205:
122:
2121:
2039:
25:
808:
2310:
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
1101:
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
1053:
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
2156:
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
1473:
3154:); 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." (
2140:
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
1785:
1928:
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
3149:
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"
2357:. 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.
1069:
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
913:, 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
1923:
1614:
2164:
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)
1831:
761:
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,
1542:
847:
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
1329:
999:
1337:
3085:. NOAA technical publications. U.S. Department of Commerce, National Oceanic and Atmospheric Administration, National Ocean Service, Charting and Geodetic Services. p. 107
3135:
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
1196:
1696:
1165:
1660:
1637:
2252:
1229:
2336:
2232:
1283:
1256:
2308:
2280:
2205:
2739:
2653:
2182:
1971:
1951:
1688:
2145:
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
3241:
1837:
780:
of millions of boundary stones should remain fixed for a long period. If their reference surface changes, the coordinates themselves also change.
564:
891:
2360:
Note that the same ellipsoid may be known by different names. It is best to mention the defining constants for unambiguous identification.
3190:
P. K. Seidelmann (Chair), et al. (2005), “Report Of The IAU/IAG Working Group On
Cartographic Coordinates And Rotational Elements: 2003,”
386:
776:, despite the fact that their main axes deviate by several hundred meters from the modern values. Another reason is a judicial one: the
2775:
554:
522:
2149:
in 1910 and adopted by the
International Union of Geodesy and Geophysics (IUGG) in 1924, which recommended it for international use.
1286:
532:
831:, or other planetary body, as opposed to a perfect, smooth, and unaltered sphere, which factors in the undulations of the bodies'
894:
in which he included a proof that a rotating self-gravitating fluid body in equilibrium takes the form of a flattened ("oblate")
3226:
3072:
89:
3114:
512:
1547:
61:
3170:
1008:
is the amount of flattening at each pole, relative to the radius at the equator. This is often expressed as a fraction 1/
940:, becomes the distance from the centre to either pole. These two lengths completely specify the shape of the ellipsoid.
492:
293:
68:
3206:
OpenGIS Implementation
Specification for Geographic information - Simple feature access - Part 1: Common architecture
3028:
3007:
592:
108:
1796:
1111:
2103:
42:
2003:
75:
1481:
146:
378:
46:
2065:
394:
1102:
radii of curvature so obtained would be related to the equatorial radius and the polar radius, respectively
57:
2068:
1468:{\displaystyle M_{0}(\varphi _{i})={\frac {a(1-e^{2})}{(1-e_{0}^{2}\sin ^{2}\varphi _{i})^{\frac {3}{2}}}}}
1544:. Then discrepancies between empirical and theoretical values of the radius of curvature can be formed as
2877:
1291:
1054:
its axis of revolution; and those parameters have been adopted also for all modern reference ellipsoids.
960:
182:
1045:
A great many ellipsoids have been used to model the Earth in the past, with different assumed values of
1989:
1663:
3136:
2852:
2798:
357:
2782:
3231:
2576:
1930:
1780:{\displaystyle \delta M_{i}\approx \delta a(\partial M/\partial a)+\delta f(\partial M/\partial f)}
871:
324:
271:
3096:
1174:
943:
In geodesy publications, however, it is common to specify the semi-major axis (equatorial radius)
3261:
2978:
2385:
2354:
2107:
762:
462:
422:
35:
3078:
2633:
1993:
1124:
1090:
between 1/3 and 1/2 (meaning that the polar diameter is between 50% and 67% of the equatorial.
442:
3104:
3080:
1642:
1619:
660:, is approximately aligned with the Earth's axis of rotation. The ellipsoid is defined by the
2596:
2556:
2536:
2516:
2237:
2146:
2142:
2133:
1201:
1058:
645:
585:
502:
252:
82:
2311:
degree zonal harmonic gravitational coefficient, that was derived from the GRS-80 value for
2916:
2314:
2210:
2158:
2152:
At the 1967 meeting of the IUGG held in
Lucerne, Switzerland, the ellipsoid called GRS-67 (
2061:
2031:
2018:
1261:
1234:
402:
352:
278:
2345:
field formula to go with it. Commonly an ellipsoidal model is part of a more encompassing
2157:
IUGG held in Moscow. It is used in Australia for the Australian Geodetic Datum and in the
8:
3048:. USGS Professional Paper 1395. Washington, D.C.: Government Printing Office. p. 17.
2887:
2476:
2285:
2257:
2077:
2013:
no longer uses simple meridian arcs or ground triangulation networks, but the methods of
828:
792:
735:
614:
314:
262:
133:
2920:
2187:
2959:
2882:
2167:
1956:
1936:
1673:
914:
618:
234:
204:
3155:
3151:
920:
The shape of an ellipsoid of revolution is determined by the shape parameters of that
3110:
3024:
3003:
2963:
2014:
1981:
844:
769:
704:
688:
683:
Many methods exist for determination of the axes of an Earth ellipsoid, ranging from
334:
319:
3256:
2951:
2924:
2847:
2496:
1985:
848:
773:
700:
692:
578:
329:
2101:
The triad is also known as Earth ellipsoidal coordinates (not to be confused with
3174:
2955:
2346:
2153:
1098:
933:
925:
903:
696:
186:
3198:
3167:
1918:{\displaystyle \partial M/\partial f\approx -2a_{0}(1-1.5\sin ^{2}\varphi _{i})}
3063:
2872:
2857:
2422:
2342:
1083:
875:
836:
768:
This is the reason for the "long life" of former reference ellipsoids like the
743:
542:
309:
283:
1231:, the solution starts from an initial approximation for the equatorial radius
3250:
2862:
1667:
121:
2867:
887:
799:
usually adapts the axes of the Earth ellipsoid to the best available data.
788:
684:
267:
3079:
National Geodetic Survey (U.S.).; National Geodetic Survey (U.S.) (1986).
2907:
Alexander, J. C. (1985). "The Numerics of Computing Geodetic Ellipsoids".
2338:, was truncated to eight significant digits in the normalization process.
2120:
676:); their radial difference is slightly more than 21 km, or 0.335% of
2942:
Heine, George (September 2013). "Euler and the Flattening of the Earth".
753:
While the mean Earth ellipsoid is the ideal basis of global geodesy, for
224:
165:
2038:
1999:
1974:
1115:
948:
840:
777:
657:
653:
649:
630:
156:
126:
2087:
1074:
is even less elliptical, with a flattening of less than 1/825, while
895:
860:
856:
634:
626:
303:
229:
169:
2928:
24:
2341:
An ellipsoidal model describes only the ellipsoid's geometry and a
852:
754:
641:
482:
299:
288:
3103:
Awange, J.L.; Grafarend, E.W.; Paláncz, B.; Zaletnyik, P. (2010).
3235:
2072:
2010:
1075:
1027:
1023:
921:
910:
899:
866:
In the context of standardization and geographic applications, a
832:
816:
807:
727:
622:
362:
219:
196:
142:
3212:
3002:
Torge, W (2001) Geodesy (3rd edition), published by de Gruyter,
2002:
is another technique for determining Earth's flattening, as per
795:
is increasingly accurate, the International Geoscientific Union
3102:
2759:
1079:
747:
472:
432:
370:
1331:
can be calculated at the latitude of each arc measurement as:
902:
rotated around its minor diameter; a shape which he termed an
1984:
observed in the radius of curvature measurements reflect the
1933:. The parameters determined are usually the semi-major axis,
1171:
For two arc measurements each at arbitrary average latitudes
824:
739:
712:
257:
178:
791:, these regional reasons are less relevant. As knowledge of
3021:
Flattening the Earth: Two Thousand Years of Map Projections
2805:
2350:
1071:
796:
559:
452:
130:
823:
is a mathematically defined surface that approximates the
1616:. Finally, corrections for the initial equatorial radius
1062:
1022:
then being the "inverse flattening". A great many other
851:
computations are performed and point coordinates such as
784:
708:
835:
due to variations in the composition and density of the
1609:{\displaystyle \delta M_{i}=M_{i}-M_{0}(\varphi _{i})}
932:, becomes the equatorial radius of the ellipsoid: the
746:, and therefore an ideal Earth ellipsoid has the same
723:
There are two types of ellipsoid: mean and reference.
2317:
2288:
2260:
2240:
2213:
2190:
2170:
1959:
1939:
1840:
1799:
1699:
1676:
1645:
1622:
1550:
1484:
1340:
1294:
1264:
1237:
1204:
1177:
1127:
1030:
but they can all be related to one or two of the set
963:
16:
Geometric figure which approximates the Earth's shape
652:(shorter diameter), which connects the geographical
734:. It refers to a theoretical coherence between the
691:or the analysis and interconnection of continental
49:. Unsourced material may be challenged and removed.
2330:
2302:
2274:
2246:
2226:
2199:
2176:
1965:
1945:
1917:
1825:
1779:
1682:
1654:
1631:
1608:
1536:
1467:
1323:
1277:
1250:
1223:
1190:
1159:
993:
155: Circle with diameter equal to the ellipse's
3248:
870:is the mathematical model used as foundation by
765:of the measurements will get small distortions.
3018:
2081:. They include geodetic latitude (north/south)
730:of the Earth's surface curvature is called the
2132:) and mean Earth radii as defined in the 1984
2115:
1826:{\displaystyle \partial M/\partial a\approx 1}
1112:Earth polar and equatorial radius of curvature
3199:https://astrogeology.usgs.gov/Projects/WGCCRE
3057:
3055:
2282:a derived quantity. The minute difference in
586:
695:. Amongst the different set of data used in
613:is a mathematical figure approximating the
3192:Celestial Mechanics and Dynamical Astronomy
3109:. Springer Berlin Heidelberg. p. 156.
3023:. University of Chicago Press. p. 82.
1078:is visibly oblate at about 1/15 and one of
3052:
1537:{\displaystyle e_{0}^{2}=2f_{0}-f_{0}^{2}}
1118:would readily follow from its definition:
593:
579:
560:Spatial Reference System Identifier (SRID)
555:International Terrestrial Reference System
3012:
2906:
1217:
109:Learn how and when to remove this message
2119:
2037:
806:
120:
3145:
3143:
3061:
2024:
881:
699:are several of special importance: the
149:as that of Earth, with north at the top
3249:
3232:Coordinate systems and transformations
3043:
2979:"Strange but True: Earth Is Not Round"
1287:Earth's meridional radius of curvature
802:
726:A data set which describes the global
3242:Coordinate Systems, Frames and Datums
2941:
783:However, for international networks,
3140:
3106:Algebraic Geodesy and Geoinformatics
2996:
2976:
738:and the meridional curvature of the
680:(which is not quite 6,400 km).
47:adding citations to reliable sources
18:
1790:where the partial derivatives are:
1324:{\displaystyle M_{0}(\varphi _{i})}
994:{\displaystyle f={\frac {a-b}{a}}.}
565:Universal Transverse Mercator (UTM)
527:European Terrestrial Ref. Sys. 1989
13:
3046:Map Projections — A Working Manual
2977:Choi, Charles Q. (12 April 2007).
2099:(also known as geodetic height).
1953:, and any of the semi-minor axis,
1852:
1841:
1811:
1800:
1768:
1757:
1736:
1725:
637:have been used as approximations.
437:Ordnance Survey Great Britain 1936
403:Discrete Global Grid and Geocoding
294:Horizontal position representation
14:
3273:
3220:
2234:and angular velocity of rotation
917:(or scalene) ellipsoid is used.
168:, 100 km (62 mi) above
2349:. For example, the older ED-50 (
2254:, making the inverse flattening
2104:ellipsoidal-harmonic coordinates
2030:This section is an excerpt from
1093:
827:, which is the truer, imperfect
353:Global Nav. Sat. Systems (GNSSs)
203:
23:
3184:
3161:
3129:
898:of revolution, generated by an
517:N. American Vertical Datum 1988
34:needs additional citations for
3037:
2970:
2935:
2900:
2459:Everest 1830 (1967 Definition)
2454:West Malaysia & Singapore
1912:
1877:
1774:
1754:
1742:
1722:
1603:
1590:
1448:
1400:
1395:
1376:
1364:
1351:
1318:
1305:
1146:
1134:
1061:, widely used for mapping and
547:Internet link to a point 2010
477:Geodetic Reference System 1980
395:Quasi-Zenith Sat. Sys. (QZSS)
1:
3213:http://www.opengeospatial.org
2893:
2353:) is based on the Hayford or
742:. The latter is close to the
537:Chinese obfuscated datum 2002
3227:Geographic coordinate system
2956:10.4169/mathhorizons.21.1.25
2442:Everest 1830 Modified (1967)
2069:orthogonal coordinate system
1662:can be solved by means of a
1191:{\displaystyle \varphi _{i}}
1086:, is highly flattened, with
868:geodesic reference ellipsoid
839:, as well as the subsequent
487:Geographic point coord. 1983
7:
2878:Planetary coordinate system
2841:
2471:Brunei & East Malaysia
2116:Historical Earth ellipsoids
703:of 1841, the international
447:Systema Koordinat 1942 goda
10:
3278:
2673:Australian National (1966)
2029:
1990:deflection of the vertical
1664:system of linear equations
507:World Geodetic System 1984
2853:Earth radius of curvature
2154:Geodetic Reference System
2095:, and ellipsoidal height
1160:{\displaystyle f=(a-b)/a}
497:North American Datum 1983
467:South American Datum 1969
3208:, Annex B.4. 2005-11-30
3194:, 91, pp. 203–215.
3044:Snyder, John P. (1987).
3019:Snyder, John P. (1993).
2690:New International (1967)
2368:Reference ellipsoid name
1931:least squares adjustment
1655:{\displaystyle \delta f}
1632:{\displaystyle \delta a}
872:spatial reference system
718:
358:Global Pos. System (GPS)
325:Spatial reference system
3168:IERS Conventions (2003)
2355:International Ellipsoid
2247:{\displaystyle \omega }
2136:revision (not to scale)
2108:ellipsoidal coordinates
1258:and for the flattening
1224:{\displaystyle i=1,\,2}
125:A scale diagram of the
2648:USSR, Russia, Romania
2332:
2304:
2276:
2248:
2228:
2207:, dynamic form factor
2201:
2178:
2137:
2058:
1994:astrogeodetic leveling
1967:
1947:
1919:
1827:
1781:
1684:
1656:
1633:
1610:
1538:
1469:
1325:
1279:
1252:
1225:
1192:
1161:
995:
812:
190:
2722:South American (1969)
2371:Equatorial radius (m)
2333:
2331:{\displaystyle J_{2}}
2305:
2277:
2249:
2229:
2227:{\displaystyle J_{2}}
2202:
2179:
2147:John Fillmore Hayford
2143:Alexander Ross Clarke
2134:World Geodetic System
2123:
2042:Geodetic coordinates
2041:
2025:Geodetic coordinates
1968:
1948:
1920:
1828:
1782:
1685:
1657:
1634:
1611:
1539:
1470:
1326:
1280:
1278:{\displaystyle f_{0}}
1253:
1251:{\displaystyle a_{0}}
1226:
1193:
1162:
996:
810:
757:networks a so-called
253:Geographical distance
124:
3062:Bomford, G. (1952).
2616:International (1924)
2315:
2286:
2258:
2238:
2211:
2188:
2168:
2159:South American Datum
2062:Geodetic coordinates
2032:Geodetic coordinates
2019:satellite gravimetry
1957:
1937:
1838:
1797:
1697:
1674:
1643:
1620:
1548:
1482:
1338:
1292:
1262:
1235:
1202:
1175:
1125:
1063:satellite navigation
961:
882:Ellipsoid parameters
732:mean Earth Ellipsoid
633:. Various different
621:for computations in
427:Sea Level Datum 1929
279:Geodetic coordinates
43:improve this article
2983:Scientific American
2921:1985SIAMR..27..241A
2888:Planetary ellipsoid
2351:European Datum 1950
2303:{\displaystyle 1/f}
2275:{\displaystyle 1/f}
2078:reference ellipsoid
1977:, or eccentricity.
1639:and the flattening
1533:
1499:
1423:
829:figure of the Earth
821:reference ellipsoid
803:Reference ellipsoid
759:reference ellipsoid
736:geographic latitude
457:European Datum 1950
415:Standards (history)
315:Reference ellipsoid
263:Figure of the Earth
134:reference ellipsoid
3173:2014-04-19 at the
2883:History of geodesy
2377:Inverse flattening
2328:
2300:
2272:
2244:
2224:
2200:{\displaystyle GM}
2197:
2174:
2138:
2059:
2004:Clairaut's theorem
1982:systematic effects
1963:
1943:
1915:
1823:
1777:
1680:
1652:
1629:
1606:
1534:
1519:
1485:
1465:
1409:
1321:
1285:. The theoretical
1275:
1248:
1221:
1188:
1157:
1024:ellipse parameters
991:
813:
793:the Earth's figure
707:of 1924, and (for
335:Vertical positions
191:
3177:(Chp. 1, page 12)
3116:978-3-642-12124-1
3082:Geodetic Glossary
2839:
2838:
2177:{\displaystyle a}
2015:satellite geodesy
1992:, as explored in
1966:{\displaystyle b}
1946:{\displaystyle a}
1683:{\displaystyle M}
1463:
1459:
986:
845:centrifugal force
711:positioning) the
705:Hayford ellipsoid
693:geodetic networks
689:satellite geodesy
644:(an ellipsoid of
603:
602:
551:
550:
330:Spatial relations
320:Satellite geodesy
275:
119:
118:
111:
93:
58:"Earth ellipsoid"
3269:
3178:
3165:
3159:
3147:
3138:
3133:
3127:
3126:
3124:
3123:
3100:
3094:
3093:
3091:
3090:
3076:
3070:
3069:
3059:
3050:
3049:
3041:
3035:
3034:
3016:
3010:
3000:
2994:
2993:
2991:
2989:
2974:
2968:
2967:
2939:
2933:
2932:
2904:
2848:Equatorial bulge
2374:Polar radius (m)
2365:
2364:
2337:
2335:
2334:
2329:
2327:
2326:
2309:
2307:
2306:
2301:
2296:
2281:
2279:
2278:
2273:
2268:
2253:
2251:
2250:
2245:
2233:
2231:
2230:
2225:
2223:
2222:
2206:
2204:
2203:
2198:
2183:
2181:
2180:
2175:
2131:
2127:
2098:
2094:
2084:
2057:
1986:geoid undulation
1972:
1970:
1969:
1964:
1952:
1950:
1949:
1944:
1924:
1922:
1921:
1916:
1911:
1910:
1898:
1897:
1876:
1875:
1851:
1832:
1830:
1829:
1824:
1810:
1786:
1784:
1783:
1778:
1767:
1735:
1712:
1711:
1689:
1687:
1686:
1681:
1661:
1659:
1658:
1653:
1638:
1636:
1635:
1630:
1615:
1613:
1612:
1607:
1602:
1601:
1589:
1588:
1576:
1575:
1563:
1562:
1543:
1541:
1540:
1535:
1532:
1527:
1515:
1514:
1498:
1493:
1474:
1472:
1471:
1466:
1464:
1462:
1461:
1460:
1452:
1446:
1445:
1433:
1432:
1422:
1417:
1398:
1394:
1393:
1371:
1363:
1362:
1350:
1349:
1330:
1328:
1327:
1322:
1317:
1316:
1304:
1303:
1284:
1282:
1281:
1276:
1274:
1273:
1257:
1255:
1254:
1249:
1247:
1246:
1230:
1228:
1227:
1222:
1197:
1195:
1194:
1189:
1187:
1186:
1166:
1164:
1163:
1158:
1153:
1089:
1082:triaxial moons,
1068:
1052:
1048:
1041:
1037:
1033:
1021:
1011:
1007:
1000:
998:
997:
992:
987:
982:
971:
953:
946:
939:
936:of the ellipse,
931:
928:of the ellipse,
849:geodetic network
811:Flattened sphere
787:positioning, or
774:Bessel ellipsoid
701:Bessel ellipsoid
697:national surveys
679:
675:
667:
595:
588:
581:
419:
418:
398:
390:
382:
374:
366:
306:
265:
207:
193:
192:
176:
163:
154:
140:
114:
107:
103:
100:
94:
92:
51:
27:
19:
3277:
3276:
3272:
3271:
3270:
3268:
3267:
3266:
3247:
3246:
3223:
3187:
3182:
3181:
3175:Wayback Machine
3166:
3162:
3148:
3141:
3134:
3130:
3121:
3119:
3117:
3101:
3097:
3088:
3086:
3077:
3073:
3060:
3053:
3042:
3038:
3031:
3017:
3013:
3001:
2997:
2987:
2985:
2975:
2971:
2940:
2936:
2929:10.1137/1027056
2905:
2901:
2896:
2844:
2571:France, Africa
2322:
2318:
2316:
2313:
2312:
2292:
2287:
2284:
2283:
2264:
2259:
2256:
2255:
2239:
2236:
2235:
2218:
2214:
2212:
2209:
2208:
2189:
2186:
2185:
2169:
2166:
2165:
2129:
2125:
2118:
2113:
2112:
2096:
2092:
2082:
2043:
2035:
2027:
1980:Regional-scale
1958:
1955:
1954:
1938:
1935:
1934:
1906:
1902:
1893:
1889:
1871:
1867:
1847:
1839:
1836:
1835:
1806:
1798:
1795:
1794:
1763:
1731:
1707:
1703:
1698:
1695:
1694:
1675:
1672:
1671:
1666:formulated via
1644:
1641:
1640:
1621:
1618:
1617:
1597:
1593:
1584:
1580:
1571:
1567:
1558:
1554:
1549:
1546:
1545:
1528:
1523:
1510:
1506:
1494:
1489:
1483:
1480:
1479:
1451:
1447:
1441:
1437:
1428:
1424:
1418:
1413:
1399:
1389:
1385:
1372:
1370:
1358:
1354:
1345:
1341:
1339:
1336:
1335:
1312:
1308:
1299:
1295:
1293:
1290:
1289:
1269:
1265:
1263:
1260:
1259:
1242:
1238:
1236:
1233:
1232:
1203:
1200:
1199:
1182:
1178:
1176:
1173:
1172:
1149:
1126:
1123:
1122:
1099:Arc measurement
1096:
1087:
1066:
1050:
1046:
1039:
1035:
1031:
1013:
1009:
1005:
972:
970:
962:
959:
958:
951:
944:
937:
934:semi-minor axis
929:
926:semi-major axis
909:In geophysics,
904:oblate spheroid
884:
805:
721:
677:
673:
665:
662:equatorial axis
619:reference frame
607:Earth ellipsoid
599:
570:
569:
416:
408:
407:
396:
388:
380:
372:
364:
348:
340:
339:
298:
248:
240:
239:
215:
189:
187:low Earth orbit
174:
172:
161:
159:
152:
150:
138:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
3275:
3265:
3264:
3262:Earth sciences
3259:
3245:
3244:
3239:
3229:
3222:
3221:External links
3219:
3218:
3217:
3216:
3215:
3203:
3202:
3201:
3186:
3183:
3180:
3179:
3160:
3139:
3128:
3115:
3095:
3071:
3051:
3036:
3029:
3011:
2995:
2969:
2934:
2915:(2): 241–247.
2898:
2897:
2895:
2892:
2891:
2890:
2885:
2880:
2875:
2873:Normal gravity
2870:
2865:
2860:
2858:Geodetic datum
2855:
2850:
2843:
2840:
2837:
2836:
2834:
2831:
2828:
2825:
2821:
2820:
2818:
2815:
2812:
2809:
2802:
2801:
2795:
2792:
2791:6,356,752.3142
2789:
2786:
2779:
2778:
2772:
2769:
2768:6,356,752.3141
2766:
2763:
2756:
2755:
2752:
2749:
2746:
2743:
2736:
2735:
2734:South America
2732:
2729:
2726:
2723:
2719:
2718:
2716:
2713:
2710:
2707:
2703:
2702:
2700:
2697:
2694:
2691:
2687:
2686:
2683:
2680:
2677:
2674:
2670:
2669:
2666:
2663:
2660:
2657:
2650:
2649:
2646:
2643:
2640:
2637:
2630:
2629:
2626:
2623:
2620:
2617:
2613:
2612:
2609:
2606:
2603:
2600:
2593:
2592:
2589:
2586:
2583:
2580:
2573:
2572:
2569:
2566:
2563:
2560:
2553:
2552:
2551:North America
2549:
2546:
2543:
2540:
2533:
2532:
2531:North America
2529:
2526:
2523:
2520:
2513:
2512:
2511:Europe, Japan
2509:
2506:
2503:
2500:
2493:
2492:
2489:
2486:
2483:
2480:
2473:
2472:
2469:
2466:
2463:
2460:
2456:
2455:
2452:
2449:
2448:6,356,103.0390
2446:
2443:
2439:
2438:
2435:
2432:
2429:
2426:
2419:
2418:
2415:
2412:
2411:6,355,862.9333
2409:
2406:
2405:Plessis (1817)
2402:
2401:
2398:
2395:
2392:
2389:
2382:
2381:
2378:
2375:
2372:
2369:
2347:geodetic datum
2343:normal gravity
2325:
2321:
2299:
2295:
2291:
2271:
2267:
2263:
2243:
2221:
2217:
2196:
2193:
2173:
2117:
2114:
2064:are a type of
2036:
2028:
2026:
2023:
1962:
1942:
1926:
1925:
1914:
1909:
1905:
1901:
1896:
1892:
1888:
1885:
1882:
1879:
1874:
1870:
1866:
1863:
1860:
1857:
1854:
1850:
1846:
1843:
1833:
1822:
1819:
1816:
1813:
1809:
1805:
1802:
1788:
1787:
1776:
1773:
1770:
1766:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1734:
1730:
1727:
1724:
1721:
1718:
1715:
1710:
1706:
1702:
1679:
1651:
1648:
1628:
1625:
1605:
1600:
1596:
1592:
1587:
1583:
1579:
1574:
1570:
1566:
1561:
1557:
1553:
1531:
1526:
1522:
1518:
1513:
1509:
1505:
1502:
1497:
1492:
1488:
1476:
1475:
1458:
1455:
1450:
1444:
1440:
1436:
1431:
1427:
1421:
1416:
1412:
1408:
1405:
1402:
1397:
1392:
1388:
1384:
1381:
1378:
1375:
1369:
1366:
1361:
1357:
1353:
1348:
1344:
1320:
1315:
1311:
1307:
1302:
1298:
1272:
1268:
1245:
1241:
1220:
1216:
1213:
1210:
1207:
1185:
1181:
1169:
1168:
1156:
1152:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1095:
1092:
1057:The ellipsoid
1002:
1001:
990:
985:
981:
978:
975:
969:
966:
954:, defined as:
890:published the
883:
880:
876:geodetic datum
843:caused by the
804:
801:
750:as the geoid.
744:mean sea level
720:
717:
611:Earth spheroid
601:
600:
598:
597:
590:
583:
575:
572:
571:
568:
567:
562:
557:
549:
548:
545:
539:
538:
535:
529:
528:
525:
519:
518:
515:
509:
508:
505:
499:
498:
495:
489:
488:
485:
479:
478:
475:
469:
468:
465:
459:
458:
455:
449:
448:
445:
439:
438:
435:
429:
428:
425:
417:
414:
413:
410:
409:
406:
405:
400:
392:
384:
376:
368:
360:
355:
349:
346:
345:
342:
341:
338:
337:
332:
327:
322:
317:
312:
310:Map projection
307:
296:
291:
286:
284:Geodetic datum
281:
276:
260:
255:
249:
246:
245:
242:
241:
238:
237:
232:
227:
222:
216:
213:
212:
209:
208:
200:
199:
173:
160:
151:
145:with the same
137:
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
3274:
3263:
3260:
3258:
3255:
3254:
3252:
3243:
3240:
3237:
3233:
3230:
3228:
3225:
3224:
3214:
3211:Web address:
3210:
3209:
3207:
3204:
3200:
3197:Web address:
3196:
3195:
3193:
3189:
3188:
3176:
3172:
3169:
3164:
3157:
3153:
3146:
3144:
3137:
3132:
3118:
3112:
3108:
3107:
3099:
3084:
3083:
3075:
3067:
3066:
3058:
3056:
3047:
3040:
3032:
3030:0-226-76747-7
3026:
3022:
3015:
3009:
3008:3-11-017072-8
3005:
2999:
2984:
2980:
2973:
2965:
2961:
2957:
2953:
2949:
2945:
2944:Math Horizons
2938:
2930:
2926:
2922:
2918:
2914:
2910:
2903:
2899:
2889:
2886:
2884:
2881:
2879:
2876:
2874:
2871:
2869:
2866:
2864:
2863:Great ellipse
2861:
2859:
2856:
2854:
2851:
2849:
2846:
2845:
2835:
2832:
2829:
2826:
2823:
2822:
2819:
2816:
2814:6,356,751.302
2813:
2810:
2807:
2804:
2803:
2800:
2796:
2794:298.257223563
2793:
2790:
2787:
2784:
2781:
2780:
2777:
2773:
2771:298.257222101
2770:
2767:
2764:
2761:
2758:
2757:
2753:
2750:
2747:
2744:
2741:
2738:
2737:
2733:
2730:
2728:6,356,774.719
2727:
2724:
2721:
2720:
2717:
2715:298.247167427
2714:
2712:6,356,774.516
2711:
2708:
2706:GRS-67 (1967)
2705:
2704:
2701:
2698:
2695:
2692:
2689:
2688:
2684:
2681:
2679:6,356,774.719
2678:
2675:
2672:
2671:
2667:
2664:
2662:6,356,759.769
2661:
2658:
2655:
2652:
2651:
2647:
2644:
2642:6,356,863.019
2641:
2638:
2635:
2632:
2631:
2627:
2624:
2622:6,356,911.946
2621:
2618:
2615:
2614:
2610:
2607:
2605:6,356,911.946
2604:
2601:
2598:
2595:
2594:
2590:
2587:
2584:
2581:
2578:
2575:
2574:
2570:
2567:
2565:6,356,514.870
2564:
2562:6,378,249.145
2561:
2558:
2555:
2554:
2550:
2547:
2544:
2541:
2538:
2535:
2534:
2530:
2527:
2524:
2521:
2518:
2515:
2514:
2510:
2507:
2505:6,356,078.963
2504:
2502:6,377,397.155
2501:
2498:
2495:
2494:
2490:
2487:
2485:6,356,256.909
2484:
2482:6,377,563.396
2481:
2478:
2475:
2474:
2470:
2467:
2465:6,356,097.550
2464:
2462:6,377,298.556
2461:
2458:
2457:
2453:
2450:
2447:
2445:6,377,304.063
2444:
2441:
2440:
2436:
2433:
2431:6,356,098.359
2430:
2428:6,377,299.365
2427:
2424:
2421:
2420:
2416:
2413:
2410:
2407:
2404:
2403:
2399:
2396:
2394:6,363,806.283
2393:
2390:
2387:
2384:
2383:
2379:
2376:
2373:
2370:
2367:
2366:
2363:
2361:
2358:
2356:
2352:
2348:
2344:
2339:
2323:
2319:
2297:
2293:
2289:
2269:
2265:
2261:
2241:
2219:
2215:
2194:
2191:
2184:, total mass
2171:
2162:
2160:
2155:
2150:
2148:
2144:
2135:
2122:
2110:
2109:
2105:
2100:
2090:
2089:
2080:
2079:
2074:
2070:
2067:
2063:
2055:
2051:
2047:
2040:
2033:
2022:
2020:
2017:, especially
2016:
2012:
2007:
2005:
2001:
1997:
1995:
1991:
1987:
1983:
1978:
1976:
1960:
1940:
1932:
1907:
1903:
1899:
1894:
1890:
1886:
1883:
1880:
1872:
1868:
1864:
1861:
1858:
1855:
1848:
1844:
1834:
1820:
1817:
1814:
1807:
1803:
1793:
1792:
1791:
1771:
1764:
1760:
1751:
1748:
1745:
1739:
1732:
1728:
1719:
1716:
1713:
1708:
1704:
1700:
1693:
1692:
1691:
1677:
1669:
1668:linearization
1665:
1649:
1646:
1626:
1623:
1598:
1594:
1585:
1581:
1577:
1572:
1568:
1564:
1559:
1555:
1551:
1529:
1524:
1520:
1516:
1511:
1507:
1503:
1500:
1495:
1490:
1486:
1456:
1453:
1442:
1438:
1434:
1429:
1425:
1419:
1414:
1410:
1406:
1403:
1390:
1386:
1382:
1379:
1373:
1367:
1359:
1355:
1346:
1342:
1334:
1333:
1332:
1313:
1309:
1300:
1296:
1288:
1270:
1266:
1243:
1239:
1218:
1214:
1211:
1208:
1205:
1183:
1179:
1154:
1150:
1143:
1140:
1137:
1131:
1128:
1121:
1120:
1119:
1117:
1114:). Then, the
1113:
1109:
1105:
1100:
1094:Determination
1091:
1085:
1081:
1077:
1073:
1064:
1060:
1055:
1043:
1029:
1025:
1020:
1016:
988:
983:
979:
976:
973:
967:
964:
957:
956:
955:
950:
941:
935:
927:
923:
918:
916:
912:
907:
905:
901:
897:
893:
889:
879:
878:definitions.
877:
873:
869:
864:
863:are defined.
862:
858:
854:
850:
846:
842:
838:
834:
830:
826:
822:
818:
809:
800:
798:
794:
790:
786:
781:
779:
775:
771:
766:
764:
760:
756:
751:
749:
745:
741:
737:
733:
729:
724:
716:
714:
710:
706:
702:
698:
694:
690:
687:up to modern
686:
685:meridian arcs
681:
671:
663:
659:
655:
651:
647:
643:
638:
636:
632:
628:
624:
620:
616:
612:
608:
596:
591:
589:
584:
582:
577:
576:
574:
573:
566:
563:
561:
558:
556:
553:
552:
546:
544:
541:
540:
536:
534:
531:
530:
526:
524:
521:
520:
516:
514:
511:
510:
506:
504:
501:
500:
496:
494:
491:
490:
486:
484:
481:
480:
476:
474:
471:
470:
466:
464:
461:
460:
456:
454:
451:
450:
446:
444:
441:
440:
436:
434:
431:
430:
426:
424:
421:
420:
412:
411:
404:
401:
399:
393:
391:
385:
383:
377:
375:
371:BeiDou (BDS)
369:
367:
361:
359:
356:
354:
351:
350:
344:
343:
336:
333:
331:
328:
326:
323:
321:
318:
316:
313:
311:
308:
305:
301:
297:
295:
292:
290:
287:
285:
282:
280:
277:
273:
272:circumference
269:
264:
261:
259:
256:
254:
251:
250:
244:
243:
236:
233:
231:
228:
226:
223:
221:
218:
217:
211:
210:
206:
202:
201:
198:
195:
194:
188:
184:
181:range of the
180:
171:
167:
158:
148:
144:
135:
132:
128:
123:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
3205:
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3185:Bibliography
3163:
3131:
3120:. Retrieved
3105:
3098:
3087:. Retrieved
3081:
3074:
3064:
3045:
3039:
3020:
3014:
2998:
2986:. Retrieved
2982:
2972:
2950:(1): 25–29.
2947:
2943:
2937:
2912:
2908:
2902:
2868:Meridian arc
2748:6,356,750.52
2699:298.24961539
2585:6,356,818.17
2434:300.80172554
2362:
2359:
2340:
2163:
2151:
2139:
2124:Equatorial (
2102:
2091:(east/west)
2086:
2076:
2060:
2053:
2049:
2045:
2008:
1998:
1979:
1927:
1789:
1477:
1170:
1107:
1103:
1097:
1056:
1044:
1026:are used in
1018:
1014:
1003:
942:
919:
908:
888:Isaac Newton
885:
867:
865:
820:
814:
789:astronautics
782:
767:
758:
752:
731:
725:
722:
682:
669:
661:
639:
617:, used as a
615:Earth's form
610:
606:
604:
347:Technologies
302: /
214:Fundamentals
147:eccentricity
129:of the 2003
105:
99:October 2016
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
2909:SIAM Review
2830:6,356,751.9
2827:6,378,136.6
2824:IERS (2003)
2696:6,356,772.2
2693:6,378,157.5
2548:293.4659980
2528:294.9786982
2525:6,356,583.8
2522:6,378,206.4
2508:299.1528128
2488:299.3249646
2408:6,376,523.0
2380:Where used
2075:based on a
2066:curvilinear
778:coordinates
715:ellipsoid.
631:geosciences
225:Geodynamics
166:Karman line
3251:Categories
3238:help page)
3122:2021-10-24
3089:2021-10-24
2894:References
2685:Australia
2634:Krassovsky
2386:Maupertuis
2128:), polar (
2000:Gravimetry
1975:flattening
1116:flattening
949:flattening
841:flattening
670:polar axis
668:) and the
658:South Pole
654:North Pole
650:minor axis
646:revolution
635:ellipsoids
629:, and the
157:minor axis
127:oblateness
69:newspapers
2964:126412032
2833:298.25642
2811:6,378,136
2788:6,378,137
2765:6,378,137
2745:6,378,135
2725:6,378,160
2709:6,378,160
2676:6,378,160
2659:6,378,145
2639:6,378,245
2619:6,378,388
2602:6,378,388
2582:6,378,200
2545:6,356,456
2542:6,378,190
2391:6,397,300
2242:ω
2088:longitude
1904:φ
1900:
1884:−
1862:−
1859:≈
1853:∂
1842:∂
1818:≈
1812:∂
1801:∂
1769:∂
1758:∂
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1310:φ
1180:φ
1141:−
1004:That is,
977:−
896:ellipsoid
892:Principia
861:elevation
857:longitude
763:reduction
627:astronomy
304:Longitude
230:Geomatics
170:sea level
3171:Archived
2842:See also
2754:USA/DoD
2668:USA/DoD
2491:Britain
2468:300.8017
2451:300.8017
2071:used in
1988:and the
1080:Saturn's
947:and the
915:triaxial
886:In 1687
853:latitude
837:interior
755:regional
648:) whose
642:spheroid
640:It is a
483:ISO 6709
381:(Europe)
379:Galileo
365:(Russia)
363:GLONASS
300:Latitude
289:Geodesic
247:Concepts
179:Altitude
3257:Geodesy
3236:SPENVIS
3156:chap. 4
3152:chap. 1
3065:Geodesy
2917:Bibcode
2817:298.257
2797:Global
2774:Global
2628:Europe
2597:Hayford
2577:Helmert
2568:293.465
2423:Everest
2417:France
2400:France
2073:geodesy
2011:geodesy
2009:Modern
1084:Telesto
1076:Jupiter
1028:geodesy
922:ellipse
911:geodesy
900:ellipse
833:gravity
817:geodesy
772:or the
770:Hayford
728:average
623:geodesy
543:Geo URI
513:NAVD 88
423:NGVD 29
397:(Japan)
389:(India)
373:(China)
235:History
220:Geodesy
197:Geodesy
143:Ellipse
83:scholar
3113:
3027:
3006:
2962:
2808:(1989)
2785:(1984)
2783:WGS-84
2762:(1979)
2760:GRS-80
2751:298.26
2742:(1972)
2740:WGS-72
2731:298.25
2682:298.25
2665:298.25
2656:(1966)
2636:(1940)
2599:(1910)
2591:Egypt
2579:(1906)
2559:(1880)
2557:Clarke
2539:(1878)
2537:Clarke
2519:(1866)
2517:Clarke
2499:(1841)
2497:Bessel
2479:(1830)
2437:India
2425:(1830)
2414:308.64
2388:(1738)
2161:1969.
1478:where
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924:. The
859:, and
748:volume
533:GCJ-02
523:ETRS89
503:WGS 84
493:NAD 83
473:GRS 80
433:OSGB36
387:NAVIC
268:radius
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2645:298.3
2588:298.3
825:geoid
740:geoid
719:Types
713:WGS84
463:SAD69
443:SK-42
258:Geoid
90:JSTOR
76:books
3111:ISBN
3025:ISBN
3004:ISBN
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1038:and
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270:and
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