4846:
4150:
4841:{\displaystyle {\begin{aligned}\Delta {X}&=\cos(\phi _{2})\cos(\lambda _{2})-\cos(\phi _{1})\cos(\lambda _{1});\\\Delta {Y}&=\cos(\phi _{2})\sin(\lambda _{2})-\cos(\phi _{1})\sin(\lambda _{1});\\\Delta {Z}&=\sin(\phi _{2})-\sin(\phi _{1});\\D_{\textrm {t}}&=R{\sqrt {(\Delta {X})^{2}+(\Delta {Y})^{2}+(\Delta {Z})^{2}}}\\&=2R{\sqrt {\sin ^{2}{\frac {\Delta \phi }{2}}+\left(\cos ^{2}{\frac {\Delta \phi }{2}}-\sin ^{2}\phi _{\textrm {m}}\right)\sin ^{2}{\frac {\Delta \lambda }{2}}}}\\&=2R{\sqrt {\left(\sin {\frac {\Delta \lambda }{2}}\cos \phi _{\textrm {m}}\right)^{2}+\left(\cos {\frac {\Delta \lambda }{2}}\sin {\frac {\Delta \phi }{2}}\right)^{2}}}.\end{aligned}}}
4863:
31:
2122:
432:
1830:
6017:
3279:
2117:{\displaystyle {\begin{aligned}D&=R{\sqrt {\left(2\sin {\frac {\Delta \phi }{2}}\,\cos {\frac {\Delta \lambda }{2}}\right)^{2}+\left(2\cos \phi _{\textrm {m}}\sin {\frac {\Delta \lambda }{2}}\right)^{2}}}\\&\approx R{\sqrt {\left(\Delta \phi \,\cos {\frac {\Delta \lambda }{2}}\right)^{2}+\left(2\cos \phi _{\textrm {m}}\sin {\frac {\Delta \lambda }{2}}\right)^{2}}}\ .\end{aligned}}}
5778:
2456:
6663:
5644:
3051:
6012:{\displaystyle D=2N\left(\phi _{\textrm {m}}\right)\arcsin {\sqrt {\left(\sin {\frac {\Delta \lambda }{2}}\cos \phi _{\textrm {m}}\right)^{2}+\left(\cos {\frac {\Delta \lambda }{2}}\sin \left({\frac {\Delta \phi }{2}}{\frac {M\left(\phi _{\textrm {m}}\right)}{N\left(\phi _{\textrm {m}}\right)}}\right)\right)^{2}}}.}
4932:.) This defect is cured in the algorithm given by Karney, who employs series which are accurate to sixth order in the flattening. This results in an algorithm which is accurate to full double precision and which converges for arbitrary pairs of points on the Earth. This algorithm is implemented in GeographicLib.
2281:
1103:
be expressed in the specified units to obtain the correct result. Where geographic coordinates are used as the argument of a trigonometric function, the values may be expressed in any angular units compatible with the method used to determine the value of the trigonometric function. Many electronic
4935:
The exact methods above are feasible when carrying out calculations on a computer. They are intended to give millimeter accuracy on lines of any length; one can use simpler formulas if one doesn't need millimeter accuracy, or if one does need millimeter accuracy but the line is short.
3687:
2788:
1221:
5450:
6398:
4075:
3274:{\displaystyle {\begin{aligned}K_{1}&=111.13209-0.56605\cos(2\phi _{\mathrm {m} })+0.00120\cos(4\phi _{\mathrm {m} });\\K_{2}&=111.41513\cos(\phi _{\mathrm {m} })-0.09455\cos(3\phi _{\mathrm {m} })+0.00012\cos(5\phi _{\mathrm {m} }).\end{aligned}}\,\!}
2266:
This approximation is very fast and produces fairly accurate result for small distances . Also, when ordering locations by distance, such as in a database query, it is faster to order by squared distance, eliminating the need for computing the square root.
2614:
6164:
5442:
2260:
3497:
2451:{\displaystyle D={\sqrt {\left(M\left(\phi _{\textrm {m}}\right)\Delta \phi \,\cos {\frac {\Delta \lambda }{2}}\right)^{2}+\left(2N\left(\phi _{\textrm {m}}\right)\cos \phi _{\textrm {m}}\sin {\frac {\Delta \lambda }{2}}\right)^{2}}}}
6298:
2894:
4875:. Geodesics follow more complicated paths than great circles and in particular, they usually don't return to their starting positions after one circuit of the Earth. This is illustrated in the figure on the right where
6884:
1299:
3574:
3408:
3794:
2662:
6775:
6375:
1117:
7040:
5720:
4870:
An ellipsoid approximates the surface of the Earth much better than a sphere or a flat surface does. The shortest distance along the surface of an ellipsoid between two points on the surface is along the
5042:
3939:
6824:
5639:{\displaystyle X=(\sigma -\sin \sigma ){\frac {\sin ^{2}P\cos ^{2}Q}{\cos ^{2}{\frac {\sigma }{2}}}}\qquad \qquad Y=(\sigma +\sin \sigma ){\frac {\cos ^{2}P\sin ^{2}Q}{\sin ^{2}{\frac {\sigma }{2}}}}}
4142:
566:
1727:
If a calculation based on latitude/longitude should be valid for all Earth positions, it should be verified that the discontinuity and the Poles are handled correctly. Another solution is to use
6403:
4155:
3056:
1835:
1122:
5772:
It has the similar form of the arc length converted from tunnel distance. Detailed formulas are given by Rapp, §6.4. It is consistent with the above-mentioned flat-surface formulae apparently.
770:
614:
7663:: Formule donnant la longueur de la géodésique joignant 2 points de l’ellipsoïde donnés par leurs coordonnées géographiques, Bulletin Géodésique, Volume 34, Number 1, April 1932, pages 77–81,
6658:{\displaystyle {\begin{aligned}\tan \phi _{1}'&={\frac {\tan \phi _{1}}{B}},\\\Delta \phi '&={\frac {\Delta \phi }{B}}{\biggl },\\\Delta \lambda '&=A\Delta \lambda ,\end{aligned}}}
6715:
3732:
3045:
1017:
3976:
2654:
2162:
966:
5281:
5234:
3559:. For a more computationally efficient implementation of the formula above, multiple applications of cosine can be replaced with a single application and use of recurrence relation for
506:
All abstractions above ignore changes in elevation. Calculation of distances which account for changes in elevation relative to the idealized surface are not discussed in this article.
3005:
492:
distance, which is unattainable if one attempted to account for every irregularity in the surface of the Earth. Common abstractions for the surface between two geographic points are:
5164:
5135:
4924:, who uses a series which is accurate to third order in the flattening of the ellipsoid, i.e., about 0.5 mm; however, the algorithm fails to converge for points that are nearly
5103:
5074:
2534:
1805:
Even over short distances, the accuracy of geographic distance calculations which assume a flat Earth depend on the method by which the latitude and longitude coordinates have been
1610:
1500:
6898:
The variation in altitude from the topographical or ground level down to the sphere's or ellipsoid's surface, also changes the scale of distance measurements. The slant distance
2972:
7061:
are each point. The first term on the right-hand side of the equation accounts for the mean elevation and the second term for the inclination. A further reduction of the above
4943:
method, the Gauss mid-latitude method, and the
Bowring method. Karl Hubeny got the expanded series of Gauss mid-latitude one represented as the correction to flat-surface one.
1524:
1470:
2947:
3874:
1793:
1395:
918:
5345:
5317:
1722:
1666:
1580:
1552:
1446:
1093:
6037:
1694:
1638:
1071:
1046:
889:
860:
1364:
5187:
5353:
3968:
2170:
831:
3550:
3528:
3415:
3339:
3312:
2920:
2504:
2482:
1331:
724:
697:
670:
643:
5744:
4972:
3842:
1764:
794:
1366:= Distance between the two points, as measured along the surface of the Earth and in the same units as the value used for radius unless specified otherwise.
1397:, appearing in some flat-surface formulae below, may induce singularity and discontinuity. It may also degrade the accuracy in the case of higher latitude.
7719:
6175:
2808:
4085:
A tunnel between points on Earth is defined by a
Cartesian line through three-dimensional space between the points of interest. The tunnel distance
7575:
390:
1235:
3817:
The shortest distance along the surface of a sphere between two points on the surface is along the great-circle which contains the two points.
3682:{\displaystyle D=R{\sqrt {\theta _{1}^{2}\;{\boldsymbol {+}}\;\theta _{2}^{2}\;\mathbf {-} \;2\theta _{1}\theta _{2}\cos(\Delta \lambda )}},}
2783:{\displaystyle D={\sqrt {(M(\phi _{\mathrm {m} })\Delta \phi )^{2}+(N(\phi _{\mathrm {m} })\cos \phi _{\mathrm {m} }\Delta \lambda )^{2}}}.}
3350:
3739:
1216:{\displaystyle {\begin{aligned}\Delta \phi &=\phi _{2}-\phi _{1};\\\Delta \lambda &=\lambda _{2}-\lambda _{1}.\end{aligned}}\,\!}
212:
7750:
380:
348:
7138:
6920:
5652:
358:
6829:
4984:
4883:, was the focus of many mathematicians and geodesists over the course of the 18th and 19th centuries with major contributions by
2525:
920:, is measured along Cartesian straight line. The geographical coordinates of the two points, as (latitude, longitude) pairs, are
6890:
to give approximations for the spheroidal distance and bearing. Detailed formulas are given by Rapp §6.5, Bowring, and Karney.
3882:
6780:
7205:
7636:
1746:
A planar approximation for the surface of the Earth may be useful over very small distances. It approximates the arc length,
338:
6309:
4088:
518:
6720:
729:
573:
488:
Calculating the distance between geographical coordinates is based on some level of abstraction; it does not provide an
318:
1735:
119:
6671:
3695:
4070:{\displaystyle D=D_{\textrm {t}}\left(1+{\frac {1}{24}}\left({\frac {D_{\textrm {t}}}{R}}\right)^{2}+\cdots \right).}
2799:
418:
3010:
4917:
4879:
is taken to be 1/50 to accentuate the effect. Finding the geodesic between two points on the Earth, the so-called
1502:) may not give the expected answer for positions near the Poles or the ±180° meridian. Consider e.g. the value of
971:
2627:
2135:
923:
5239:
5192:
3844:
on a sphere about the size of the Earth. That article includes an example of the calculation. For example, from
7464:
7372:
7338:
2609:{\displaystyle \left(\cos \phi _{\textrm {m}}\sin {\frac {\Delta \lambda }{2}}\Delta \phi \right)^{2}\approx 0}
204:
2979:
7611:
7050:
5140:
5111:
220:
5079:
5050:
1585:
1475:
7726:
7356:
2952:
1409:
3342:
2521:
7695:
1505:
1451:
7676:
Lambert, W. D (1942). "The distance between two widely separated points on the surface of the earth".
2927:
4857:
4144:
is the great-circle chord length and may be calculated as follows for the corresponding unit sphere:
3850:
1769:
1377:
894:
183:
7592:
Bowring, B. R. (1981). "The direct and inverse problems for short geodesic lines on the ellipsoid".
6159:{\displaystyle A={\sqrt {1+e'^{2}\cos ^{4}\phi _{1}}},\quad B={\sqrt {1+e'^{2}\cos ^{2}\phi _{1}}},}
5322:
5294:
5106:
4978:
4920:, software libraries, standalone utilities, and online tools. The most widely used algorithm is by
1699:
1643:
1557:
1529:
1427:
7787:
7352:
7245:[Geometrical determination of the perpendicular to the meridian drawn by Jacques Cassini].
4896:
4880:
1401:
465:
150:
97:
5044:, for better accuracy. They give accuracy on the order of 10 meters over thousands of kilometers.
1809:
onto the plane. The projection of latitude and longitude coordinates onto a plane is the realm of
1076:
7422:"Direct and Inverse Solutions of Geodesics on the Ellipsoid with application of nested equations"
7098:
6887:
4951:
Historically, the long-line formulae were derived in the form of expansion series with regard to
1671:
1615:
440:
288:
248:
1226:
It is not important whether the result is positive or negative when used in the formulae below.
7238:
7108:
7103:
5437:{\displaystyle P={\frac {\beta _{1}+\beta _{2}}{2}}\qquad Q={\frac {\beta _{2}-\beta _{1}}{2}}}
4939:
The short-line methods have been studied by several researchers. Rapp, Chap. 6, describes the
4929:
4884:
3811:
2255:{\displaystyle D=R{\sqrt {(\Delta \phi )^{2}+(\cos(\phi _{\mathrm {m} })\Delta \lambda )^{2}}}}
1054:
1022:
865:
836:
268:
7264:
7242:
3492:{\displaystyle K_{2}=\cos(\phi _{\mathrm {m} })N(\phi _{\mathrm {m} }){\frac {\pi }{180}}\,\!}
1343:
7458:
7332:
7260:
7093:
5284:
5172:
4888:
3821:
3805:
3560:
833:
is the minimum distance along the surface of sphere/ellipsoid calculated between two points,
411:
328:
3947:
3736:
For a latitude measured in degrees, the colatitude in radians may be calculated as follows:
2802:
prescribes the following formulae for distances not exceeding 475 kilometres (295 mi):
811:
7803:
7569:
7497:
7304:
7066:
7062:
3533:
3511:
3317:
3290:
2903:
2487:
2465:
1824:
This formula takes into account the variation in distance between meridians with latitude:
1314:
702:
675:
648:
621:
228:
178:
104:
7142:
4910:
8:
7780:
7660:
7523:
3556:
1799:
140:
88:
7501:
7308:
1108:. The calculator mode must be compatible with the units used for geometric coordinates.
7513:
7487:
7366:
7320:
7294:
7243:"Détermination géometrique de la perpendiculaire à la méridienne tracée par M. Cassini"
7058:
6381:
5729:
4957:
4862:
3827:
1749:
1417:
779:
60:
30:
7744:
7642:
7632:
7517:
7417:
7324:
5288:
4921:
4900:
1405:
464:
The formulae in this article calculate distances between points which are defined by
160:
145:
3504:
Note that the expressions in the FCC formula are derived from the truncation of the
1798:
The shortest distance between two points in plane is a
Cartesian straight line. The
7813:
7557:
7505:
7436:
7312:
7282:
6903:
4892:
477:
404:
155:
7185:
7378:
7285:(2010) . "The calculation of longitude and latitude from geodesic measurements".
7088:
7078:
3505:
6293:{\displaystyle e'^{2}={\frac {a^{2}-b^{2}}{b^{2}}}={\frac {f(2-f)}{(1-f)^{2}}}.}
7808:
7624:
7440:
7421:
4940:
2889:{\displaystyle D={\sqrt {(K_{1}\Delta \phi )^{2}+(K_{2}\Delta \lambda )^{2}}},}
1806:
1421:
1413:
1104:
calculators allow calculations of trigonometric functions in either degrees or
1096:
431:
368:
135:
109:
7509:
6886:. The resulting problem on the sphere may be solved using the techniques for
3810:
If one is willing to accept a possible error of 0.5%, one can use formulas of
1741:
7797:
7646:
5167:
2624:
The above is furthermore simplified by approximating sinusoidal functions of
2132:
The above is furthermore simplified by approximating sinusoidal functions of
1334:
1111:
Differences in latitude and longitude are labeled and calculated as follows:
7164:
6031:, with latitude and longitude represented as φ′ and λ′. Define
1816:
The formulae presented in this section provide varying degrees of accuracy.
7316:
7113:
7083:
1305:
436:
93:
1294:{\displaystyle \phi _{\mathrm {m} }={\frac {\phi _{1}+\phi _{2}}{2}}.\,\!}
7213:
Code of
Federal Regulations (Annual Edition). Title 47: Telecommunication
7118:
1810:
50:
7556:
Rapp, R, H (1991). Geometric
Geodesy, Part I (Report). Ohio Start Univ.
1369:
515:
Tunnel-distance based approximations: Flat surface, Gauss-mid-latitude;
7664:
7537:
6907:
4952:
797:
7561:
1795:, or omits the conversion between arc and chord lengths shown below.
7139:"The British Cartographic Society > How long is the UK coastline?"
4925:
473:
129:
55:
7380:
7361:. Vol. 1. St. Louis: Aeronautical Chart and Information Center.
4916:
Methods for computing the geodesic distance are widely available in
3403:{\displaystyle K_{1}=M(\phi _{\mathrm {m} }){\frac {\pi }{180}}\,\!}
509:
7394:
4872:
3824:
article gives the formula for calculating the shortest arch length
3789:{\displaystyle \theta ={\frac {\pi }{180}}(90^{\circ }-\phi ).\,\!}
1728:
469:
454:
308:
125:
114:
7492:
7299:
7293:(8). . Translated by C. F. F. Karney & R. E. Deakin: 852–861.
3341:
are in units of kilometers per arc degree. They are derived from
3007:
must be in units compatible with the method used for determining
1420:
are highly curved near the Poles. Hence, the above equations for
188:
45:
22:
7035:{\displaystyle S-s=-0.5(h_{1}+h_{2})s/R-0.5(h_{1}-h_{2})^{2}/s}
5750:
3814:
on the sphere that best approximates the surface of the Earth.
1105:
1099:. In the given forms of the formulae below, one or more values
298:
258:
196:
5715:{\textstyle D=a{\bigl (}\sigma -{\tfrac {f}{2}}(X+Y){\bigr )}}
1802:
is used to calculate the distance between points in a plane.
7265:"Analyse des triangles tracées sur la surface d'un sphéroïde"
1333:= 6,371.009 kilometers = 3,958.761 statute miles = 3,440.069
776:
The theoretical estimations of error are added in above and
458:
83:
6879:{\displaystyle \Delta \lambda '=\lambda _{2}'-\lambda _{1}'}
5767:
5037:{\displaystyle \beta =\arctan \left((1-f)\tan \phi \right)}
2270:
1742:
Flat-surface approximation formulae for very short distance
385:
278:
3934:{\displaystyle D=2R\arcsin {\frac {D_{\textrm {t}}}{2R}}.}
2619:
1582:
are on either side of the ±180° meridian, or the value of
6819:{\displaystyle \Delta \lambda =\lambda _{2}-\lambda _{1}}
1819:
7696:"GeographicLib: Geodesics on an ellipsoid of revolution"
7247:
Mémoires de l'Académie Royale des
Sciences de Paris 1733
2127:
1019:
respectively. Which of the two points is designated as
6587:
6370:{\displaystyle R'={\frac {\sqrt {1+e'^{2}}}{B^{2}}}a.}
5679:
5655:
5144:
5115:
5083:
5054:
4977:
Lambert's formulae use the first-order correction and
4137:{\displaystyle D_{\textrm {t}}=2R\sin {\frac {D}{2R}}}
3567:
1229:"Mean latitude" is labeled and calculated as follows:
561:{\displaystyle |\Delta D_{\text{error}}|\propto D^{3}}
7478:
Karney, C. F. F. (2013). "Algorithms for geodesics".
6923:
6832:
6783:
6723:
6674:
6401:
6312:
6178:
6040:
5781:
5732:
5453:
5356:
5325:
5297:
5242:
5195:
5175:
5143:
5114:
5082:
5053:
4987:
4960:
4153:
4091:
3979:
3950:
3885:
3853:
3830:
3742:
3698:
3577:
3536:
3514:
3418:
3353:
3320:
3293:
3054:
3013:
2982:
2955:
2930:
2906:
2811:
2665:
2630:
2537:
2490:
2468:
2284:
2275:
The above formula is extended for ellipsoidal Earth:
2173:
2138:
1833:
1772:
1752:
1702:
1674:
1646:
1618:
1588:
1560:
1532:
1508:
1478:
1454:
1430:
1380:
1370:
Singularities and discontinuity of latitude/longitude
1346:
1317:
1238:
1120:
1079:
1057:
1025:
974:
926:
897:
868:
839:
814:
782:
732:
705:
678:
651:
624:
576:
521:
7766:
Torge & Müller (2012) Geodesy, De
Gruyter, p.249
7358:
Mathematical and
Physical Theories of Higher Geodesy
3499:= kilometers per arc degree of longitude difference;
7614:, Österreichische Zeitschrift für Vermessungswesen.
6770:{\displaystyle \Delta \phi '=\phi _{2}'-\phi _{1}'}
6022:
4946:
3410:= kilometers per arc degree of latitude difference;
7034:
6878:
6818:
6769:
6709:
6657:
6369:
6292:
6158:
6011:
5738:
5714:
5638:
5436:
5339:
5311:
5275:
5228:
5181:
5158:
5129:
5097:
5068:
5036:
4966:
4840:
4136:
4069:
3962:
3933:
3868:
3836:
3788:
3726:
3681:
3544:
3522:
3491:
3402:
3333:
3306:
3273:
3039:
2999:
2966:
2941:
2914:
2888:
2782:
2648:
2608:
2498:
2476:
2450:
2254:
2156:
2116:
1787:
1758:
1716:
1688:
1660:
1632:
1604:
1574:
1546:
1518:
1494:
1464:
1440:
1389:
1358:
1325:
1293:
1215:
1087:
1065:
1048:is not important for the calculation of distance.
1040:
1011:
960:
912:
883:
854:
825:
788:
765:{\displaystyle \Delta |D_{\text{error}}|\propto D}
764:
718:
691:
664:
637:
609:{\displaystyle |\Delta D_{\text{error}}|\propto D}
608:
560:
6906:length) between two points can be reduced to the
6609:
6499:
5746:is the equatorial radius of the chosen spheroid.
5347:being the same on the sphere as on the spheroid.
3785:
3541:
3519:
3488:
3399:
3270:
3036:
2996:
2963:
2938:
2911:
2495:
2473:
1713:
1685:
1657:
1629:
1601:
1571:
1543:
1515:
1491:
1461:
1437:
1355:
1322:
1290:
1212:
1084:
1062:
1037:
1008:
957:
891:. Whereas, the tunnel distance, or chord length,
880:
851:
822:
510:Classification of Formulae based on Approximation
7795:
7612:Entwicklung der Gauss'schen Mittelbreitenformeln
6710:{\displaystyle \Delta \phi =\phi _{2}-\phi _{1}}
5757:0 North 0 West to 40 North 120 West, 12.6 meters
3727:{\displaystyle \theta ={\frac {\pi }{2}}-\phi .}
1738:does not have discontinuities or singularities.
618:higher-order approximations based on Ellipsoid:
16:Distance measured along the surface of the Earth
7688:
4851:
6027:Bowring maps the points to a sphere of radius
3040:{\displaystyle \cos \phi _{\mathrm {m} };\,\!}
5707:
5667:
4913:. Rapp provides a good summary of this work.
3799:
1374:The approximation of sinusoidal functions of
1095:coordinates on maps are usually expressed in
1012:{\displaystyle (\phi _{2},\lambda _{2}),\,\!}
570:0-th-order approximation: Spherical surface;
476:. This distance is an element in solving the
412:
7587:
7585:
7574:: CS1 maint: multiple names: authors list (
7267:[Analysis of spheroidal triangles].
7206:"Reference points and distance computations"
3692:where the colatitude values are in radians:
2649:{\displaystyle {\frac {\Delta \lambda }{2}}}
2157:{\displaystyle {\frac {\Delta \lambda }{2}}}
1308:of the Earth for the calculations below is:
961:{\displaystyle (\phi _{1},\lambda _{1})\,\!}
5276:{\displaystyle (\beta _{2},\;\lambda _{2})}
5229:{\displaystyle (\beta _{1},\;\lambda _{1})}
7762:
7760:
6392:.) The spherical coordinates are given by
5336:
5308:
5259:
5212:
3632:
3626:
3610:
3604:
1734:instead of latitude/longitude, since this
419:
405:
386:Spatial Reference System Identifier (SRID)
381:International Terrestrial Reference System
7582:
7551:
7549:
7547:
7491:
7399:(Technical report). Ohio State University
7298:
7269:Mémoires de l'Institut National de France
6169:where the second eccentricity squared is
5768:Gauss mid-latitude method for short lines
3784:
3540:
3518:
3487:
3398:
3269:
3035:
2995:
2962:
2937:
2910:
2494:
2472:
2328:
2007:
1883:
1612:("mean latitude") for the two positions (
1352:
1321:
1289:
1211:
1083:
1061:
1036:
1007:
956:
879:
850:
821:
7416:
7259:
7237:
7183:
7162:
4861:
3000:{\displaystyle \phi _{\mathrm {m} }\,\!}
2271:Ellipsoidal Earth approximation formulae
645:: Andoyer(1932); Andoyer-Lambert(1942),
430:
7757:
7675:
7623:
7591:
7351:
5159:{\displaystyle \scriptstyle \beta _{2}}
5130:{\displaystyle \scriptstyle \beta _{1}}
7796:
7749:: CS1 maint: archived copy as title (
7544:
7535:
7477:
7281:
6893:
5098:{\displaystyle \scriptstyle \phi _{2}}
5069:{\displaystyle \scriptstyle \phi _{1}}
2656:, justified except for high latitude:
2531:It is derived by the approximation of
2164:, justified except for high latitude:
1820:Spherical Earth approximation formulae
1605:{\displaystyle \phi _{\mathrm {m} }\!}
1495:{\displaystyle \phi _{\mathrm {m} }\!}
5753:spheroid Lambert's formula is off by
2620:In the case of medium or low latitude
2128:In the case of medium or low latitude
7665:https://doi.org/10.1007%2FBF03030136
7555:
7392:
7271:(in French) (1st semester): 130–161.
7186:"Navigation on the spheroidal earth"
3568:Polar coordinate flat-Earth formula
2967:{\displaystyle \Delta \lambda \,\!}
391:Universal Transverse Mercator (UTM)
353:European Terrestrial Ref. Sys. 1989
13:
7383:, Vol. 1 (Teubner, Leipzig, 1880).
6833:
6784:
6724:
6675:
6642:
6621:
6598:
6552:
6485:
6464:
5922:
5896:
5837:
4806:
4785:
4726:
4679:
4618:
4582:
4528:
4504:
4480:
4384:
4271:
4158:
4080:
3665:
3468:
3447:
3379:
3253:
3220:
3187:
3136:
3103:
3026:
2989:
2956:
2931:
2865:
2833:
2759:
2753:
2732:
2698:
2689:
2634:
2583:
2571:
2423:
2338:
2322:
2234:
2225:
2188:
2142:
2079:
2017:
2001:
1955:
1893:
1871:
1595:
1509:
1485:
1455:
1431:
1416:. Also, planar projections of the
1381:
1245:
1168:
1125:
733:
582:
527:
457:measured along the surface of the
263:Ordnance Survey Great Britain 1936
229:Discrete Global Grid and Geocoding
120:Horizontal position representation
14:
7825:
7773:
7678:J. Washington Academy of Sciences
1519:{\displaystyle \Delta \lambda \!}
1465:{\displaystyle \Delta \lambda \!}
478:second (inverse) geodetic problem
6023:Bowring's method for short lines
5285:the Great-circle distance method
4947:Lambert's formula for long lines
4918:geographical information systems
3628:
3606:
2942:{\displaystyle \Delta \phi \,\!}
2793:
2526:Geographic coordinate conversion
1304:Unless specified otherwise, the
672:: Andoyer-Lambert-Thomas(1970),
461:, or the shortest arch length.
179:Global Nav. Sat. Systems (GNSSs)
29:
7712:
7669:
7654:
7617:
7604:
7529:
7471:
7410:
6098:
5545:
5544:
5395:
4866:Geodesic on an oblate ellipsoid
3869:{\displaystyle D_{\textrm {t}}}
1788:{\displaystyle D_{\textrm {t}}}
1408:(longitude is undefined) and a
1390:{\displaystyle \Delta \lambda }
913:{\displaystyle D_{\textrm {t}}}
803:
483:
343:N. American Vertical Datum 1988
7386:
7345:
7275:
7253:
7231:
7198:
7177:
7156:
7131:
7015:
6988:
6968:
6942:
6604:
6567:
6558:
6549:
6275:
6262:
6257:
6245:
5702:
5690:
5570:
5552:
5478:
5460:
5340:{\displaystyle \lambda _{2}\;}
5312:{\displaystyle \lambda _{1}\;}
5270:
5243:
5223:
5196:
5189:in radians between two points
5017:
5005:
4537:
4525:
4513:
4501:
4489:
4477:
4443:
4430:
4418:
4405:
4374:
4361:
4352:
4339:
4327:
4314:
4305:
4292:
4261:
4248:
4239:
4226:
4214:
4201:
4192:
4179:
3778:
3759:
3671:
3662:
3474:
3459:
3453:
3438:
3385:
3370:
3259:
3241:
3226:
3208:
3193:
3178:
3142:
3124:
3109:
3091:
2872:
2852:
2840:
2820:
2766:
2738:
2723:
2717:
2705:
2695:
2680:
2674:
2241:
2231:
2216:
2207:
2195:
2185:
1717:{\displaystyle \lambda _{2}\!}
1661:{\displaystyle \lambda _{1}\!}
1575:{\displaystyle \lambda _{2}\!}
1547:{\displaystyle \lambda _{1}\!}
1441:{\displaystyle \Delta \phi \!}
1001:
975:
953:
927:
752:
737:
596:
578:
541:
523:
373:Internet link to a point 2010
303:Geodetic Reference System 1980
221:Quasi-Zenith Sat. Sys. (QZSS)
1:
7124:
7051:azimuthal radius of curvature
5763:40N 0W to 40N 60W, 0.85 meter
363:Chinese obfuscated datum 2002
7788:online geodesic bibliography
7700:geographiclib.sourceforge.io
7069:length is often negligible.
5760:0N 0W to 40N 60W, 6.6 meters
5047:First convert the latitudes
4852:Ellipsoidal-surface formulae
1418:circles of constant latitude
1088:{\displaystyle \lambda \,\!}
313:Geographic point coord. 1983
7:
7072:
3343:radii of curvature of Earth
2522:radii of curvature of Earth
2513:and its perpendicular, or "
1689:{\displaystyle \phi _{2}\!}
1633:{\displaystyle \phi _{1}\!}
1526:("east displacement") when
273:Systema Koordinat 1942 goda
10:
7830:
7781:online geodesic calculator
7463:: CS1 maint: postscript (
7449:. Addendum: Survey Review
7441:10.1179/sre.1975.23.176.88
7396:Geometric Geodesy, Part II
7393:Rapp, R. H. (March 1993).
7371:: CS1 maint: postscript (
7337:: CS1 maint: postscript (
7049:is evaluated from Earth's
4855:
3803:
3800:Spherical-surface formulae
1766:, to the tunnel distance,
333:World Geodetic System 1984
7783:(based on GeographicLib).
7631:. Cambridge Univ. Press.
7536:Karney, C. F. F. (2013).
7510:10.1007/s00190-012-0578-z
7287:Astronomische Nachrichten
6910:on the ellipsoid surface
4858:Geodesics on an ellipsoid
2922:= Distance in kilometers;
1066:{\displaystyle \phi \,\!}
1041:{\displaystyle P_{1}\,\!}
884:{\displaystyle P_{2}\,\!}
855:{\displaystyle P_{1}\,\!}
323:North American Datum 1983
293:South American Datum 1969
6303:The spherical radius is
4881:inverse geodetic problem
1359:{\displaystyle D_{\,}\!}
466:geographical coordinates
184:Global Pos. System (GPS)
151:Spatial reference system
7377:English translation of
7099:Great-circle navigation
6888:great-circle navigation
5182:{\displaystyle \sigma }
4899:English translation of
2528:" for their formulas).
7629:Calculus of Variations
7317:10.1002/asna.201011352
7219:(208). October 1, 2016
7104:Ground sample distance
7036:
6880:
6820:
6771:
6711:
6659:
6371:
6294:
6160:
6013:
5740:
5716:
5640:
5438:
5341:
5313:
5277:
5230:
5183:
5160:
5131:
5099:
5070:
5038:
4968:
4867:
4842:
4138:
4071:
3964:
3963:{\displaystyle D\ll R}
3935:
3870:
3838:
3812:spherical trigonometry
3790:
3728:
3683:
3546:
3524:
3493:
3404:
3335:
3308:
3275:
3041:
3001:
2968:
2943:
2916:
2890:
2784:
2650:
2610:
2500:
2478:
2452:
2256:
2158:
2118:
1789:
1760:
1718:
1690:
1662:
1634:
1606:
1576:
1548:
1520:
1496:
1466:
1442:
1391:
1360:
1327:
1295:
1217:
1089:
1067:
1042:
1013:
962:
914:
885:
856:
827:
826:{\displaystyle D,\,\!}
790:
766:
720:
693:
666:
639:
610:
562:
443:
7594:Surveying and Mapping
7249:(in French): 406–416.
7184:Williams, E. (2002).
7163:Williams, E. (2013).
7094:Great-circle distance
7037:
6881:
6821:
6772:
6712:
6660:
6384:of the ellipsoid at φ
6372:
6295:
6161:
6014:
5741:
5717:
5641:
5439:
5342:
5314:
5278:
5231:
5184:
5166:. Then calculate the
5161:
5132:
5105:of the two points to
5100:
5071:
5039:
4969:
4928:. (For details, see
4865:
4843:
4139:
4072:
3965:
3944:For short distances (
3936:
3871:
3839:
3822:great-circle distance
3806:Great-circle distance
3791:
3729:
3684:
3561:Chebyshev polynomials
3547:
3545:{\displaystyle N\,\!}
3525:
3523:{\displaystyle M\,\!}
3494:
3405:
3336:
3334:{\displaystyle K_{2}}
3309:
3307:{\displaystyle K_{1}}
3276:
3042:
3002:
2969:
2944:
2917:
2915:{\displaystyle D\,\!}
2891:
2785:
2651:
2611:
2501:
2499:{\displaystyle N\,\!}
2479:
2477:{\displaystyle M\,\!}
2453:
2257:
2159:
2119:
1790:
1761:
1719:
1691:
1663:
1635:
1607:
1577:
1549:
1521:
1497:
1472:) and mean latitude (
1467:
1443:
1392:
1361:
1328:
1326:{\displaystyle R\,\!}
1296:
1218:
1090:
1068:
1043:
1014:
963:
915:
886:
857:
828:
791:
767:
721:
719:{\displaystyle f^{6}}
694:
692:{\displaystyle f^{3}}
667:
665:{\displaystyle f^{2}}
640:
638:{\displaystyle f^{1}}
611:
563:
447:Geographical distance
434:
79:Geographical distance
7165:"Aviation Formulary"
7067:ellipsoidal geodesic
7063:Earth normal section
6921:
6830:
6781:
6721:
6672:
6399:
6310:
6176:
6038:
5779:
5730:
5653:
5451:
5354:
5323:
5295:
5240:
5193:
5173:
5141:
5112:
5080:
5051:
4985:
4958:
4151:
4089:
3977:
3948:
3883:
3851:
3828:
3740:
3696:
3575:
3534:
3512:
3416:
3351:
3318:
3291:
3052:
3011:
2980:
2953:
2928:
2904:
2809:
2663:
2628:
2616:in the square root.
2535:
2488:
2466:
2282:
2171:
2136:
1831:
1770:
1750:
1700:
1672:
1644:
1616:
1586:
1558:
1530:
1506:
1476:
1452:
1428:
1424:latitude/longitude (
1378:
1344:
1315:
1236:
1118:
1077:
1055:
1023:
972:
924:
895:
866:
837:
812:
780:
730:
703:
676:
649:
622:
574:
519:
502:Ellipsoidal surface.
253:Sea Level Datum 1929
105:Geodetic coordinates
7610:Hubeny, K. (1954).
7502:2013JGeod..87...43K
7309:2010AN....331..852K
7109:Vincenty's formulae
7059:ellipsoidal heights
6894:Altitude correction
6875:
6859:
6766:
6750:
6424:
5291:), with longitudes
4930:Vincenty's formulae
3625:
3603:
3557:reference ellipsoid
1800:Pythagorean theorem
283:European Datum 1950
241:Standards (history)
141:Reference ellipsoid
89:Figure of the Earth
7480:Journal of Geodesy
7453:(180): 294 (1976).
7032:
6876:
6863:
6847:
6816:
6767:
6754:
6738:
6707:
6655:
6653:
6596:
6412:
6382:Gaussian curvature
6367:
6290:
6156:
6009:
5736:
5712:
5688:
5636:
5434:
5337:
5309:
5283:on a sphere using
5273:
5226:
5179:
5156:
5155:
5127:
5126:
5095:
5094:
5066:
5065:
5034:
4964:
4868:
4838:
4836:
4134:
4067:
3960:
3931:
3866:
3834:
3786:
3724:
3679:
3611:
3589:
3542:
3520:
3508:expansion form of
3489:
3400:
3331:
3304:
3271:
3267:
3037:
2997:
2964:
2939:
2912:
2886:
2780:
2646:
2606:
2496:
2474:
2448:
2252:
2154:
2114:
2112:
1785:
1756:
1714:
1686:
1658:
1630:
1602:
1572:
1544:
1516:
1492:
1462:
1438:
1387:
1356:
1323:
1291:
1213:
1209:
1085:
1063:
1038:
1009:
958:
910:
881:
852:
823:
786:
762:
716:
699:: Vincenty(1975),
689:
662:
635:
606:
558:
499:Spherical surface;
444:
161:Vertical positions
7638:978-1-107-64083-2
7522:– (open access).
6595:
6547:
6495:
6455:
6359:
6348:
6285:
6234:
6151:
6093:
6004:
5986:
5977:
5952:
5932:
5906:
5863:
5847:
5806:
5739:{\displaystyle a}
5687:
5634:
5631:
5542:
5539:
5432:
5393:
5289:haversine formula
5107:reduced latitudes
4967:{\displaystyle f}
4829:
4816:
4795:
4752:
4736:
4691:
4689:
4654:
4628:
4592:
4546:
4461:
4132:
4100:
4041:
4034:
4017:
3994:
3926:
3914:
3862:
3837:{\displaystyle D}
3757:
3713:
3674:
3485:
3396:
2881:
2775:
2644:
2581:
2558:
2446:
2433:
2410:
2388:
2348:
2314:
2250:
2152:
2106:
2102:
2089:
2066:
2027:
1978:
1965:
1942:
1903:
1881:
1781:
1759:{\displaystyle D}
1284:
906:
789:{\displaystyle f}
772:on the hemisphere
748:
592:
537:
451:geodetic distance
429:
428:
377:
376:
156:Spatial relations
146:Satellite geodesy
101:
7821:
7767:
7764:
7755:
7754:
7748:
7740:
7738:
7737:
7731:
7725:. Archived from
7724:
7716:
7710:
7709:
7707:
7706:
7692:
7686:
7685:
7673:
7667:
7658:
7652:
7650:
7621:
7615:
7608:
7602:
7601:
7589:
7580:
7579:
7573:
7565:
7553:
7542:
7541:
7533:
7527:
7521:
7495:
7475:
7469:
7468:
7462:
7454:
7448:
7447:
7426:
7414:
7408:
7407:
7405:
7404:
7390:
7384:
7376:
7370:
7362:
7349:
7343:
7342:
7336:
7328:
7302:
7279:
7273:
7272:
7257:
7251:
7250:
7235:
7229:
7228:
7226:
7224:
7210:
7202:
7196:
7195:
7193:
7192:
7181:
7175:
7174:
7172:
7171:
7160:
7154:
7153:
7151:
7150:
7141:. Archived from
7135:
7041:
7039:
7038:
7033:
7028:
7023:
7022:
7013:
7012:
7000:
6999:
6978:
6967:
6966:
6954:
6953:
6885:
6883:
6882:
6877:
6871:
6855:
6843:
6825:
6823:
6822:
6817:
6815:
6814:
6802:
6801:
6776:
6774:
6773:
6768:
6762:
6746:
6734:
6716:
6714:
6713:
6708:
6706:
6705:
6693:
6692:
6664:
6662:
6661:
6656:
6654:
6631:
6613:
6612:
6597:
6588:
6582:
6581:
6548:
6546:
6545:
6544:
6531:
6530:
6529:
6528:
6511:
6503:
6502:
6496:
6491:
6483:
6474:
6456:
6451:
6450:
6449:
6433:
6420:
6376:
6374:
6373:
6368:
6360:
6358:
6357:
6347:
6346:
6345:
6326:
6325:
6320:
6299:
6297:
6296:
6291:
6286:
6284:
6283:
6282:
6260:
6240:
6235:
6233:
6232:
6223:
6222:
6221:
6209:
6208:
6198:
6193:
6192:
6191:
6165:
6163:
6162:
6157:
6152:
6150:
6149:
6137:
6136:
6127:
6126:
6125:
6106:
6094:
6092:
6091:
6079:
6078:
6069:
6068:
6067:
6048:
6018:
6016:
6015:
6010:
6005:
6003:
6002:
5997:
5993:
5992:
5988:
5987:
5985:
5984:
5980:
5979:
5978:
5975:
5960:
5959:
5955:
5954:
5953:
5950:
5935:
5933:
5928:
5920:
5907:
5902:
5894:
5877:
5876:
5871:
5867:
5866:
5865:
5864:
5861:
5848:
5843:
5835:
5821:
5813:
5809:
5808:
5807:
5804:
5745:
5743:
5742:
5737:
5721:
5719:
5718:
5713:
5711:
5710:
5689:
5680:
5671:
5670:
5645:
5643:
5642:
5637:
5635:
5633:
5632:
5624:
5619:
5618:
5608:
5601:
5600:
5585:
5584:
5574:
5543:
5541:
5540:
5532:
5527:
5526:
5516:
5509:
5508:
5493:
5492:
5482:
5443:
5441:
5440:
5435:
5433:
5428:
5427:
5426:
5414:
5413:
5403:
5394:
5389:
5388:
5387:
5375:
5374:
5364:
5346:
5344:
5343:
5338:
5335:
5334:
5318:
5316:
5315:
5310:
5307:
5306:
5282:
5280:
5279:
5274:
5269:
5268:
5255:
5254:
5235:
5233:
5232:
5227:
5222:
5221:
5208:
5207:
5188:
5186:
5185:
5180:
5165:
5163:
5162:
5157:
5154:
5153:
5136:
5134:
5133:
5128:
5125:
5124:
5104:
5102:
5101:
5096:
5093:
5092:
5075:
5073:
5072:
5067:
5064:
5063:
5043:
5041:
5040:
5035:
5033:
5029:
4979:reduced latitude
4973:
4971:
4970:
4965:
4907:, 241–254 (1825)
4847:
4845:
4844:
4839:
4837:
4830:
4828:
4827:
4822:
4818:
4817:
4812:
4804:
4796:
4791:
4783:
4766:
4765:
4760:
4756:
4755:
4754:
4753:
4750:
4737:
4732:
4724:
4710:
4696:
4692:
4690:
4685:
4677:
4672:
4671:
4662:
4658:
4657:
4656:
4655:
4652:
4642:
4641:
4629:
4624:
4616:
4611:
4610:
4593:
4588:
4580:
4575:
4574:
4565:
4551:
4547:
4545:
4544:
4535:
4521:
4520:
4511:
4497:
4496:
4487:
4476:
4464:
4463:
4462:
4459:
4442:
4441:
4417:
4416:
4391:
4373:
4372:
4351:
4350:
4326:
4325:
4304:
4303:
4278:
4260:
4259:
4238:
4237:
4213:
4212:
4191:
4190:
4165:
4143:
4141:
4140:
4135:
4133:
4131:
4120:
4103:
4102:
4101:
4098:
4076:
4074:
4073:
4068:
4063:
4059:
4052:
4051:
4046:
4042:
4037:
4036:
4035:
4032:
4025:
4018:
4010:
3997:
3996:
3995:
3992:
3969:
3967:
3966:
3961:
3940:
3938:
3937:
3932:
3927:
3925:
3917:
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3912:
3905:
3875:
3873:
3872:
3867:
3865:
3864:
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3860:
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3835:
3795:
3793:
3792:
3787:
3771:
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3725:
3714:
3706:
3688:
3686:
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3644:
3631:
3624:
3619:
3609:
3602:
3597:
3588:
3551:
3549:
3548:
3543:
3529:
3527:
3526:
3521:
3498:
3496:
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3490:
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3478:
3473:
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3428:
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3409:
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3340:
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3329:
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3311:
3310:
3305:
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3280:
3278:
3277:
3272:
3268:
3258:
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3256:
3225:
3224:
3223:
3192:
3191:
3190:
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3160:
3141:
3140:
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3108:
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3106:
3068:
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3046:
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3038:
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3029:
3006:
3004:
3003:
2998:
2994:
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2992:
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2965:
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2940:
2921:
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2895:
2893:
2892:
2887:
2882:
2880:
2879:
2864:
2863:
2848:
2847:
2832:
2831:
2819:
2789:
2787:
2786:
2781:
2776:
2774:
2773:
2758:
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2737:
2736:
2735:
2713:
2712:
2694:
2693:
2692:
2673:
2655:
2653:
2652:
2647:
2645:
2640:
2632:
2615:
2613:
2612:
2607:
2599:
2598:
2593:
2589:
2582:
2577:
2569:
2561:
2560:
2559:
2556:
2505:
2503:
2502:
2497:
2483:
2481:
2480:
2475:
2457:
2455:
2454:
2449:
2447:
2445:
2444:
2439:
2435:
2434:
2429:
2421:
2413:
2412:
2411:
2408:
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2360:
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2344:
2336:
2321:
2317:
2316:
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2261:
2259:
2258:
2253:
2251:
2249:
2248:
2230:
2229:
2228:
2203:
2202:
2184:
2163:
2161:
2160:
2155:
2153:
2148:
2140:
2123:
2121:
2120:
2115:
2113:
2104:
2103:
2101:
2100:
2095:
2091:
2090:
2085:
2077:
2069:
2068:
2067:
2064:
2039:
2038:
2033:
2029:
2028:
2023:
2015:
1994:
1983:
1979:
1977:
1976:
1971:
1967:
1966:
1961:
1953:
1945:
1944:
1943:
1940:
1915:
1914:
1909:
1905:
1904:
1899:
1891:
1882:
1877:
1869:
1852:
1794:
1792:
1791:
1786:
1784:
1783:
1782:
1779:
1765:
1763:
1762:
1757:
1723:
1721:
1720:
1715:
1712:
1711:
1695:
1693:
1692:
1687:
1684:
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1667:
1665:
1664:
1659:
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1655:
1639:
1637:
1636:
1631:
1628:
1627:
1611:
1609:
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1600:
1599:
1598:
1581:
1579:
1578:
1573:
1570:
1569:
1553:
1551:
1550:
1545:
1542:
1541:
1525:
1523:
1522:
1517:
1501:
1499:
1498:
1493:
1490:
1489:
1488:
1471:
1469:
1468:
1463:
1447:
1445:
1444:
1439:
1396:
1394:
1393:
1388:
1365:
1363:
1362:
1357:
1354:
1353:
1332:
1330:
1329:
1324:
1300:
1298:
1297:
1292:
1285:
1280:
1279:
1278:
1266:
1265:
1255:
1250:
1249:
1248:
1222:
1220:
1219:
1214:
1210:
1203:
1202:
1190:
1189:
1160:
1159:
1147:
1146:
1094:
1092:
1091:
1086:
1072:
1070:
1069:
1064:
1047:
1045:
1044:
1039:
1035:
1034:
1018:
1016:
1015:
1010:
1000:
999:
987:
986:
967:
965:
964:
959:
952:
951:
939:
938:
919:
917:
916:
911:
909:
908:
907:
904:
890:
888:
887:
882:
878:
877:
861:
859:
858:
853:
849:
848:
832:
830:
829:
824:
795:
793:
792:
787:
771:
769:
768:
763:
755:
750:
749:
746:
740:
725:
723:
722:
717:
715:
714:
698:
696:
695:
690:
688:
687:
671:
669:
668:
663:
661:
660:
644:
642:
641:
636:
634:
633:
615:
613:
612:
607:
599:
594:
593:
590:
581:
567:
565:
564:
559:
557:
556:
544:
539:
538:
535:
526:
421:
414:
407:
245:
244:
224:
216:
208:
200:
192:
132:
91:
33:
19:
18:
7829:
7828:
7824:
7823:
7822:
7820:
7819:
7818:
7794:
7793:
7776:
7771:
7770:
7765:
7758:
7742:
7741:
7735:
7733:
7729:
7722:
7720:"Archived copy"
7718:
7717:
7713:
7704:
7702:
7694:
7693:
7689:
7674:
7670:
7659:
7655:
7639:
7622:
7618:
7609:
7605:
7590:
7583:
7567:
7566:
7554:
7545:
7538:"GeographicLib"
7534:
7530:
7476:
7472:
7456:
7455:
7445:
7443:
7424:
7415:
7411:
7402:
7400:
7391:
7387:
7364:
7363:
7350:
7346:
7330:
7329:
7280:
7276:
7261:Legendre, A. M.
7258:
7254:
7239:Clairaut, A. C.
7236:
7232:
7222:
7220:
7208:
7204:
7203:
7199:
7190:
7188:
7182:
7178:
7169:
7167:
7161:
7157:
7148:
7146:
7137:
7136:
7132:
7127:
7089:Spherical Earth
7079:Arc measurement
7075:
7024:
7018:
7014:
7008:
7004:
6995:
6991:
6974:
6962:
6958:
6949:
6945:
6922:
6919:
6918:
6896:
6867:
6851:
6836:
6831:
6828:
6827:
6810:
6806:
6797:
6793:
6782:
6779:
6778:
6758:
6742:
6727:
6722:
6719:
6718:
6701:
6697:
6688:
6684:
6673:
6670:
6669:
6652:
6651:
6632:
6624:
6618:
6617:
6608:
6607:
6586:
6577:
6573:
6540:
6536:
6532:
6524:
6520:
6516:
6512:
6510:
6498:
6497:
6484:
6482:
6475:
6467:
6461:
6460:
6445:
6441:
6434:
6432:
6425:
6416:
6402:
6400:
6397:
6396:
6387:
6353:
6349:
6341:
6337:
6333:
6324:
6313:
6311:
6308:
6307:
6278:
6274:
6261:
6241:
6239:
6228:
6224:
6217:
6213:
6204:
6200:
6199:
6197:
6187:
6183:
6179:
6177:
6174:
6173:
6145:
6141:
6132:
6128:
6121:
6117:
6113:
6105:
6087:
6083:
6074:
6070:
6063:
6059:
6055:
6047:
6039:
6036:
6035:
6025:
5998:
5974:
5973:
5969:
5965:
5961:
5949:
5948:
5944:
5940:
5936:
5934:
5921:
5919:
5918:
5914:
5895:
5893:
5886:
5882:
5881:
5872:
5860:
5859:
5855:
5836:
5834:
5827:
5823:
5822:
5820:
5803:
5802:
5798:
5794:
5780:
5777:
5776:
5770:
5731:
5728:
5727:
5706:
5705:
5678:
5666:
5665:
5654:
5651:
5650:
5623:
5614:
5610:
5609:
5596:
5592:
5580:
5576:
5575:
5573:
5531:
5522:
5518:
5517:
5504:
5500:
5488:
5484:
5483:
5481:
5452:
5449:
5448:
5422:
5418:
5409:
5405:
5404:
5402:
5383:
5379:
5370:
5366:
5365:
5363:
5355:
5352:
5351:
5330:
5326:
5324:
5321:
5320:
5302:
5298:
5296:
5293:
5292:
5264:
5260:
5250:
5246:
5241:
5238:
5237:
5217:
5213:
5203:
5199:
5194:
5191:
5190:
5174:
5171:
5170:
5149:
5145:
5142:
5139:
5138:
5120:
5116:
5113:
5110:
5109:
5088:
5084:
5081:
5078:
5077:
5059:
5055:
5052:
5049:
5048:
5004:
5000:
4986:
4983:
4982:
4959:
4956:
4955:
4949:
4860:
4854:
4835:
4834:
4823:
4805:
4803:
4784:
4782:
4775:
4771:
4770:
4761:
4749:
4748:
4744:
4725:
4723:
4716:
4712:
4711:
4709:
4694:
4693:
4678:
4676:
4667:
4663:
4651:
4650:
4646:
4637:
4633:
4617:
4615:
4606:
4602:
4601:
4597:
4581:
4579:
4570:
4566:
4564:
4549:
4548:
4540:
4536:
4531:
4516:
4512:
4507:
4492:
4488:
4483:
4475:
4465:
4458:
4457:
4453:
4450:
4449:
4437:
4433:
4412:
4408:
4392:
4387:
4381:
4380:
4368:
4364:
4346:
4342:
4321:
4317:
4299:
4295:
4279:
4274:
4268:
4267:
4255:
4251:
4233:
4229:
4208:
4204:
4186:
4182:
4166:
4161:
4154:
4152:
4149:
4148:
4124:
4119:
4097:
4096:
4092:
4090:
4087:
4086:
4083:
4081:Tunnel distance
4047:
4031:
4030:
4026:
4024:
4020:
4019:
4009:
4002:
3998:
3991:
3990:
3986:
3978:
3975:
3974:
3949:
3946:
3945:
3918:
3911:
3910:
3906:
3904:
3884:
3881:
3880:
3859:
3858:
3854:
3852:
3849:
3848:
3846:tunnel distance
3829:
3826:
3825:
3808:
3802:
3766:
3762:
3749:
3741:
3738:
3737:
3705:
3697:
3694:
3693:
3650:
3646:
3640:
3636:
3627:
3620:
3615:
3605:
3598:
3593:
3587:
3576:
3573:
3572:
3570:
3535:
3532:
3531:
3513:
3510:
3509:
3506:binomial series
3477:
3467:
3466:
3462:
3446:
3445:
3441:
3423:
3419:
3417:
3414:
3413:
3388:
3378:
3377:
3373:
3358:
3354:
3352:
3349:
3348:
3325:
3321:
3319:
3316:
3315:
3298:
3294:
3292:
3289:
3288:
3266:
3265:
3252:
3251:
3247:
3219:
3218:
3214:
3186:
3185:
3181:
3162:
3156:
3152:
3149:
3148:
3135:
3134:
3130:
3102:
3101:
3097:
3069:
3063:
3059:
3055:
3053:
3050:
3049:
3025:
3024:
3020:
3012:
3009:
3008:
2988:
2987:
2983:
2981:
2978:
2977:
2974:are in degrees;
2954:
2951:
2950:
2929:
2926:
2925:
2905:
2902:
2901:
2875:
2871:
2859:
2855:
2843:
2839:
2827:
2823:
2818:
2810:
2807:
2806:
2796:
2769:
2765:
2752:
2751:
2747:
2731:
2730:
2726:
2708:
2704:
2688:
2687:
2683:
2672:
2664:
2661:
2660:
2633:
2631:
2629:
2626:
2625:
2622:
2594:
2570:
2568:
2555:
2554:
2550:
2543:
2539:
2538:
2536:
2533:
2532:
2489:
2486:
2485:
2467:
2464:
2463:
2440:
2422:
2420:
2407:
2406:
2402:
2385:
2384:
2380:
2376:
2369:
2365:
2364:
2355:
2337:
2335:
2311:
2310:
2306:
2302:
2298:
2294:
2293:
2291:
2283:
2280:
2279:
2273:
2244:
2240:
2224:
2223:
2219:
2198:
2194:
2183:
2172:
2169:
2168:
2141:
2139:
2137:
2134:
2133:
2130:
2111:
2110:
2096:
2078:
2076:
2063:
2062:
2058:
2048:
2044:
2043:
2034:
2016:
2014:
2000:
1996:
1995:
1993:
1981:
1980:
1972:
1954:
1952:
1939:
1938:
1934:
1924:
1920:
1919:
1910:
1892:
1890:
1870:
1868:
1858:
1854:
1853:
1851:
1841:
1834:
1832:
1829:
1828:
1822:
1778:
1777:
1773:
1771:
1768:
1767:
1751:
1748:
1747:
1744:
1707:
1703:
1701:
1698:
1697:
1679:
1675:
1673:
1670:
1669:
1651:
1647:
1645:
1642:
1641:
1623:
1619:
1617:
1614:
1613:
1594:
1593:
1589:
1587:
1584:
1583:
1565:
1561:
1559:
1556:
1555:
1537:
1533:
1531:
1528:
1527:
1507:
1504:
1503:
1484:
1483:
1479:
1477:
1474:
1473:
1453:
1450:
1449:
1429:
1426:
1425:
1379:
1376:
1375:
1372:
1351:
1347:
1345:
1342:
1341:
1316:
1313:
1312:
1274:
1270:
1261:
1257:
1256:
1254:
1244:
1243:
1239:
1237:
1234:
1233:
1208:
1207:
1198:
1194:
1185:
1181:
1174:
1165:
1164:
1155:
1151:
1142:
1138:
1131:
1121:
1119:
1116:
1115:
1078:
1075:
1074:
1056:
1053:
1052:
1030:
1026:
1024:
1021:
1020:
995:
991:
982:
978:
973:
970:
969:
947:
943:
934:
930:
925:
922:
921:
903:
902:
898:
896:
893:
892:
873:
869:
867:
864:
863:
844:
840:
838:
835:
834:
813:
810:
809:
806:
781:
778:
777:
751:
745:
741:
736:
731:
728:
727:
726:: Kaney(2011);
710:
706:
704:
701:
700:
683:
679:
677:
674:
673:
656:
652:
650:
647:
646:
629:
625:
623:
620:
619:
595:
589:
585:
577:
575:
572:
571:
552:
548:
540:
534:
530:
522:
520:
517:
516:
512:
486:
425:
396:
395:
242:
234:
233:
222:
214:
206:
198:
190:
174:
166:
165:
124:
74:
66:
65:
41:
17:
12:
11:
5:
7827:
7817:
7816:
7811:
7806:
7792:
7791:
7784:
7775:
7774:External links
7772:
7769:
7768:
7756:
7711:
7687:
7668:
7653:
7637:
7625:Forsyth, A. R.
7616:
7603:
7581:
7543:
7528:
7470:
7435:(176): 88–93.
7420:(April 1975).
7409:
7385:
7353:Helmert, F. R.
7344:
7274:
7252:
7230:
7197:
7176:
7155:
7129:
7128:
7126:
7123:
7122:
7121:
7116:
7111:
7106:
7101:
7096:
7091:
7086:
7081:
7074:
7071:
7065:length to the
7043:
7042:
7031:
7027:
7021:
7017:
7011:
7007:
7003:
6998:
6994:
6990:
6987:
6984:
6981:
6977:
6973:
6970:
6965:
6961:
6957:
6952:
6948:
6944:
6941:
6938:
6935:
6932:
6929:
6926:
6895:
6892:
6874:
6870:
6866:
6862:
6858:
6854:
6850:
6846:
6842:
6839:
6835:
6813:
6809:
6805:
6800:
6796:
6792:
6789:
6786:
6765:
6761:
6757:
6753:
6749:
6745:
6741:
6737:
6733:
6730:
6726:
6704:
6700:
6696:
6691:
6687:
6683:
6680:
6677:
6666:
6665:
6650:
6647:
6644:
6641:
6638:
6635:
6633:
6630:
6627:
6623:
6620:
6619:
6616:
6611:
6606:
6603:
6600:
6594:
6591:
6585:
6580:
6576:
6572:
6569:
6566:
6563:
6560:
6557:
6554:
6551:
6543:
6539:
6535:
6527:
6523:
6519:
6515:
6509:
6506:
6501:
6494:
6490:
6487:
6481:
6478:
6476:
6473:
6470:
6466:
6463:
6462:
6459:
6454:
6448:
6444:
6440:
6437:
6431:
6428:
6426:
6423:
6419:
6415:
6411:
6408:
6405:
6404:
6385:
6378:
6377:
6366:
6363:
6356:
6352:
6344:
6340:
6336:
6332:
6329:
6323:
6319:
6316:
6301:
6300:
6289:
6281:
6277:
6273:
6270:
6267:
6264:
6259:
6256:
6253:
6250:
6247:
6244:
6238:
6231:
6227:
6220:
6216:
6212:
6207:
6203:
6196:
6190:
6186:
6182:
6167:
6166:
6155:
6148:
6144:
6140:
6135:
6131:
6124:
6120:
6116:
6112:
6109:
6104:
6101:
6097:
6090:
6086:
6082:
6077:
6073:
6066:
6062:
6058:
6054:
6051:
6046:
6043:
6024:
6021:
6020:
6019:
6008:
6001:
5996:
5991:
5983:
5972:
5968:
5964:
5958:
5947:
5943:
5939:
5931:
5927:
5924:
5917:
5913:
5910:
5905:
5901:
5898:
5892:
5889:
5885:
5880:
5875:
5870:
5858:
5854:
5851:
5846:
5842:
5839:
5833:
5830:
5826:
5819:
5816:
5812:
5801:
5797:
5793:
5790:
5787:
5784:
5769:
5766:
5765:
5764:
5761:
5758:
5735:
5724:
5723:
5709:
5704:
5701:
5698:
5695:
5692:
5686:
5683:
5677:
5674:
5669:
5664:
5661:
5658:
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5646:
5630:
5627:
5622:
5617:
5613:
5607:
5604:
5599:
5595:
5591:
5588:
5583:
5579:
5572:
5569:
5566:
5563:
5560:
5557:
5554:
5551:
5548:
5538:
5535:
5530:
5525:
5521:
5515:
5512:
5507:
5503:
5499:
5496:
5491:
5487:
5480:
5477:
5474:
5471:
5468:
5465:
5462:
5459:
5456:
5445:
5444:
5431:
5425:
5421:
5417:
5412:
5408:
5401:
5398:
5392:
5386:
5382:
5378:
5373:
5369:
5362:
5359:
5333:
5329:
5305:
5301:
5272:
5267:
5263:
5258:
5253:
5249:
5245:
5225:
5220:
5216:
5211:
5206:
5202:
5198:
5178:
5152:
5148:
5123:
5119:
5091:
5087:
5062:
5058:
5032:
5028:
5025:
5022:
5019:
5016:
5013:
5010:
5007:
5003:
4999:
4996:
4993:
4990:
4963:
4948:
4945:
4902:Astron. Nachr.
4856:Main article:
4853:
4850:
4849:
4848:
4833:
4826:
4821:
4815:
4811:
4808:
4802:
4799:
4794:
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4778:
4774:
4769:
4764:
4759:
4747:
4743:
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4735:
4731:
4728:
4722:
4719:
4715:
4708:
4705:
4702:
4699:
4697:
4695:
4688:
4684:
4681:
4675:
4670:
4666:
4661:
4649:
4645:
4640:
4636:
4632:
4627:
4623:
4620:
4614:
4609:
4605:
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4596:
4591:
4587:
4584:
4578:
4573:
4569:
4563:
4560:
4557:
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4530:
4527:
4524:
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4515:
4510:
4506:
4503:
4500:
4495:
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4482:
4479:
4474:
4471:
4468:
4466:
4456:
4452:
4451:
4448:
4445:
4440:
4436:
4432:
4429:
4426:
4423:
4420:
4415:
4411:
4407:
4404:
4401:
4398:
4395:
4393:
4390:
4386:
4383:
4382:
4379:
4376:
4371:
4367:
4363:
4360:
4357:
4354:
4349:
4345:
4341:
4338:
4335:
4332:
4329:
4324:
4320:
4316:
4313:
4310:
4307:
4302:
4298:
4294:
4291:
4288:
4285:
4282:
4280:
4277:
4273:
4270:
4269:
4266:
4263:
4258:
4254:
4250:
4247:
4244:
4241:
4236:
4232:
4228:
4225:
4222:
4219:
4216:
4211:
4207:
4203:
4200:
4197:
4194:
4189:
4185:
4181:
4178:
4175:
4172:
4169:
4167:
4164:
4160:
4157:
4156:
4130:
4127:
4123:
4118:
4115:
4112:
4109:
4106:
4095:
4082:
4079:
4078:
4077:
4066:
4062:
4058:
4055:
4050:
4045:
4040:
4029:
4023:
4016:
4013:
4008:
4005:
4001:
3989:
3985:
3982:
3959:
3956:
3953:
3942:
3941:
3930:
3924:
3921:
3909:
3903:
3900:
3897:
3894:
3891:
3888:
3857:
3833:
3804:Main article:
3801:
3798:
3797:
3796:
3783:
3780:
3777:
3774:
3769:
3765:
3761:
3756:
3753:
3748:
3745:
3734:
3723:
3720:
3717:
3712:
3709:
3704:
3701:
3678:
3673:
3670:
3667:
3664:
3661:
3658:
3653:
3649:
3643:
3639:
3635:
3630:
3623:
3618:
3614:
3608:
3601:
3596:
3592:
3586:
3583:
3580:
3569:
3566:
3565:
3564:
3539:
3517:
3502:
3501:
3500:
3484:
3481:
3476:
3470:
3465:
3461:
3458:
3455:
3449:
3444:
3440:
3437:
3434:
3431:
3426:
3422:
3411:
3395:
3392:
3387:
3381:
3376:
3372:
3369:
3366:
3361:
3357:
3328:
3324:
3301:
3297:
3284:
3283:
3282:
3281:
3264:
3261:
3255:
3250:
3246:
3243:
3240:
3237:
3234:
3231:
3228:
3222:
3217:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3189:
3184:
3180:
3177:
3174:
3171:
3168:
3165:
3163:
3159:
3155:
3151:
3150:
3147:
3144:
3138:
3133:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3105:
3100:
3096:
3093:
3090:
3087:
3084:
3081:
3078:
3075:
3072:
3070:
3066:
3062:
3058:
3057:
3047:
3034:
3028:
3023:
3019:
3016:
2991:
2986:
2975:
2961:
2958:
2936:
2933:
2923:
2909:
2896:
2885:
2878:
2874:
2870:
2867:
2862:
2858:
2854:
2851:
2846:
2842:
2838:
2835:
2830:
2826:
2822:
2817:
2814:
2795:
2792:
2791:
2790:
2779:
2772:
2768:
2764:
2761:
2755:
2750:
2746:
2743:
2740:
2734:
2729:
2725:
2722:
2719:
2716:
2711:
2707:
2703:
2700:
2697:
2691:
2686:
2682:
2679:
2676:
2671:
2668:
2643:
2639:
2636:
2621:
2618:
2605:
2602:
2597:
2592:
2588:
2585:
2580:
2576:
2573:
2567:
2564:
2553:
2549:
2546:
2542:
2493:
2471:
2460:
2459:
2443:
2438:
2432:
2428:
2425:
2419:
2416:
2405:
2401:
2398:
2394:
2383:
2379:
2375:
2372:
2368:
2363:
2358:
2353:
2347:
2343:
2340:
2334:
2331:
2327:
2324:
2320:
2309:
2305:
2301:
2297:
2290:
2287:
2272:
2269:
2264:
2263:
2247:
2243:
2239:
2236:
2233:
2227:
2222:
2218:
2215:
2212:
2209:
2206:
2201:
2197:
2193:
2190:
2187:
2182:
2179:
2176:
2151:
2147:
2144:
2129:
2126:
2125:
2124:
2109:
2099:
2094:
2088:
2084:
2081:
2075:
2072:
2061:
2057:
2054:
2051:
2047:
2042:
2037:
2032:
2026:
2022:
2019:
2013:
2010:
2006:
2003:
1999:
1992:
1989:
1986:
1984:
1982:
1975:
1970:
1964:
1960:
1957:
1951:
1948:
1937:
1933:
1930:
1927:
1923:
1918:
1913:
1908:
1902:
1898:
1895:
1889:
1886:
1880:
1876:
1873:
1867:
1864:
1861:
1857:
1850:
1847:
1844:
1842:
1840:
1837:
1836:
1821:
1818:
1776:
1755:
1743:
1740:
1736:representation
1710:
1706:
1682:
1678:
1654:
1650:
1626:
1622:
1597:
1592:
1568:
1564:
1540:
1536:
1514:
1511:
1487:
1482:
1460:
1457:
1436:
1433:
1400:Longitude has
1386:
1383:
1371:
1368:
1350:
1339:
1338:
1335:nautical miles
1320:
1302:
1301:
1288:
1283:
1277:
1273:
1269:
1264:
1260:
1253:
1247:
1242:
1224:
1223:
1206:
1201:
1197:
1193:
1188:
1184:
1180:
1177:
1175:
1173:
1170:
1167:
1166:
1163:
1158:
1154:
1150:
1145:
1141:
1137:
1134:
1132:
1130:
1127:
1124:
1123:
1082:
1073:and longitude
1060:
1033:
1029:
1006:
1003:
998:
994:
990:
985:
981:
977:
955:
950:
946:
942:
937:
933:
929:
901:
876:
872:
847:
843:
820:
817:
808:Arc distance,
805:
802:
800:of the Earth.
785:
774:
773:
761:
758:
754:
744:
739:
735:
713:
709:
686:
682:
659:
655:
632:
628:
616:
605:
602:
598:
588:
584:
580:
568:
555:
551:
547:
543:
533:
529:
525:
511:
508:
504:
503:
500:
497:
485:
482:
435:View from the
427:
426:
424:
423:
416:
409:
401:
398:
397:
394:
393:
388:
383:
375:
374:
371:
365:
364:
361:
355:
354:
351:
345:
344:
341:
335:
334:
331:
325:
324:
321:
315:
314:
311:
305:
304:
301:
295:
294:
291:
285:
284:
281:
275:
274:
271:
265:
264:
261:
255:
254:
251:
243:
240:
239:
236:
235:
232:
231:
226:
218:
210:
202:
194:
186:
181:
175:
172:
171:
168:
167:
164:
163:
158:
153:
148:
143:
138:
136:Map projection
133:
122:
117:
112:
110:Geodetic datum
107:
102:
86:
81:
75:
72:
71:
68:
67:
64:
63:
58:
53:
48:
42:
39:
38:
35:
34:
26:
25:
15:
9:
6:
4:
3:
2:
7826:
7815:
7812:
7810:
7807:
7805:
7802:
7801:
7799:
7789:
7785:
7782:
7778:
7777:
7763:
7761:
7752:
7746:
7732:on 2014-08-27
7728:
7721:
7715:
7701:
7697:
7691:
7684:(5): 125–130.
7683:
7679:
7672:
7666:
7662:
7661:Henri Andoyer
7657:
7648:
7644:
7640:
7634:
7630:
7626:
7620:
7613:
7607:
7600:(2): 135–141.
7599:
7595:
7588:
7586:
7577:
7571:
7563:
7559:
7552:
7550:
7548:
7539:
7532:
7525:
7519:
7515:
7511:
7507:
7503:
7499:
7494:
7489:
7485:
7481:
7474:
7466:
7460:
7452:
7442:
7438:
7434:
7430:
7429:Survey Review
7423:
7419:
7413:
7398:
7397:
7389:
7382:
7381:
7374:
7368:
7360:
7359:
7354:
7348:
7340:
7334:
7326:
7322:
7318:
7314:
7310:
7306:
7301:
7296:
7292:
7288:
7284:
7283:Bessel, F. W.
7278:
7270:
7266:
7262:
7256:
7248:
7244:
7240:
7234:
7218:
7214:
7207:
7201:
7187:
7180:
7166:
7159:
7145:on 2012-05-22
7144:
7140:
7134:
7130:
7120:
7117:
7115:
7112:
7110:
7107:
7105:
7102:
7100:
7097:
7095:
7092:
7090:
7087:
7085:
7082:
7080:
7077:
7076:
7070:
7068:
7064:
7060:
7056:
7052:
7048:
7029:
7025:
7019:
7009:
7005:
7001:
6996:
6992:
6985:
6982:
6979:
6975:
6971:
6963:
6959:
6955:
6950:
6946:
6939:
6936:
6933:
6930:
6927:
6924:
6917:
6916:
6915:
6913:
6909:
6905:
6901:
6891:
6889:
6872:
6868:
6864:
6860:
6856:
6852:
6848:
6844:
6840:
6837:
6811:
6807:
6803:
6798:
6794:
6790:
6787:
6763:
6759:
6755:
6751:
6747:
6743:
6739:
6735:
6731:
6728:
6702:
6698:
6694:
6689:
6685:
6681:
6678:
6648:
6645:
6639:
6636:
6634:
6628:
6625:
6614:
6601:
6592:
6589:
6583:
6578:
6574:
6570:
6564:
6561:
6555:
6541:
6537:
6533:
6525:
6521:
6517:
6513:
6507:
6504:
6492:
6488:
6479:
6477:
6471:
6468:
6457:
6452:
6446:
6442:
6438:
6435:
6429:
6427:
6421:
6417:
6413:
6409:
6406:
6395:
6394:
6393:
6391:
6383:
6364:
6361:
6354:
6350:
6342:
6338:
6334:
6330:
6327:
6321:
6317:
6314:
6306:
6305:
6304:
6287:
6279:
6271:
6268:
6265:
6254:
6251:
6248:
6242:
6236:
6229:
6225:
6218:
6214:
6210:
6205:
6201:
6194:
6188:
6184:
6180:
6172:
6171:
6170:
6153:
6146:
6142:
6138:
6133:
6129:
6122:
6118:
6114:
6110:
6107:
6102:
6099:
6095:
6088:
6084:
6080:
6075:
6071:
6064:
6060:
6056:
6052:
6049:
6044:
6041:
6034:
6033:
6032:
6030:
6006:
5999:
5994:
5989:
5981:
5970:
5966:
5962:
5956:
5945:
5941:
5937:
5929:
5925:
5915:
5911:
5908:
5903:
5899:
5890:
5887:
5883:
5878:
5873:
5868:
5856:
5852:
5849:
5844:
5840:
5831:
5828:
5824:
5817:
5814:
5810:
5799:
5795:
5791:
5788:
5785:
5782:
5775:
5774:
5773:
5762:
5759:
5756:
5755:
5754:
5752:
5747:
5733:
5699:
5696:
5693:
5684:
5681:
5675:
5672:
5662:
5659:
5656:
5649:
5648:
5628:
5625:
5620:
5615:
5611:
5605:
5602:
5597:
5593:
5589:
5586:
5581:
5577:
5567:
5564:
5561:
5558:
5555:
5549:
5546:
5536:
5533:
5528:
5523:
5519:
5513:
5510:
5505:
5501:
5497:
5494:
5489:
5485:
5475:
5472:
5469:
5466:
5463:
5457:
5454:
5447:
5446:
5429:
5423:
5419:
5415:
5410:
5406:
5399:
5396:
5390:
5384:
5380:
5376:
5371:
5367:
5360:
5357:
5350:
5349:
5348:
5331:
5327:
5303:
5299:
5290:
5286:
5265:
5261:
5256:
5251:
5247:
5218:
5214:
5209:
5204:
5200:
5176:
5169:
5168:central angle
5150:
5146:
5121:
5117:
5108:
5089:
5085:
5060:
5056:
5045:
5030:
5026:
5023:
5020:
5014:
5011:
5008:
5001:
4997:
4994:
4991:
4988:
4980:
4975:
4961:
4954:
4944:
4942:
4937:
4933:
4931:
4927:
4923:
4919:
4914:
4912:
4908:
4906:
4903:
4898:
4894:
4890:
4886:
4882:
4878:
4874:
4864:
4859:
4831:
4824:
4819:
4813:
4809:
4800:
4797:
4792:
4788:
4779:
4776:
4772:
4767:
4762:
4757:
4745:
4741:
4738:
4733:
4729:
4720:
4717:
4713:
4706:
4703:
4700:
4698:
4686:
4682:
4673:
4668:
4664:
4659:
4647:
4643:
4638:
4634:
4630:
4625:
4621:
4612:
4607:
4603:
4598:
4594:
4589:
4585:
4576:
4571:
4567:
4561:
4558:
4555:
4553:
4541:
4532:
4522:
4517:
4508:
4498:
4493:
4484:
4472:
4469:
4467:
4454:
4446:
4438:
4434:
4427:
4424:
4421:
4413:
4409:
4402:
4399:
4396:
4394:
4388:
4377:
4369:
4365:
4358:
4355:
4347:
4343:
4336:
4333:
4330:
4322:
4318:
4311:
4308:
4300:
4296:
4289:
4286:
4283:
4281:
4275:
4264:
4256:
4252:
4245:
4242:
4234:
4230:
4223:
4220:
4217:
4209:
4205:
4198:
4195:
4187:
4183:
4176:
4173:
4170:
4168:
4162:
4147:
4146:
4145:
4128:
4125:
4121:
4116:
4113:
4110:
4107:
4104:
4093:
4064:
4060:
4056:
4053:
4048:
4043:
4038:
4027:
4021:
4014:
4011:
4006:
4003:
3999:
3987:
3983:
3980:
3973:
3972:
3971:
3957:
3954:
3951:
3928:
3922:
3919:
3907:
3901:
3898:
3895:
3892:
3889:
3886:
3879:
3878:
3877:
3855:
3847:
3831:
3823:
3818:
3815:
3813:
3807:
3781:
3775:
3772:
3767:
3763:
3754:
3751:
3746:
3743:
3735:
3721:
3718:
3715:
3710:
3707:
3702:
3699:
3691:
3690:
3689:
3676:
3668:
3659:
3656:
3651:
3647:
3641:
3637:
3633:
3621:
3616:
3612:
3599:
3594:
3590:
3584:
3581:
3578:
3562:
3558:
3555:
3552:, set to the
3537:
3515:
3507:
3503:
3482:
3479:
3463:
3456:
3442:
3435:
3432:
3429:
3424:
3420:
3412:
3393:
3390:
3374:
3367:
3364:
3359:
3355:
3347:
3346:
3344:
3326:
3322:
3299:
3295:
3286:
3285:
3262:
3248:
3244:
3238:
3235:
3232:
3229:
3215:
3211:
3205:
3202:
3199:
3196:
3182:
3175:
3172:
3169:
3166:
3164:
3157:
3153:
3145:
3131:
3127:
3121:
3118:
3115:
3112:
3098:
3094:
3088:
3085:
3082:
3079:
3076:
3073:
3071:
3064:
3060:
3048:
3032:
3021:
3017:
3014:
2984:
2976:
2959:
2934:
2924:
2907:
2900:
2899:
2897:
2883:
2876:
2868:
2860:
2856:
2849:
2844:
2836:
2828:
2824:
2815:
2812:
2805:
2804:
2803:
2801:
2794:FCC's formula
2777:
2770:
2762:
2748:
2744:
2741:
2727:
2720:
2714:
2709:
2701:
2684:
2677:
2669:
2666:
2659:
2658:
2657:
2641:
2637:
2617:
2603:
2600:
2595:
2590:
2586:
2578:
2574:
2565:
2562:
2551:
2547:
2544:
2540:
2529:
2527:
2523:
2519:
2517:
2512:
2510:
2491:
2469:
2441:
2436:
2430:
2426:
2417:
2414:
2403:
2399:
2396:
2392:
2381:
2377:
2373:
2370:
2366:
2361:
2356:
2351:
2345:
2341:
2332:
2329:
2325:
2318:
2307:
2303:
2299:
2295:
2288:
2285:
2278:
2277:
2276:
2268:
2245:
2237:
2220:
2213:
2210:
2204:
2199:
2191:
2180:
2177:
2174:
2167:
2166:
2165:
2149:
2145:
2107:
2097:
2092:
2086:
2082:
2073:
2070:
2059:
2055:
2052:
2049:
2045:
2040:
2035:
2030:
2024:
2020:
2011:
2008:
2004:
1997:
1990:
1987:
1985:
1973:
1968:
1962:
1958:
1949:
1946:
1935:
1931:
1928:
1925:
1921:
1916:
1911:
1906:
1900:
1896:
1887:
1884:
1878:
1874:
1865:
1862:
1859:
1855:
1848:
1845:
1843:
1838:
1827:
1826:
1825:
1817:
1814:
1812:
1808:
1803:
1801:
1796:
1774:
1753:
1739:
1737:
1733:
1731:
1725:
1708:
1704:
1680:
1676:
1652:
1648:
1624:
1620:
1590:
1566:
1562:
1538:
1534:
1512:
1480:
1458:
1434:
1423:
1419:
1415:
1414:180° meridian
1411:
1410:discontinuity
1407:
1403:
1402:singularities
1398:
1384:
1367:
1348:
1336:
1318:
1311:
1310:
1309:
1307:
1286:
1281:
1275:
1271:
1267:
1262:
1258:
1251:
1240:
1232:
1231:
1230:
1227:
1204:
1199:
1195:
1191:
1186:
1182:
1178:
1176:
1171:
1161:
1156:
1152:
1148:
1143:
1139:
1135:
1133:
1128:
1114:
1113:
1112:
1109:
1107:
1102:
1098:
1080:
1058:
1049:
1031:
1027:
1004:
996:
992:
988:
983:
979:
948:
944:
940:
935:
931:
899:
874:
870:
845:
841:
818:
815:
801:
799:
783:
759:
756:
742:
711:
707:
684:
680:
657:
653:
630:
626:
617:
603:
600:
586:
569:
553:
549:
545:
531:
514:
513:
507:
501:
498:
496:Flat surface;
495:
494:
493:
491:
481:
479:
475:
471:
467:
462:
460:
456:
452:
448:
442:
438:
433:
422:
417:
415:
410:
408:
403:
402:
400:
399:
392:
389:
387:
384:
382:
379:
378:
372:
370:
367:
366:
362:
360:
357:
356:
352:
350:
347:
346:
342:
340:
337:
336:
332:
330:
327:
326:
322:
320:
317:
316:
312:
310:
307:
306:
302:
300:
297:
296:
292:
290:
287:
286:
282:
280:
277:
276:
272:
270:
267:
266:
262:
260:
257:
256:
252:
250:
247:
246:
238:
237:
230:
227:
225:
219:
217:
211:
209:
203:
201:
197:BeiDou (BDS)
195:
193:
187:
185:
182:
180:
177:
176:
170:
169:
162:
159:
157:
154:
152:
149:
147:
144:
142:
139:
137:
134:
131:
127:
123:
121:
118:
116:
113:
111:
108:
106:
103:
99:
98:circumference
95:
90:
87:
85:
82:
80:
77:
76:
70:
69:
62:
59:
57:
54:
52:
49:
47:
44:
43:
37:
36:
32:
28:
27:
24:
21:
20:
7734:. Retrieved
7727:the original
7714:
7703:. Retrieved
7699:
7690:
7681:
7677:
7671:
7656:
7628:
7619:
7606:
7597:
7593:
7531:
7486:(1): 43–55.
7483:
7479:
7473:
7459:cite journal
7450:
7444:. Retrieved
7432:
7428:
7418:Vincenty, T.
7412:
7401:. Retrieved
7395:
7388:
7379:
7357:
7347:
7333:cite journal
7290:
7286:
7277:
7268:
7255:
7246:
7233:
7221:. Retrieved
7216:
7212:
7200:
7189:. Retrieved
7179:
7168:. Retrieved
7158:
7147:. Retrieved
7143:the original
7133:
7114:Meridian arc
7084:Earth radius
7054:
7046:
7044:
6911:
6899:
6897:
6667:
6389:
6379:
6302:
6168:
6028:
6026:
5771:
5748:
5725:
5046:
4976:
4950:
4938:
4934:
4915:
4904:
4901:
4876:
4869:
4084:
3943:
3845:
3819:
3816:
3809:
3571:
3553:
3345:as follows:
2797:
2623:
2530:
2515:
2514:
2508:
2507:
2461:
2274:
2265:
2131:
1823:
1815:
1804:
1797:
1745:
1729:
1726:
1399:
1373:
1340:
1303:
1228:
1225:
1110:
1100:
1050:
807:
804:Nomenclature
775:
505:
489:
487:
484:Introduction
468:in terms of
463:
450:
446:
445:
437:Swabian Jura
173:Technologies
128: /
78:
40:Fundamentals
7804:Cartography
7570:cite report
7119:Scale (map)
3554:Clarke 1866
2524:(See also "
1811:cartography
1668:=45°) and (
51:Geodynamics
7798:Categories
7736:2014-08-26
7705:2024-08-04
7562:1811/24333
7446:2009-07-11
7403:2011-08-01
7223:8 November
7191:2023-11-28
7170:2024-06-23
7149:2008-12-06
7125:References
6908:arc length
4953:flattening
798:flattening
7647:250050479
7518:119310141
7493:1109.4448
7367:cite book
7355:(1964) .
7325:118760590
7300:0908.1824
7002:−
6983:−
6937:−
6928:−
6865:λ
6861:−
6849:λ
6838:λ
6834:Δ
6808:λ
6804:−
6795:λ
6788:λ
6785:Δ
6756:ϕ
6752:−
6740:ϕ
6729:ϕ
6725:Δ
6699:ϕ
6695:−
6686:ϕ
6679:ϕ
6676:Δ
6646:λ
6643:Δ
6626:λ
6622:Δ
6602:ϕ
6599:Δ
6575:ϕ
6565:
6556:ϕ
6553:Δ
6489:ϕ
6486:Δ
6469:ϕ
6465:Δ
6443:ϕ
6439:
6414:ϕ
6410:
6269:−
6252:−
6211:−
6143:ϕ
6139:
6085:ϕ
6081:
5971:ϕ
5946:ϕ
5926:ϕ
5923:Δ
5912:
5900:λ
5897:Δ
5891:
5857:ϕ
5853:
5841:λ
5838:Δ
5832:
5818:
5800:ϕ
5676:−
5673:σ
5626:σ
5621:
5603:
5587:
5568:σ
5565:
5556:σ
5534:σ
5529:
5511:
5495:
5476:σ
5473:
5467:−
5464:σ
5420:β
5416:−
5407:β
5381:β
5368:β
5328:λ
5300:λ
5262:λ
5248:β
5215:λ
5201:β
5177:σ
5147:β
5118:β
5086:ϕ
5057:ϕ
5027:ϕ
5024:
5012:−
4998:
4989:β
4926:antipodal
4810:ϕ
4807:Δ
4801:
4789:λ
4786:Δ
4780:
4746:ϕ
4742:
4730:λ
4727:Δ
4721:
4683:λ
4680:Δ
4674:
4648:ϕ
4644:
4631:−
4622:ϕ
4619:Δ
4613:
4586:ϕ
4583:Δ
4577:
4529:Δ
4505:Δ
4481:Δ
4435:ϕ
4428:
4422:−
4410:ϕ
4403:
4385:Δ
4366:λ
4359:
4344:ϕ
4337:
4331:−
4319:λ
4312:
4297:ϕ
4290:
4272:Δ
4253:λ
4246:
4231:ϕ
4224:
4218:−
4206:λ
4199:
4184:ϕ
4177:
4159:Δ
4117:
4057:⋯
3955:≪
3902:
3776:ϕ
3773:−
3768:∘
3752:π
3744:θ
3719:ϕ
3716:−
3708:π
3700:θ
3669:λ
3666:Δ
3660:
3648:θ
3638:θ
3629:−
3613:θ
3591:θ
3480:π
3464:ϕ
3443:ϕ
3436:
3391:π
3375:ϕ
3249:ϕ
3239:
3216:ϕ
3206:
3197:−
3183:ϕ
3176:
3170:111.41513
3132:ϕ
3122:
3099:ϕ
3089:
3080:−
3077:111.13209
3022:ϕ
3018:
2985:ϕ
2960:λ
2957:Δ
2935:ϕ
2932:Δ
2869:λ
2866:Δ
2837:ϕ
2834:Δ
2763:λ
2760:Δ
2749:ϕ
2745:
2728:ϕ
2702:ϕ
2699:Δ
2685:ϕ
2638:λ
2635:Δ
2601:≈
2587:ϕ
2584:Δ
2575:λ
2572:Δ
2566:
2552:ϕ
2548:
2511:eridional
2427:λ
2424:Δ
2418:
2404:ϕ
2400:
2382:ϕ
2342:λ
2339:Δ
2333:
2326:ϕ
2323:Δ
2308:ϕ
2238:λ
2235:Δ
2221:ϕ
2214:
2192:ϕ
2189:Δ
2146:λ
2143:Δ
2083:λ
2080:Δ
2074:
2060:ϕ
2056:
2021:λ
2018:Δ
2012:
2005:ϕ
2002:Δ
1988:≈
1959:λ
1956:Δ
1950:
1936:ϕ
1932:
1897:λ
1894:Δ
1888:
1875:ϕ
1872:Δ
1866:
1807:projected
1724:=−135°).
1705:λ
1677:ϕ
1649:λ
1621:ϕ
1591:ϕ
1563:λ
1535:λ
1513:λ
1510:Δ
1481:ϕ
1459:λ
1456:Δ
1435:ϕ
1432:Δ
1385:λ
1382:Δ
1272:ϕ
1259:ϕ
1241:ϕ
1196:λ
1192:−
1183:λ
1172:λ
1169:Δ
1153:ϕ
1149:−
1140:ϕ
1129:ϕ
1126:Δ
1081:λ
1059:ϕ
1051:Latitude
993:λ
980:ϕ
945:λ
932:ϕ
757:∝
734:Δ
601:∝
583:Δ
546:∝
528:Δ
474:longitude
130:Longitude
56:Geomatics
7745:cite web
7627:(1927).
7263:(1806).
7241:(1735).
7073:See also
6873:′
6857:′
6841:′
6764:′
6748:′
6732:′
6629:′
6522:′
6472:′
6422:′
6390:R′
6339:′
6318:′
6185:′
6119:′
6061:′
6029:R′
4941:Puissant
4922:Vincenty
4889:Legendre
4885:Clairaut
4873:geodesic
2506:are the
1412:at the ±
470:latitude
455:distance
309:ISO 6709
207:(Europe)
205:Galileo
191:(Russia)
189:GLONASS
126:Latitude
115:Geodesic
73:Concepts
7814:Geodesy
7540:. 1.32.
7524:Addenda
7498:Bibcode
7305:Bibcode
5749:On the
4897:Helmert
3233:0.00012
3200:0.09455
3116:0.00120
3083:0.56605
1732:-vector
1404:at the
1106:radians
1097:degrees
796:is the
453:is the
439:to the
369:Geo URI
339:NAVD 88
249:NGVD 29
223:(Japan)
215:(India)
199:(China)
61:History
46:Geodesy
23:Geodesy
7645:
7635:
7516:
7323:
7045:where
6668:where
5815:arcsin
5751:GRS 80
5726:where
4995:arctan
4911:Errata
4895:, and
4893:Bessel
3899:arcsin
3287:Where
2898:where
2462:where
2105:
1696:=89°,
1640:=89°,
1306:radius
359:GCJ-02
349:ETRS89
329:WGS 84
319:NAD 83
299:GRS 80
259:OSGB36
213:NAVIC
94:radius
7809:Earth
7730:(PDF)
7723:(PDF)
7514:S2CID
7488:arXiv
7425:(PDF)
7321:S2CID
7295:arXiv
7209:(PDF)
6904:chord
6388:is 1/
6380:(The
2518:ormal
1422:delta
1406:Poles
747:error
591:error
536:error
490:exact
459:Earth
289:SAD69
269:SK-42
84:Geoid
7751:link
7643:OCLC
7633:ISBN
7576:link
7465:link
7373:link
7339:link
7225:2017
7057:are
7053:and
6914:as:
5319:and
5236:and
3820:The
3530:and
3314:and
2949:and
2798:The
2484:and
1554:and
1101:must
968:and
862:and
472:and
441:Alps
279:ED50
96:and
7786:An
7779:An
7558:hdl
7506:doi
7437:doi
7313:doi
7291:331
6986:0.5
6940:0.5
6562:sin
6436:tan
6407:tan
6130:cos
6072:cos
5909:sin
5888:cos
5850:cos
5829:sin
5612:sin
5594:sin
5578:cos
5562:sin
5520:cos
5502:cos
5486:sin
5470:sin
5137:,
5021:tan
4798:sin
4777:cos
4739:cos
4718:sin
4665:sin
4635:sin
4604:cos
4568:sin
4425:sin
4400:sin
4356:sin
4334:cos
4309:sin
4287:cos
4243:cos
4221:cos
4196:cos
4174:cos
4114:sin
3970:),
3876:,
3755:180
3657:cos
3483:180
3433:cos
3394:180
3236:cos
3203:cos
3173:cos
3119:cos
3086:cos
3015:cos
2800:FCC
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