3313:ϕ 3296:π 3285:ϕ 3279:μ 3268:radians: 3198:β 3195:⁡ 3175:β 3172:⁡ 3033:ϕ 3030:⁡ 3018:− 3004:⁡ 2996:− 2980:ϕ 2977:⁡ 2959:− 2946:⁡ 2938:− 2924:ϕ 2918:β 2836:θ 2832:− 2828:ϕ 2795:ϕ 2792:⁡ 2753:− 2733:⁡ 2725:− 2705:ϕ 2699:θ 2657:ϕ 2654:⁡ 2635:− 2621:⁡ 2613:− 2597:ϕ 2594:⁡ 2573:− 2557:⁡ 2549:− 2535:ϕ 2529:θ 2361:ϕ 2358:⁡ 2335:− 2325:∘ 2315:ϕ 2312:⁡ 2303:π 2283:Δ 2248:ϕ 2242:⁡ 2230:ϕ 2224:⁡ 2215:− 2191:Δ 2167:ϕ 2147:ϕ 2105:ϕ 2102:⁡ 2079:− 2065:∘ 2040:− 2026:π 2006:Δ 1856:ϕ 1831:ϕ 1828:δ 1812:− 1803:ϕ 1800:⁡ 1777:− 1750:− 1733:ϕ 1730:δ 1723:ϕ 1708:ϕ 1699:δ 1635:π 1557:ϕ 1537:− 1524:ϕ 1520:⁡ 1497:− 1483:ϕ 1474:∫ 1455:− 1434:ϕ 1419:ϕ 1407:ϕ 1398:∫ 1388:ϕ 1285:ϕ 1249:ϕ 1238:∘ 1230:π 1219:ϕ 1175:assumed. 1040:λ 1020:ϕ 907:.0 m 817:ellipsoid 780:− 760:− 728:− 693:− 569:ellipsoid 332:meridians 305:λ 285:ϕ 151:graticule 105:longitude 101:parallels 89:the Earth 69:geography 61:parallels 53:meridians 49:longitude 44:graticule 9111:Latitude 9096:See also 9059:Dymaxion 8999:Gnomonic 8934:Robinson 8839:Mercator 8816:Gnomonic 8808:Gnomonic 8643:Behrmann 8550:Mercator 8422:Gnomonic 8404:(planar) 8379:American 8149:Behrmann 8107:Mercator 7214:Archived 7201:Archived 7184:Archived 7015:(1779). 6790:(1921). 6610:25 April 6354:Altitude 6347:See also 6337:ecliptic 5893:−10.02′ 5873:−11.55′ 5744:Authalic 5420:For the 5061:and the 3496:′ 3463:′ 3350:is (see 3226:Vincenty 1674: km 1560:′ 1527:′ 1437:′ 1422:′ 581:triaxial 573:spheroid 553:In 1687 485:and the 455:solstice 424:ecliptic 73:latitude 41:Earth's 9156:Geodesy 8972:HEALPix 8871:Littrow 8482:Wiechel 8384:Chinese 8328:Conical 8192:Central 8187:Cassini 8164:Lambert 8061:History 7376:Equator 7340:Equator 7303:Equator 7054:Bibcode 6979:Bibcode 6899:Bibcode 6850:Bibcode 6636:Geodesy 6398:Geodesy 5913:−5.79′ 5890:−10.01′ 5870:−11.54′ 5853:−9.98′ 5833:−5.76′ 5017:is the 3266:⁠ 3252:⁠ 3063:ellipse 2386:(NGA). 2218:559.822 2212:132.954 1992:⁠ 1978:⁠ 965: m 941:exactly 933:298.257 928:⁠ 914:⁠ 909:exactly 884:⁠ 870:⁠ 865:inverse 857:⁠ 845:⁠ 835:. 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6260:Geoid 6248:Ocean 5930:0.00′ 5927:0.00′ 5924:0.00′ 5921:0.00′ 5810:0.00′ 5807:0.00′ 5804:0.00′ 5801:0.00′ 5701:sin 2 5034:small 3520:) is 2471:below 2451:below 2265:WGS84 2236:1.175 1686:small 1662:WGS84 1202:then 1088:below 976:0.006 889:WGS84 604:WGS84 596:geoid 440:epoch 328:poles 134:geoid 99:, or 85:south 81:north 75:is a 51:, or 9064:ISEA 8066:List 7778:170° 7773:160° 7768:140° 7763:130° 7758:110° 7753:100° 7718:170° 7713:160° 7708:140° 7703:130° 7698:110° 7693:100° 7658:175° 7653:165° 7648:155° 7643:145° 7638:135° 7633:125° 7628:115° 7623:105° 7568:175° 7563:165° 7558:155° 7553:145° 7548:135° 7543:125° 7538:115° 7533:105° 7477:180° 7471:150° 7466:120° 7443:180° 7437:150° 7432:120° 7158:ISBN 6803:Note 6640:ISBN 6612:2020 6565:2020 6153:and 5713:= . 5597:and 5368:tanh 5293:tanh 5256:sinh 5049:The 4955:tanh 4859:tanh 4822:sinh 4810:sinh 4419:The 4309:tanh 4133:and 4084:tanh 3670:"), 3662:The 3242:The 3065:is: 2868:The 2503:The 2455:only 2287:long 1889:long 1670:.965 1660:For 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6134:β 6132:, 6130:u 6108:λ 6100:O 6096:P 6092:r 6088:) 6086:λ 6082:θ 6080:, 6078:r 6055:θ 6048:) 6046:λ 6042:θ 6040:, 6038:r 6012:N 6008:) 6006:h 6004:, 6002:λ 6000:, 5998:ɸ 5988:P 5981:) 5979:h 5977:, 5975:λ 5973:, 5971:ɸ 5791:ϕ 5787:θ 5778:ϕ 5774:χ 5765:ϕ 5761:μ 5752:ϕ 5748:ξ 5739:ϕ 5735:β 5727:ϕ 5719:ϕ 5711:ζ 5707:C 5703:ϕ 5695:ϕ 5691:ζ 5687:f 5680:ϕ 5676:β 5672:ξ 5668:μ 5664:χ 5660:θ 5638:) 5636:b 5632:a 5628:b 5624:a 5620:n 5568:. 5564:) 5558:( 5547:1 5536:= 5533:) 5527:( 5511:χ 5507:ψ 5490:. 5486:) 5480:( 5468:2 5464:E 5459:= 5456:) 5450:( 5447:y 5434:ϕ 5430:y 5426:E 5401:. 5398:) 5386:e 5383:( 5375:1 5364:e 5358:) 5352:( 5344:1 5333:= 5323:) 5311:e 5308:( 5300:1 5289:e 5283:) 5271:( 5263:1 5252:= 5241:] 5226:e 5223:+ 5220:1 5206:e 5200:1 5194:[ 5182:2 5179:e 5174:+ 5170:] 5165:) 5159:2 5151:+ 5146:4 5137:( 5126:[ 5116:= 5109:) 5103:( 5083:λ 5079:ψ 5071:λ 5067:ψ 5055:ψ 5015:) 5013:x 4989:] 4985:) 4973:e 4970:( 4962:1 4951:e 4945:) 4939:( 4931:1 4919:[ 4909:= 4898:] 4893:) 4889:) 4877:e 4874:( 4866:1 4855:e 4849:) 4837:( 4829:1 4817:( 4806:[ 4797:1 4786:= 4774:2 4762:] 4755:2 4752:e 4746:) 4731:e 4728:+ 4725:1 4711:e 4705:1 4699:( 4693:) 4687:4 4679:+ 4674:2 4665:( 4654:[ 4645:1 4634:2 4631:= 4619:2 4605:2 4602:1 4596:] 4590:e 4585:) 4570:e 4567:+ 4564:1 4550:e 4544:1 4538:( 4532:) 4514:1 4500:+ 4497:1 4491:( 4486:[ 4476:1 4465:2 4462:= 4455:) 4449:( 4425:χ 4392:. 4385:2 4379:p 4374:q 4367:a 4364:= 4359:q 4355:R 4324:e 4316:1 4303:e 4297:2 4293:e 4286:1 4280:+ 4277:1 4274:= 4263:) 4257:e 4254:+ 4251:1 4246:e 4240:1 4234:( 4221:e 4218:2 4211:2 4207:e 4200:1 4191:1 4188:= 4180:) 4175:2 4167:( 4163:q 4160:= 4154:p 4149:q 4114:) 4102:e 4099:( 4091:1 4078:e 4072:2 4068:e 4061:1 4055:+ 4041:2 4031:2 4027:e 4020:1 4005:) 3999:2 3995:e 3988:1 3984:( 3977:= 3966:) 3951:e 3948:+ 3945:1 3931:e 3925:1 3919:( 3906:e 3903:2 3896:2 3892:e 3885:1 3865:2 3855:2 3851:e 3844:1 3829:) 3823:2 3819:e 3812:1 3808:( 3801:= 3794:) 3788:( 3785:q 3754:) 3746:p 3741:q 3736:) 3730:( 3727:q 3721:( 3712:1 3701:= 3698:) 3692:( 3672:ξ 3612:p 3607:m 3603:2 3597:= 3594:R 3568:. 3563:) 3558:2 3550:( 3546:m 3543:= 3537:p 3532:m 3501:, 3489:d 3481:2 3478:3 3468:) 3451:2 3441:2 3437:e 3430:1 3426:( 3414:0 3405:) 3399:2 3395:e 3388:1 3384:( 3380:a 3377:= 3374:) 3368:( 3365:m 3348:ϕ 3326:p 3321:m 3316:) 3310:( 3307:m 3299:2 3291:= 3288:) 3282:( 3263:2 3260:/ 3256:π 3248:μ 3202:; 3189:b 3186:= 3183:z 3179:, 3166:a 3163:= 3160:p 3134:. 3130:1 3127:= 3120:2 3116:b 3110:2 3106:z 3100:+ 3093:2 3089:a 3083:2 3079:p 3059:z 3055:p 3037:) 3024:) 3021:f 3015:1 3012:( 3008:( 2999:1 2988:= 2984:) 2967:2 2963:e 2956:1 2950:( 2941:1 2930:= 2927:) 2921:( 2905:) 2903:ϕ 2901:( 2899:u 2894:ϕ 2890:P 2886:a 2882:Q 2878:β 2862:β 2817:N 2799:) 2783:h 2780:+ 2777:N 2772:h 2769:+ 2764:2 2760:) 2756:f 2750:1 2747:( 2744:N 2737:( 2728:1 2717:= 2714:) 2711:h 2708:, 2702:( 2686:h 2666:. 2661:) 2646:2 2642:) 2638:f 2632:1 2629:( 2625:( 2616:1 2605:= 2601:) 2587:) 2581:2 2577:e 2570:1 2566:( 2561:( 2552:1 2541:= 2538:) 2532:( 2516:ϕ 2512:θ 2499:) 2497:θ 2493:ϕ 2467:e 2463:a 2402:θ 2398:ϕ 2353:2 2343:2 2339:e 2332:1 2306:a 2297:= 2292:1 2245:4 2233:+ 2227:2 2205:= 2200:1 2118:2 2115:3 2109:) 2097:2 2087:2 2083:e 2076:1 2072:( 2054:) 2048:2 2044:e 2037:1 2033:( 2029:a 2020:= 2015:1 1986:/ 1982:π 1884:Δ 1871:Δ 1820:2 1817:3 1807:) 1795:2 1785:2 1781:e 1774:1 1770:( 1764:) 1758:2 1754:e 1747:1 1743:( 1739:a 1736:= 1726:) 1720:( 1717:M 1714:= 1711:) 1705:( 1702:m 1643:) 1638:2 1630:( 1626:m 1623:= 1617:p 1612:m 1585:) 1583:ϕ 1581:( 1579:M 1553:d 1545:2 1542:3 1532:) 1515:2 1505:2 1501:e 1494:1 1490:( 1478:0 1469:) 1463:2 1459:e 1452:1 1448:( 1444:a 1441:= 1430:d 1426:) 1415:( 1412:M 1402:0 1394:= 1391:) 1385:( 1382:m 1369:ϕ 1365:ϕ 1344:R 1340:R 1336:R 1328:R 1308:s 1305:n 1302:a 1299:i 1296:d 1293:a 1290:r 1281:R 1278:= 1272:s 1269:e 1266:e 1263:r 1260:g 1257:e 1254:d 1245:R 1225:= 1222:) 1216:( 1213:m 1200:) 1198:ϕ 1196:( 1194:m 1185:ϕ 1128:. 1126:θ 1121:g 1118:ϕ 1112:c 1109:ϕ 1100:q 1096:ψ 1092:θ 1068:ϕ 971:e 954:6 950:b 924:f 920:/ 917:1 900:6 896:a 880:f 876:/ 873:1 851:/ 848:1 841:e 837:f 833:e 829:f 825:b 821:a 796:. 788:2 784:e 777:1 772:a 769:= 766:) 763:f 757:1 754:( 751:a 748:= 745:b 741:, 736:2 732:f 725:f 722:2 719:= 714:2 710:e 705:, 700:a 696:b 690:a 684:= 681:f 668:e 660:f 652:b 644:a 628:z 624:a 436:i 432:i 428:i 193:φ 189:ϕ 187:( 83:– 34:. 20:)

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