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:
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 one 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
4437:
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 23.43607°). 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:
17:
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.
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:
18:Conformal latitude
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
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:. Both
813:ellipse
636:ellipse
592:geodesy
468:tropics
340:Equator
263:geodesy
197:degrees
93:Equator
57:equator
8991:Planar
8959:Hybrid
8866:Hammer
8798:Werner
8739:Hammer
8704:Albers
8620:Werner
8597:Werner
8477:Hammer
8472:Aitoff
8391:Werner
8336:Albers
8212:Miller
8071:Portal
7160:
7072:
7021:Oevres
6997:
6917:
6868:
6642:
6563:24 May
6296:zenith
6090:where
5933:0.00′
5910:−5.78′
5907:−4.34′
5904:−3.86′
5901:−2.89′
5887:−7.51′
5884:−6.67′
5881:−5.00′
5867:−8.66′
5864:−7.70′
5861:−5.77′
5850:−9.98′
5847:−7.49′
5844:−6.66′
5841:−5.00′
5830:−5.76′
5827:−4.32′
5824:−3.84′
5821:−2.88′
5813:0.00′
5653:inline
5422:normal
5009:where
3770:where
2815:where
2518:) is:
1576:where
1326:where
565:oblate
472:zenith
163:height
159:normal
155:actual
143:sphere
117:normal
8861:Craig
8778:Conic
8584:Bonne
8364:Bonne
8006:45x90
7999:45x90
7992:45x90
7985:45x90
7112:arXiv
7070:S2CID
7044:arXiv
6995:S2CID
6969:arXiv
6915:S2CID
6889:arXiv
6866:S2CID
6840:arXiv
6797:(PDF)
6604:(PDF)
6339:(see
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6248:Ocean
5930:0.00′
5927:0.00′
5924:0.00′
5921:0.00′
5810:0.00′
5807:0.00′
5804:0.00′
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5034:small
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2451:below
2265:WGS84
2236:1.175
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1662:WGS84
1202:then
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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°
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7158:ISBN
6803:Note
6640:ISBN
6612:2020
6565:2020
6153:and
5713:= .
5597:and
5368:tanh
5293:tanh
5256:sinh
5049:The
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4822:sinh
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1670:.965
1660:For
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961:.314
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831:and
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