216:
133:
2132:
2050:
36:
819:
2321:
seen between GRS-80 and WGS-84 results from an unintentional truncation in the latter's defining constants: while the WGS-84 was designed to adhere closely to the GRS-80, incidentally the WGS-84 derived flattening turned out to differ slightly from the GRS-80 flattening because the normalized second
1112:
is the historical method of determining the ellipsoid. Two meridian arc measurements will allow the derivation of two parameters required to specify a reference ellipsoid. For example, if the measurements were hypothetically performed exactly over the equator plane and either geographical pole, the
1064:
as well as different assumed positions of the center and different axis orientations relative to the solid Earth. Starting in the late twentieth century, improved measurements of satellite orbits and star positions have provided extremely accurate determinations of the Earth's center of mass and of
2167:
1967) in the listing was recommended for adoption. The new ellipsoid was not recommended to replace the
International Ellipsoid (1924), but was advocated for use where a greater degree of accuracy is required. It became a part of the GRS-67 which was approved and adopted at the 1971 meeting of the
1484:
3165:); note further that "ITRF solutions are specified by Cartesian equatorial coordinates X, Y and Z. If needed, they can be transformed to geographical coordinates (λ, φ, h) referred to an ellipsoid. In this case the GRS80 ellipsoid is recommended." (
2151:
The reference ellipsoid models listed below have had utility in geodetic work and many are still in use. The older ellipsoids are named for the individual who derived them and the year of development is given. In 1887 the
English surveyor Colonel
1796:
1939:
Longer arcs with multiple intermediate-latitude determinations can completely determine the ellipsoid that best fits the surveyed region. In practice, multiple arc measurements are used to determine the ellipsoid parameters by the method of
3160:
Note that the current best estimates, given by the IERS Conventions, "should not be mistaken for conventional values, such as those of the
Geodetic Reference System GRS80 ... which are, for example, used to express geographic coordinates"
2368:. WGS-84 is peculiar in that the same name is used for both the complete geodetic reference system and its component ellipsoidal model. Nevertheless, the two concepts—ellipsoidal model and geodetic reference system—remain distinct.
1080:
close to 1/300 (more precisely, 1/298.257223563, by definition), corresponding to a difference of the major and minor semi-axes of approximately 21 km (13 miles) (more precisely, 21.3846857548205 km). For comparison, Earth's
924:, and related areas, the word 'ellipsoid' is understood to mean 'oblate ellipsoid of revolution', and the older term 'oblate spheroid' is hardly used. For bodies that cannot be well approximated by an ellipsoid of revolution a
1934:
1625:
2175:
The GRS-80 (Geodetic
Reference System 1980) as approved and adopted by the IUGG at its Canberra, Australia meeting of 1979 is based on the equatorial radius (semi-major axis of Earth ellipsoid)
1842:
772:
may be the better choice. When geodetic measurements have to be computed on a mathematical reference surface, this surface should have a similar curvature as the regional geoid; otherwise,
1553:
858:
from the rotation of these massive objects (for planetary bodies that do rotate). Because of their relative simplicity, reference ellipsoids are used as a preferred surface on which
1340:
1010:
1348:
3096:. NOAA technical publications. U.S. Department of Commerce, National Oceanic and Atmospheric Administration, National Ocean Service, Charting and Geodetic Services. p. 107
3146:
NIMA Technical Report TR8350.2, "Department of
Defense World Geodetic System 1984, Its Definition and Relationships With Local Geodetic Systems", Third Edition, 4 July 1997
1207:
1707:
1176:
1671:
1648:
2263:
1240:
2347:
2243:
1294:
1267:
2319:
2291:
2216:
2750:
2664:
2193:
1982:
1962:
1699:
2156:
CB FRS RE was awarded the Gold Medal of the Royal
Society for his work in determining the figure of the Earth. The international ellipsoid was developed by
3252:
1848:
791:
of millions of boundary stones should remain fixed for a long period. If their reference surface changes, the coordinates themselves also change.
575:
902:
2371:
Note that the same ellipsoid may be known by different names. It is best to mention the defining constants for unambiguous identification.
3201:
P. K. Seidelmann (Chair), et al. (2005), “Report Of The IAU/IAG Working Group On
Cartographic Coordinates And Rotational Elements: 2003,”
397:
787:, despite the fact that their main axes deviate by several hundred meters from the modern values. Another reason is a judicial one: the
2786:
565:
533:
2160:
in 1910 and adopted by the
International Union of Geodesy and Geophysics (IUGG) in 1924, which recommended it for international use.
1297:
543:
842:, or other planetary body, as opposed to a perfect, smooth, and unaltered sphere, which factors in the undulations of the bodies'
905:
in which he included a proof that a rotating self-gravitating fluid body in equilibrium takes the form of a flattened ("oblate")
3237:
3083:
100:
3125:
523:
1558:
72:
3181:
1019:
is the amount of flattening at each pole, relative to the radius at the equator. This is often expressed as a fraction 1/
17:
951:, becomes the distance from the centre to either pole. These two lengths completely specify the shape of the ellipsoid.
503:
304:
79:
3217:
OpenGIS Implementation
Specification for Geographic information - Simple feature access - Part 1: Common architecture
3039:
3018:
603:
119:
1807:
1122:
2114:
53:
2014:
86:
1492:
157:
389:
57:
2076:
405:
1113:
radii of curvature so obtained would be related to the equatorial radius and the polar radius, respectively
68:
2079:
1479:{\displaystyle M_{0}(\varphi _{i})={\frac {a(1-e^{2})}{(1-e_{0}^{2}\sin ^{2}\varphi _{i})^{\frac {3}{2}}}}}
1555:. Then discrepancies between empirical and theoretical values of the radius of curvature can be formed as
2888:
1302:
1065:
its axis of revolution; and those parameters have been adopted also for all modern reference ellipsoids.
971:
193:
1056:
A great many ellipsoids have been used to model the Earth in the past, with different assumed values of
2000:
1674:
3147:
2863:
2809:
368:
2793:
3242:
2587:
1941:
1791:{\displaystyle \delta M_{i}\approx \delta a(\partial M/\partial a)+\delta f(\partial M/\partial f)}
882:
335:
282:
3107:
1185:
954:
In geodesy publications, however, it is common to specify the semi-major axis (equatorial radius)
3272:
2989:
2396:
2365:
2118:
773:
473:
433:
46:
3089:
2644:
2004:
1135:
1101:
between 1/3 and 1/2 (meaning that the polar diameter is between 50% and 67% of the equatorial.
453:
3115:
3091:
1653:
1630:
671:, is approximately aligned with the Earth's axis of rotation. The ellipsoid is defined by the
2607:
2567:
2547:
2527:
2248:
2157:
2153:
2144:
1212:
1069:
656:
596:
513:
263:
93:
2322:
degree zonal harmonic gravitational coefficient, that was derived from the GRS-80 value for
2927:
2325:
2221:
2169:
2163:
At the 1967 meeting of the IUGG held in
Lucerne, Switzerland, the ellipsoid called GRS-67 (
2072:
2042:
2029:
1272:
1245:
413:
363:
289:
2356:
field formula to go with it. Commonly an ellipsoidal model is part of a more encompassing
2168:
IUGG held in Moscow. It is used in Australia for the Australian Geodetic Datum and in the
8:
3059:. USGS Professional Paper 1395. Washington, D.C.: Government Printing Office. p. 17.
2898:
2487:
2296:
2268:
2088:
2024:
no longer uses simple meridian arcs or ground triangulation networks, but the methods of
839:
803:
746:
625:
325:
273:
144:
2931:
2198:
2970:
2893:
2178:
1967:
1947:
1684:
925:
629:
245:
215:
3166:
3162:
931:
The shape of an ellipsoid of revolution is determined by the shape parameters of that
3121:
3035:
3014:
2974:
2025:
1992:
855:
780:
715:
699:
694:
Many methods exist for determination of the axes of an Earth ellipsoid, ranging from
345:
330:
3267:
2962:
2935:
2858:
2507:
1996:
859:
784:
711:
703:
589:
340:
2112:
The triad is also known as Earth ellipsoidal coordinates (not to be confused with
3185:
2966:
2357:
2164:
1109:
944:
936:
914:
707:
197:
3209:
3178:
1929:{\displaystyle \partial M/\partial f\approx -2a_{0}(1-1.5\sin ^{2}\varphi _{i})}
3074:
2883:
2868:
2433:
2353:
1094:
886:
847:
779:
This is the reason for the "long life" of former reference ellipsoids like the
754:
553:
320:
294:
1242:, the solution starts from an initial approximation for the equatorial radius
3261:
2873:
1678:
132:
2878:
898:
810:
usually adapts the axes of the Earth ellipsoid to the best available data.
799:
695:
278:
3090:
National Geodetic Survey (U.S.).; National Geodetic Survey (U.S.) (1986).
2918:
Alexander, J. C. (1985). "The Numerics of Computing Geodetic Ellipsoids".
2349:, was truncated to eight significant digits in the normalization process.
2131:
687:); their radial difference is slightly more than 21 km, or 0.335% of
2953:
Heine, George (September 2013). "Euler and the Flattening of the Earth".
764:
While the mean Earth ellipsoid is the ideal basis of global geodesy, for
235:
176:
2049:
2010:
1985:
1126:
959:
851:
788:
668:
664:
660:
641:
167:
137:
2098:
1085:
is even less elliptical, with a flattening of less than 1/825, while
906:
871:
867:
645:
637:
314:
240:
180:
2939:
35:
2352:
An ellipsoidal model describes only the ellipsoid's geometry and a
863:
765:
652:
493:
310:
299:
3114:
Awange, J.L.; Grafarend, E.W.; Paláncz, B.; Zaletnyik, P. (2010).
3246:
2083:
2021:
1086:
1038:
1034:
932:
921:
910:
877:
In the context of standardization and geographic applications, a
843:
827:
818:
738:
633:
373:
230:
207:
153:
3223:
3013:
Torge, W (2001) Geodesy (3rd edition), published by de Gruyter,
2013:
is another technique for determining Earth's flattening, as per
806:
is increasingly accurate, the International Geoscientific Union
3113:
2770:
1090:
758:
483:
443:
381:
1342:
can be calculated at the latitude of each arc measurement as:
913:
rotated around its minor diameter; a shape which he termed an
1995:
observed in the radius of curvature measurements reflect the
1944:. The parameters determined are usually the semi-major axis,
1182:
For two arc measurements each at arbitrary average latitudes
835:
750:
723:
268:
189:
802:, these regional reasons are less relevant. As knowledge of
3032:
Flattening the Earth: Two Thousand Years of Map Projections
2816:
2361:
1082:
807:
570:
463:
141:
834:
is a mathematically defined surface that approximates the
1627:. Finally, corrections for the initial equatorial radius
1073:
1033:
then being the "inverse flattening". A great many other
862:
computations are performed and point coordinates such as
795:
719:
846:
due to variations in the composition and density of the
1620:{\displaystyle \delta M_{i}=M_{i}-M_{0}(\varphi _{i})}
943:, becomes the equatorial radius of the ellipsoid: the
757:, and therefore an ideal Earth ellipsoid has the same
734:
There are two types of ellipsoid: mean and reference.
2328:
2299:
2271:
2251:
2224:
2201:
2181:
1970:
1950:
1851:
1810:
1710:
1687:
1656:
1633:
1561:
1495:
1351:
1305:
1275:
1248:
1215:
1188:
1138:
1041:
but they can all be related to one or two of the set
974:
27:
Geometric figure which approximates the Earth's shape
663:(shorter diameter), which connects the geographical
745:. It refers to a theoretical coherence between the
702:or the analysis and interconnection of continental
60:. Unsourced material may be challenged and removed.
2341:
2313:
2285:
2257:
2237:
2210:
2187:
1976:
1956:
1928:
1836:
1790:
1693:
1665:
1642:
1619:
1547:
1478:
1334:
1288:
1261:
1234:
1201:
1170:
1004:
166: Circle with diameter equal to the ellipse's
3259:
881:is the mathematical model used as foundation by
776:of the measurements will get small distortions.
3029:
2092:. They include geodetic latitude (north/south)
741:of the Earth's surface curvature is called the
2143:) and mean Earth radii as defined in the 1984
2126:
1837:{\displaystyle \partial M/\partial a\approx 1}
1123:Earth polar and equatorial radius of curvature
3210:https://astrogeology.usgs.gov/Projects/WGCCRE
3068:
3066:
2293:a derived quantity. The minute difference in
597:
706:. Amongst the different set of data used in
624:is a mathematical figure approximating the
3203:Celestial Mechanics and Dynamical Astronomy
3120:. Springer Berlin Heidelberg. p. 156.
3034:. University of Chicago Press. p. 82.
1089:is visibly oblate at about 1/15 and one of
3063:
1548:{\displaystyle e_{0}^{2}=2f_{0}-f_{0}^{2}}
1129:would readily follow from its definition:
604:
590:
571:Spatial Reference System Identifier (SRID)
566:International Terrestrial Reference System
3023:
2917:
1228:
120:Learn how and when to remove this message
2130:
2048:
817:
131:
3156:
3154:
3072:
2035:
892:
710:are several of special importance: the
160:as that of Earth, with north at the top
14:
3260:
3243:Coordinate systems and transformations
3054:
2990:"Strange but True: Earth Is Not Round"
1298:Earth's meridional radius of curvature
813:
737:A data set which describes the global
3253:Coordinate Systems, Frames and Datums
2952:
794:However, for international networks,
3151:
3117:Algebraic Geodesy and Geoinformatics
3007:
2987:
749:and the meridional curvature of the
691:(which is not quite 6,400 km).
58:adding citations to reliable sources
29:
1801:where the partial derivatives are:
1335:{\displaystyle M_{0}(\varphi _{i})}
1005:{\displaystyle f={\frac {a-b}{a}}.}
576:Universal Transverse Mercator (UTM)
538:European Terrestrial Ref. Sys. 1989
24:
3057:Map Projections — A Working Manual
2988:Choi, Charles Q. (12 April 2007).
2110:(also known as geodetic height).
1964:, and any of the semi-minor axis,
1863:
1852:
1822:
1811:
1779:
1768:
1747:
1736:
648:have been used as approximations.
448:Ordnance Survey Great Britain 1936
414:Discrete Global Grid and Geocoding
305:Horizontal position representation
25:
3284:
3231:
2245:and angular velocity of rotation
928:(or scalene) ellipsoid is used.
179:, 100 km (62 mi) above
2360:. For example, the older ED-50 (
2265:, making the inverse flattening
2115:ellipsoidal-harmonic coordinates
2041:This section is an excerpt from
1104:
838:, which is the truer, imperfect
364:Global Nav. Sat. Systems (GNSSs)
214:
34:
3195:
3172:
3140:
909:of revolution, generated by an
528:N. American Vertical Datum 1988
45:needs additional citations for
3048:
2981:
2946:
2911:
2470:Everest 1830 (1967 Definition)
2465:West Malaysia & Singapore
1923:
1888:
1785:
1765:
1753:
1733:
1614:
1601:
1459:
1411:
1406:
1387:
1375:
1362:
1329:
1316:
1157:
1145:
1072:, widely used for mapping and
558:Internet link to a point 2010
488:Geodetic Reference System 1980
406:Quasi-Zenith Sat. Sys. (QZSS)
13:
1:
3224:http://www.opengeospatial.org
2904:
2364:) is based on the Hayford or
753:. The latter is close to the
548:Chinese obfuscated datum 2002
3238:Geographic coordinate system
2967:10.4169/mathhorizons.21.1.25
2453:Everest 1830 Modified (1967)
2080:orthogonal coordinate system
1673:can be solved by means of a
1202:{\displaystyle \varphi _{i}}
1097:, is highly flattened, with
879:geodesic reference ellipsoid
850:, as well as the subsequent
498:Geographic point coord. 1983
7:
2889:Planetary coordinate system
2852:
2482:Brunei & East Malaysia
2127:Historical Earth ellipsoids
714:of 1841, the international
458:Systema Koordinat 1942 goda
10:
3289:
2684:Australian National (1966)
2040:
2001:deflection of the vertical
1675:system of linear equations
518:World Geodetic System 1984
2864:Earth radius of curvature
2165:Geodetic Reference System
2106:, and ellipsoidal height
1171:{\displaystyle f=(a-b)/a}
508:North American Datum 1983
478:South American Datum 1969
3219:, Annex B.4. 2005-11-30
3205:, 91, pp. 203–215.
3055:Snyder, John P. (1987).
3030:Snyder, John P. (1993).
2701:New International (1967)
2379:Reference ellipsoid name
1942:least squares adjustment
1666:{\displaystyle \delta f}
1643:{\displaystyle \delta a}
883:spatial reference system
729:
369:Global Pos. System (GPS)
336:Spatial reference system
3179:IERS Conventions (2003)
2366:International Ellipsoid
2258:{\displaystyle \omega }
2147:revision (not to scale)
2119:ellipsoidal coordinates
1269:and for the flattening
1235:{\displaystyle i=1,\,2}
136:A scale diagram of the
2659:USSR, Russia, Romania
2343:
2315:
2287:
2259:
2239:
2218:, dynamic form factor
2212:
2189:
2148:
2069:
2005:astrogeodetic leveling
1978:
1958:
1930:
1838:
1792:
1695:
1667:
1644:
1621:
1549:
1480:
1336:
1290:
1263:
1236:
1203:
1172:
1006:
823:
201:
2733:South American (1969)
2382:Equatorial radius (m)
2344:
2342:{\displaystyle J_{2}}
2316:
2288:
2260:
2240:
2238:{\displaystyle J_{2}}
2213:
2190:
2158:John Fillmore Hayford
2154:Alexander Ross Clarke
2145:World Geodetic System
2134:
2053:Geodetic coordinates
2052:
2036:Geodetic coordinates
1979:
1959:
1931:
1839:
1793:
1696:
1668:
1645:
1622:
1550:
1481:
1337:
1291:
1289:{\displaystyle f_{0}}
1264:
1262:{\displaystyle a_{0}}
1237:
1204:
1173:
1007:
821:
768:networks a so-called
264:Geographical distance
135:
3073:Bomford, G. (1952).
2627:International (1924)
2326:
2297:
2269:
2249:
2222:
2199:
2179:
2170:South American Datum
2073:Geodetic coordinates
2043:Geodetic coordinates
2030:satellite gravimetry
1968:
1948:
1849:
1808:
1708:
1685:
1654:
1631:
1559:
1493:
1349:
1303:
1273:
1246:
1213:
1186:
1136:
1074:satellite navigation
972:
893:Ellipsoid parameters
743:mean Earth Ellipsoid
644:. Various different
632:for computations in
438:Sea Level Datum 1929
290:Geodetic coordinates
54:improve this article
2994:Scientific American
2932:1985SIAMR..27..241A
2899:Planetary ellipsoid
2362:European Datum 1950
2314:{\displaystyle 1/f}
2286:{\displaystyle 1/f}
2089:reference ellipsoid
1988:, or eccentricity.
1650:and the flattening
1544:
1510:
1434:
840:figure of the Earth
832:reference ellipsoid
814:Reference ellipsoid
770:reference ellipsoid
747:geographic latitude
468:European Datum 1950
426:Standards (history)
326:Reference ellipsoid
274:Figure of the Earth
145:reference ellipsoid
18:Reference ellipsoid
3184:2014-04-19 at the
2894:History of geodesy
2388:Inverse flattening
2339:
2311:
2283:
2255:
2235:
2211:{\displaystyle GM}
2208:
2185:
2149:
2070:
2015:Clairaut's theorem
1993:systematic effects
1974:
1954:
1926:
1834:
1788:
1691:
1663:
1640:
1617:
1545:
1530:
1496:
1476:
1420:
1332:
1296:. The theoretical
1286:
1259:
1232:
1199:
1168:
1035:ellipse parameters
1002:
824:
804:the Earth's figure
718:of 1924, and (for
346:Vertical positions
202:
3188:(Chp. 1, page 12)
3127:978-3-642-12124-1
3093:Geodetic Glossary
2850:
2849:
2188:{\displaystyle a}
2026:satellite geodesy
2003:, as explored in
1977:{\displaystyle b}
1957:{\displaystyle a}
1694:{\displaystyle M}
1474:
1470:
997:
856:centrifugal force
722:positioning) the
716:Hayford ellipsoid
704:geodetic networks
700:satellite geodesy
655:(an ellipsoid of
614:
613:
562:
561:
341:Spatial relations
331:Satellite geodesy
286:
130:
129:
122:
104:
69:"Earth ellipsoid"
16:(Redirected from
3280:
3189:
3176:
3170:
3158:
3149:
3144:
3138:
3137:
3135:
3134:
3111:
3105:
3104:
3102:
3101:
3087:
3081:
3080:
3070:
3061:
3060:
3052:
3046:
3045:
3027:
3021:
3011:
3005:
3004:
3002:
3000:
2985:
2979:
2978:
2950:
2944:
2943:
2915:
2859:Equatorial bulge
2385:Polar radius (m)
2376:
2375:
2348:
2346:
2345:
2340:
2338:
2337:
2320:
2318:
2317:
2312:
2307:
2292:
2290:
2289:
2284:
2279:
2264:
2262:
2261:
2256:
2244:
2242:
2241:
2236:
2234:
2233:
2217:
2215:
2214:
2209:
2194:
2192:
2191:
2186:
2142:
2138:
2109:
2105:
2095:
2068:
1997:geoid undulation
1983:
1981:
1980:
1975:
1963:
1961:
1960:
1955:
1935:
1933:
1932:
1927:
1922:
1921:
1909:
1908:
1887:
1886:
1862:
1843:
1841:
1840:
1835:
1821:
1797:
1795:
1794:
1789:
1778:
1746:
1723:
1722:
1700:
1698:
1697:
1692:
1672:
1670:
1669:
1664:
1649:
1647:
1646:
1641:
1626:
1624:
1623:
1618:
1613:
1612:
1600:
1599:
1587:
1586:
1574:
1573:
1554:
1552:
1551:
1546:
1543:
1538:
1526:
1525:
1509:
1504:
1485:
1483:
1482:
1477:
1475:
1473:
1472:
1471:
1463:
1457:
1456:
1444:
1443:
1433:
1428:
1409:
1405:
1404:
1382:
1374:
1373:
1361:
1360:
1341:
1339:
1338:
1333:
1328:
1327:
1315:
1314:
1295:
1293:
1292:
1287:
1285:
1284:
1268:
1266:
1265:
1260:
1258:
1257:
1241:
1239:
1238:
1233:
1208:
1206:
1205:
1200:
1198:
1197:
1177:
1175:
1174:
1169:
1164:
1100:
1093:triaxial moons,
1079:
1063:
1059:
1052:
1048:
1044:
1032:
1022:
1018:
1011:
1009:
1008:
1003:
998:
993:
982:
964:
957:
950:
947:of the ellipse,
942:
939:of the ellipse,
860:geodetic network
822:Flattened sphere
798:positioning, or
785:Bessel ellipsoid
712:Bessel ellipsoid
708:national surveys
690:
686:
678:
606:
599:
592:
430:
429:
409:
401:
393:
385:
377:
317:
276:
218:
204:
203:
187:
174:
165:
151:
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
3288:
3287:
3283:
3282:
3281:
3279:
3278:
3277:
3258:
3257:
3234:
3198:
3193:
3192:
3186:Wayback Machine
3177:
3173:
3159:
3152:
3145:
3141:
3132:
3130:
3128:
3112:
3108:
3099:
3097:
3088:
3084:
3071:
3064:
3053:
3049:
3042:
3028:
3024:
3012:
3008:
2998:
2996:
2986:
2982:
2951:
2947:
2940:10.1137/1027056
2916:
2912:
2907:
2855:
2582:France, Africa
2333:
2329:
2327:
2324:
2323:
2303:
2298:
2295:
2294:
2275:
2270:
2267:
2266:
2250:
2247:
2246:
2229:
2225:
2223:
2220:
2219:
2200:
2197:
2196:
2180:
2177:
2176:
2140:
2136:
2129:
2124:
2123:
2107:
2103:
2093:
2054:
2046:
2038:
1991:Regional-scale
1969:
1966:
1965:
1949:
1946:
1945:
1917:
1913:
1904:
1900:
1882:
1878:
1858:
1850:
1847:
1846:
1817:
1809:
1806:
1805:
1774:
1742:
1718:
1714:
1709:
1706:
1705:
1686:
1683:
1682:
1677:formulated via
1655:
1652:
1651:
1632:
1629:
1628:
1608:
1604:
1595:
1591:
1582:
1578:
1569:
1565:
1560:
1557:
1556:
1539:
1534:
1521:
1517:
1505:
1500:
1494:
1491:
1490:
1462:
1458:
1452:
1448:
1439:
1435:
1429:
1424:
1410:
1400:
1396:
1383:
1381:
1369:
1365:
1356:
1352:
1350:
1347:
1346:
1323:
1319:
1310:
1306:
1304:
1301:
1300:
1280:
1276:
1274:
1271:
1270:
1253:
1249:
1247:
1244:
1243:
1214:
1211:
1210:
1193:
1189:
1187:
1184:
1183:
1160:
1137:
1134:
1133:
1110:Arc measurement
1107:
1098:
1077:
1061:
1057:
1050:
1046:
1042:
1024:
1020:
1016:
983:
981:
973:
970:
969:
962:
955:
948:
945:semi-minor axis
940:
937:semi-major axis
920:In geophysics,
915:oblate spheroid
895:
816:
732:
688:
684:
676:
673:equatorial axis
630:reference frame
618:Earth ellipsoid
610:
581:
580:
427:
419:
418:
407:
399:
391:
383:
375:
359:
351:
350:
309:
259:
251:
250:
226:
200:
198:low Earth orbit
185:
183:
172:
170:
163:
161:
149:
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
3286:
3276:
3275:
3273:Earth sciences
3270:
3256:
3255:
3250:
3240:
3233:
3232:External links
3230:
3229:
3228:
3227:
3226:
3214:
3213:
3212:
3197:
3194:
3191:
3190:
3171:
3150:
3139:
3126:
3106:
3082:
3062:
3047:
3040:
3022:
3006:
2980:
2945:
2926:(2): 241–247.
2909:
2908:
2906:
2903:
2902:
2901:
2896:
2891:
2886:
2884:Normal gravity
2881:
2876:
2871:
2869:Geodetic datum
2866:
2861:
2854:
2851:
2848:
2847:
2845:
2842:
2839:
2836:
2832:
2831:
2829:
2826:
2823:
2820:
2813:
2812:
2806:
2803:
2802:6,356,752.3142
2800:
2797:
2790:
2789:
2783:
2780:
2779:6,356,752.3141
2777:
2774:
2767:
2766:
2763:
2760:
2757:
2754:
2747:
2746:
2745:South America
2743:
2740:
2737:
2734:
2730:
2729:
2727:
2724:
2721:
2718:
2714:
2713:
2711:
2708:
2705:
2702:
2698:
2697:
2694:
2691:
2688:
2685:
2681:
2680:
2677:
2674:
2671:
2668:
2661:
2660:
2657:
2654:
2651:
2648:
2641:
2640:
2637:
2634:
2631:
2628:
2624:
2623:
2620:
2617:
2614:
2611:
2604:
2603:
2600:
2597:
2594:
2591:
2584:
2583:
2580:
2577:
2574:
2571:
2564:
2563:
2562:North America
2560:
2557:
2554:
2551:
2544:
2543:
2542:North America
2540:
2537:
2534:
2531:
2524:
2523:
2522:Europe, Japan
2520:
2517:
2514:
2511:
2504:
2503:
2500:
2497:
2494:
2491:
2484:
2483:
2480:
2477:
2474:
2471:
2467:
2466:
2463:
2460:
2459:6,356,103.0390
2457:
2454:
2450:
2449:
2446:
2443:
2440:
2437:
2430:
2429:
2426:
2423:
2422:6,355,862.9333
2420:
2417:
2416:Plessis (1817)
2413:
2412:
2409:
2406:
2403:
2400:
2393:
2392:
2389:
2386:
2383:
2380:
2358:geodetic datum
2354:normal gravity
2336:
2332:
2310:
2306:
2302:
2282:
2278:
2274:
2254:
2232:
2228:
2207:
2204:
2184:
2128:
2125:
2075:are a type of
2047:
2039:
2037:
2034:
1973:
1953:
1937:
1936:
1925:
1920:
1916:
1912:
1907:
1903:
1899:
1896:
1893:
1890:
1885:
1881:
1877:
1874:
1871:
1868:
1865:
1861:
1857:
1854:
1844:
1833:
1830:
1827:
1824:
1820:
1816:
1813:
1799:
1798:
1787:
1784:
1781:
1777:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1745:
1741:
1738:
1735:
1732:
1729:
1726:
1721:
1717:
1713:
1690:
1662:
1659:
1639:
1636:
1616:
1611:
1607:
1603:
1598:
1594:
1590:
1585:
1581:
1577:
1572:
1568:
1564:
1542:
1537:
1533:
1529:
1524:
1520:
1516:
1513:
1508:
1503:
1499:
1487:
1486:
1469:
1466:
1461:
1455:
1451:
1447:
1442:
1438:
1432:
1427:
1423:
1419:
1416:
1413:
1408:
1403:
1399:
1395:
1392:
1389:
1386:
1380:
1377:
1372:
1368:
1364:
1359:
1355:
1331:
1326:
1322:
1318:
1313:
1309:
1283:
1279:
1256:
1252:
1231:
1227:
1224:
1221:
1218:
1196:
1192:
1180:
1179:
1167:
1163:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1106:
1103:
1068:The ellipsoid
1013:
1012:
1001:
996:
992:
989:
986:
980:
977:
965:, defined as:
901:published the
894:
891:
887:geodetic datum
854:caused by the
815:
812:
761:as the geoid.
755:mean sea level
731:
728:
622:Earth spheroid
612:
611:
609:
608:
601:
594:
586:
583:
582:
579:
578:
573:
568:
560:
559:
556:
550:
549:
546:
540:
539:
536:
530:
529:
526:
520:
519:
516:
510:
509:
506:
500:
499:
496:
490:
489:
486:
480:
479:
476:
470:
469:
466:
460:
459:
456:
450:
449:
446:
440:
439:
436:
428:
425:
424:
421:
420:
417:
416:
411:
403:
395:
387:
379:
371:
366:
360:
357:
356:
353:
352:
349:
348:
343:
338:
333:
328:
323:
321:Map projection
318:
307:
302:
297:
295:Geodetic datum
292:
287:
271:
266:
260:
257:
256:
253:
252:
249:
248:
243:
238:
233:
227:
224:
223:
220:
219:
211:
210:
184:
171:
162:
156:with the same
148:
128:
127:
42:
40:
33:
26:
9:
6:
4:
3:
2:
3285:
3274:
3271:
3269:
3266:
3265:
3263:
3254:
3251:
3248:
3244:
3241:
3239:
3236:
3235:
3225:
3222:Web address:
3221:
3220:
3218:
3215:
3211:
3208:Web address:
3207:
3206:
3204:
3200:
3199:
3187:
3183:
3180:
3175:
3168:
3164:
3157:
3155:
3148:
3143:
3129:
3123:
3119:
3118:
3110:
3095:
3094:
3086:
3078:
3077:
3069:
3067:
3058:
3051:
3043:
3041:0-226-76747-7
3037:
3033:
3026:
3020:
3019:3-11-017072-8
3016:
3010:
2995:
2991:
2984:
2976:
2972:
2968:
2964:
2960:
2956:
2955:Math Horizons
2949:
2941:
2937:
2933:
2929:
2925:
2921:
2914:
2910:
2900:
2897:
2895:
2892:
2890:
2887:
2885:
2882:
2880:
2877:
2875:
2874:Great ellipse
2872:
2870:
2867:
2865:
2862:
2860:
2857:
2856:
2846:
2843:
2840:
2837:
2834:
2833:
2830:
2827:
2825:6,356,751.302
2824:
2821:
2818:
2815:
2814:
2811:
2807:
2805:298.257223563
2804:
2801:
2798:
2795:
2792:
2791:
2788:
2784:
2782:298.257222101
2781:
2778:
2775:
2772:
2769:
2768:
2764:
2761:
2758:
2755:
2752:
2749:
2748:
2744:
2741:
2739:6,356,774.719
2738:
2735:
2732:
2731:
2728:
2726:298.247167427
2725:
2723:6,356,774.516
2722:
2719:
2717:GRS-67 (1967)
2716:
2715:
2712:
2709:
2706:
2703:
2700:
2699:
2695:
2692:
2690:6,356,774.719
2689:
2686:
2683:
2682:
2678:
2675:
2673:6,356,759.769
2672:
2669:
2666:
2663:
2662:
2658:
2655:
2653:6,356,863.019
2652:
2649:
2646:
2643:
2642:
2638:
2635:
2633:6,356,911.946
2632:
2629:
2626:
2625:
2621:
2618:
2616:6,356,911.946
2615:
2612:
2609:
2606:
2605:
2601:
2598:
2595:
2592:
2589:
2586:
2585:
2581:
2578:
2576:6,356,514.870
2575:
2573:6,378,249.145
2572:
2569:
2566:
2565:
2561:
2558:
2555:
2552:
2549:
2546:
2545:
2541:
2538:
2535:
2532:
2529:
2526:
2525:
2521:
2518:
2516:6,356,078.963
2515:
2513:6,377,397.155
2512:
2509:
2506:
2505:
2501:
2498:
2496:6,356,256.909
2495:
2493:6,377,563.396
2492:
2489:
2486:
2485:
2481:
2478:
2476:6,356,097.550
2475:
2473:6,377,298.556
2472:
2469:
2468:
2464:
2461:
2458:
2456:6,377,304.063
2455:
2452:
2451:
2447:
2444:
2442:6,356,098.359
2441:
2439:6,377,299.365
2438:
2435:
2432:
2431:
2427:
2424:
2421:
2418:
2415:
2414:
2410:
2407:
2405:6,363,806.283
2404:
2401:
2398:
2395:
2394:
2390:
2387:
2384:
2381:
2378:
2377:
2374:
2372:
2369:
2367:
2363:
2359:
2355:
2350:
2334:
2330:
2308:
2304:
2300:
2280:
2276:
2272:
2252:
2230:
2226:
2205:
2202:
2195:, total mass
2182:
2173:
2171:
2166:
2161:
2159:
2155:
2146:
2133:
2121:
2120:
2116:
2111:
2101:
2100:
2091:
2090:
2085:
2081:
2078:
2074:
2066:
2062:
2058:
2051:
2044:
2033:
2031:
2028:, especially
2027:
2023:
2018:
2016:
2012:
2008:
2006:
2002:
1998:
1994:
1989:
1987:
1971:
1951:
1943:
1918:
1914:
1910:
1905:
1901:
1897:
1894:
1891:
1883:
1879:
1875:
1872:
1869:
1866:
1859:
1855:
1845:
1831:
1828:
1825:
1818:
1814:
1804:
1803:
1802:
1782:
1775:
1771:
1762:
1759:
1756:
1750:
1743:
1739:
1730:
1727:
1724:
1719:
1715:
1711:
1704:
1703:
1702:
1688:
1680:
1679:linearization
1676:
1660:
1657:
1637:
1634:
1609:
1605:
1596:
1592:
1588:
1583:
1579:
1575:
1570:
1566:
1562:
1540:
1535:
1531:
1527:
1522:
1518:
1514:
1511:
1506:
1501:
1497:
1467:
1464:
1453:
1449:
1445:
1440:
1436:
1430:
1425:
1421:
1417:
1414:
1401:
1397:
1393:
1390:
1384:
1378:
1370:
1366:
1357:
1353:
1345:
1344:
1343:
1324:
1320:
1311:
1307:
1299:
1281:
1277:
1254:
1250:
1229:
1225:
1222:
1219:
1216:
1194:
1190:
1165:
1161:
1154:
1151:
1148:
1142:
1139:
1132:
1131:
1130:
1128:
1125:). Then, the
1124:
1120:
1116:
1111:
1105:Determination
1102:
1096:
1092:
1088:
1084:
1075:
1071:
1066:
1054:
1040:
1036:
1031:
1027:
999:
994:
990:
987:
984:
978:
975:
968:
967:
966:
961:
952:
946:
938:
934:
929:
927:
923:
918:
916:
912:
908:
904:
900:
890:
889:definitions.
888:
884:
880:
875:
874:are defined.
873:
869:
865:
861:
857:
853:
849:
845:
841:
837:
833:
829:
820:
811:
809:
805:
801:
797:
792:
790:
786:
782:
777:
775:
771:
767:
762:
760:
756:
752:
748:
744:
740:
735:
727:
725:
721:
717:
713:
709:
705:
701:
698:up to modern
697:
696:meridian arcs
692:
682:
674:
670:
666:
662:
658:
654:
649:
647:
643:
639:
635:
631:
627:
623:
619:
607:
602:
600:
595:
593:
588:
587:
585:
584:
577:
574:
572:
569:
567:
564:
563:
557:
555:
552:
551:
547:
545:
542:
541:
537:
535:
532:
531:
527:
525:
522:
521:
517:
515:
512:
511:
507:
505:
502:
501:
497:
495:
492:
491:
487:
485:
482:
481:
477:
475:
472:
471:
467:
465:
462:
461:
457:
455:
452:
451:
447:
445:
442:
441:
437:
435:
432:
431:
423:
422:
415:
412:
410:
404:
402:
396:
394:
388:
386:
382:BeiDou (BDS)
380:
378:
372:
370:
367:
365:
362:
361:
355:
354:
347:
344:
342:
339:
337:
334:
332:
329:
327:
324:
322:
319:
316:
312:
308:
306:
303:
301:
298:
296:
293:
291:
288:
284:
283:circumference
280:
275:
272:
270:
267:
265:
262:
261:
255:
254:
247:
244:
242:
239:
237:
234:
232:
229:
228:
222:
221:
217:
213:
212:
209:
206:
205:
199:
195:
192:range of the
191:
182:
178:
169:
159:
155:
146:
143:
139:
134:
124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
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71: –
70:
66:
65:Find sources:
59:
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48:
43:This article
41:
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31:
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3216:
3202:
3196:Bibliography
3174:
3142:
3131:. Retrieved
3116:
3109:
3098:. Retrieved
3092:
3085:
3075:
3056:
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3031:
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3009:
2997:. Retrieved
2993:
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2958:
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2879:Meridian arc
2759:6,356,750.52
2710:298.24961539
2596:6,356,818.17
2445:300.80172554
2373:
2370:
2351:
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2162:
2150:
2135:Equatorial (
2113:
2102:(east/west)
2097:
2087:
2071:
2064:
2060:
2056:
2019:
2009:
1990:
1938:
1800:
1488:
1181:
1118:
1114:
1108:
1067:
1055:
1037:are used in
1029:
1025:
1014:
953:
930:
919:
899:Isaac Newton
896:
878:
876:
831:
825:
800:astronautics
793:
778:
769:
763:
742:
736:
733:
693:
680:
672:
650:
628:, used as a
626:Earth's form
621:
617:
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358:Technologies
313: /
225:Fundamentals
158:eccentricity
140:of the 2003
116:
110:October 2016
107:
97:
90:
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76:
64:
52:Please help
47:verification
44:
2920:SIAM Review
2841:6,356,751.9
2838:6,378,136.6
2835:IERS (2003)
2707:6,356,772.2
2704:6,378,157.5
2559:293.4659980
2539:294.9786982
2536:6,356,583.8
2533:6,378,206.4
2519:299.1528128
2499:299.3249646
2419:6,376,523.0
2391:Where used
2086:based on a
2077:curvilinear
789:coordinates
726:ellipsoid.
642:geosciences
236:Geodynamics
177:Karman line
3262:Categories
3249:help page)
3133:2021-10-24
3100:2021-10-24
2905:References
2696:Australia
2645:Krassovsky
2397:Maupertuis
2139:), polar (
2011:Gravimetry
1986:flattening
1127:flattening
960:flattening
852:flattening
681:polar axis
679:) and the
669:South Pole
665:North Pole
661:minor axis
657:revolution
646:ellipsoids
640:, and the
168:minor axis
138:oblateness
80:newspapers
2975:126412032
2844:298.25642
2822:6,378,136
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2776:6,378,137
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2687:6,378,160
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2253:ω
2099:longitude
1915:φ
1911:
1895:−
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1870:≈
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1780:∂
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988:−
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903:Principia
872:elevation
868:longitude
774:reduction
638:astronomy
315:Longitude
241:Geomatics
181:sea level
3182:Archived
2853:See also
2765:USA/DoD
2679:USA/DoD
2502:Britain
2479:300.8017
2462:300.8017
2082:used in
1999:and the
1091:Saturn's
958:and the
926:triaxial
897:In 1687
864:latitude
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766:regional
659:) whose
653:spheroid
651:It is a
494:ISO 6709
392:(Europe)
390:Galileo
376:(Russia)
374:GLONASS
311:Latitude
300:Geodesic
258:Concepts
190:Altitude
3268:Geodesy
3247:SPENVIS
3167:chap. 4
3163:chap. 1
3076:Geodesy
2928:Bibcode
2828:298.257
2808:Global
2785:Global
2639:Europe
2608:Hayford
2588:Helmert
2579:293.465
2434:Everest
2428:France
2411:France
2084:geodesy
2022:geodesy
2020:Modern
1095:Telesto
1087:Jupiter
1039:geodesy
933:ellipse
922:geodesy
911:ellipse
844:gravity
828:geodesy
783:or the
781:Hayford
739:average
634:geodesy
554:Geo URI
524:NAVD 88
434:NGVD 29
408:(Japan)
400:(India)
384:(China)
246:History
231:Geodesy
208:Geodesy
154:Ellipse
94:scholar
3124:
3038:
3017:
2973:
2819:(1989)
2796:(1984)
2794:WGS-84
2773:(1979)
2771:GRS-80
2762:298.26
2753:(1972)
2751:WGS-72
2742:298.25
2693:298.25
2676:298.25
2667:(1966)
2647:(1940)
2610:(1910)
2602:Egypt
2590:(1906)
2570:(1880)
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2550:(1878)
2548:Clarke
2530:(1866)
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2510:(1841)
2508:Bessel
2490:(1830)
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2172:1969.
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870:, and
759:volume
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836:geoid
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730:Types
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454:SK-42
269:Geoid
101:JSTOR
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3122:ISBN
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830:, a
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