54:
42:
956:
753:
742:
758:
664:
204:
634:
494:
561:
459:
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525:
303:
951:{\displaystyle {\begin{aligned}{\frac {b}{a}}&=1-f={\frac {1-n}{1+n}},\\e^{2}&=2f-f^{2}={\frac {4n}{(1+n)^{2}}},\\f&=1-{\sqrt {1-e^{2}}},\end{aligned}}}
239:
326:
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1174:
979:
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423:
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157:
137:
114:
659:
1277:
1218:
117:
53:
1380:
1236:
35:
999:
982:
1260:
Lapaine, Miljenko (2017). "Basics of
Geodesy for Map Projections". In Lapaine, Miljenko; Usery, E. Lynn (eds.).
1050:. U.S. Geological Survey Professional Paper. Vol. 1395. Washington, D.C.: U.S. Government Printing Office.
165:
1390:
1100:
596:
470:
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41:
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1024:
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31:
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1265:
1187:
1084:
1051:
1004:
994:
216:
1120:
505:
308:
283:
256:
27:
Measure of compression between circle to ellipse or sphere to an ellipsoid of revolution
1327:
Uber die
Berechnung der geographischen Langen und Breiten aus geodatischen Vermessungen
1210:
1159:
964:
737:{\displaystyle {\begin{aligned}f={\frac {2n}{1+n}},\\n={\frac {f}{2-f}}.\end{aligned}}}
575:
408:
366:
346:
142:
122:
99:
1374:
1312:
1096:
242:
1304:
1073:"Transformation of the Geodetic Horizontal Control to Another Reference Ellipsoid"
1269:
1191:
1149:
1104:
1088:
1045:
1019:
1232:(Technical report). Ohio State Univ. Dept. of Geodetic Science and Surveying.
1072:
747:
The flattenings are related to other parameters of the ellipse. For example,
81:
30:"Ellipticity" redirects here. For ellipticity in differential calculus, see
1338:
1243:
17:
383:
is the smaller (semiminor axis). All flattenings are zero for a circle (
85:
1009:
77:
1264:. Lecture Notes in Geoinformation and Cartography. pp. 327–343.
73:
69:
1343:
The calculation of longitude and latitude from geodesic measurements
1295:
1227:
1056:
1354:
52:
40:
1341:, translated into English by C. F. F. Karney and R. E. Deakin as
638:
Used in geodetic calculations as a small expansion parameter.
332:, sometimes only given a symbol, or sometimes called the
399:
1285:
Karney, Charles F.F. (2023). "On auxiliary latitudes".
363:
is the larger dimension (e.g. semimajor axis), whereas
1162:
1123:
967:
756:
662:
599:
578:
534:
508:
473:
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411:
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311:
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219:
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145:
125:
102:
1168:
1134:
973:
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628:
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519:
488:
453:
417:
375:
355:
320:
297:
268:
233:
198:
151:
131:
108:
485:
1154:(Technical report). MIT Lincoln Lab. p. 84.
241:in each case; for the ellipse, this is also its
61:compressed to an oblate ellipsoid of revolution.
653:The flattenings can be related to each-other:
8:
1000:Eccentricity (mathematics) § Ellipses
1294:
1161:
1122:
1055:
966:
933:
921:
889:
862:
853:
827:
790:
761:
757:
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709:
673:
663:
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577:
535:
533:
507:
484:
474:
472:
433:
431:
410:
368:
348:
310:
285:
258:
253:There are three variants: the flattening
223:
218:
175:
167:
159:of the resulting ellipse or ellipsoid is
144:
124:
101:
1036:
96:. The usual notation for flattening is
1207:Coordinate Systems and Map Projections
280:, as well as two other "flattenings"
88:) respectively. Other terms used are
68:is a measure of the compression of a
7:
199:{\displaystyle f={\frac {a-b}{a}}.}
116:and its definition in terms of the
1209:(2nd ed.). Oxford; New York:
1184:Practical Geodesy: Using Computers
629:{\displaystyle {\frac {a-b}{a+b}}}
489:{\displaystyle {\frac {1}{f}}\,\!}
25:
1047:Map Projections: A Working Manual
1349:331(8), 852–861 (2010), E-print
556:{\displaystyle {\frac {a-b}{b}}}
454:{\displaystyle {\frac {a-b}{a}}}
1077:Studia Geophysica et Geodaetica
886:
873:
1:
1305:10.1080/00396265.2023.2217604
1205:Maling, Derek Hylton (1992).
1270:10.1007/978-3-319-51835-0_13
76:along a diameter to form an
1192:10.1007/978-3-642-60584-0_3
1182:Hooijberg, Maarten (1997).
36:Flattening (disambiguation)
1412:
1237:"The Mercator Projections"
1148:Taff, Laurence G. (1980).
328:each sometimes called the
29:
1262:Choosing a Map Projection
1229:Geometric Geodesy, Part I
1226:Rapp, Richard H. (1991).
570:Third flattening
569:
499:
403:(First) flattening
402:
49:compressed to an ellipse.
1186:. Springer. p. 41.
1151:An Astronomical Glossary
1044:Snyder, John P. (1987).
467:are specified by giving
1089:10.1023/A:1019881431482
1071:Tenzer, Róbert (2002).
1339:10.1002/asna.201011352
1242:. §5.2. Archived from
1170:
1136:
975:
952:
738:
630:
586:
557:
521:
490:
463:Fundamental. Geodetic
455:
419:
377:
357:
322:
299:
270:
235:
200:
153:
133:
110:
62:
50:
34:. For other uses, see
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1137:
976:
953:
739:
631:
587:
558:
522:
491:
456:
420:
378:
358:
323:
300:
276:sometimes called the
271:
236:
201:
154:
134:
111:
56:
44:
1325:F. W. Bessel, 1825,
1235:Osborne, P. (2008).
1160:
1121:
1015:Planetary flattening
965:
754:
660:
597:
576:
532:
506:
471:
465:reference ellipsoids
430:
409:
367:
347:
309:
284:
257:
217:
166:
143:
123:
100:
1381:Celestial mechanics
1363:1825AN......4..241B
234:{\displaystyle b/a}
57:A sphere of radius
45:A circle of radius
1333:, 4(86), 241–254,
1166:
1135:{\displaystyle f'}
1132:
1025:Roundness (object)
971:
948:
946:
734:
732:
626:
582:
553:
520:{\displaystyle f'}
517:
500:Second flattening
486:
451:
415:
373:
353:
343:In the following,
321:{\displaystyle n,}
318:
298:{\displaystyle f'}
295:
269:{\displaystyle f,}
266:
231:
211:compression factor
196:
149:
129:
106:
63:
51:
1279:978-3-319-51834-3
1178:second flattening
1169:{\displaystyle n}
1144:second flattening
974:{\displaystyle e}
939:
896:
814:
769:
725:
694:
642:
641:
624:
585:{\displaystyle n}
551:
482:
449:
418:{\displaystyle f}
376:{\displaystyle b}
356:{\displaystyle a}
334:second flattening
330:second flattening
191:
152:{\displaystyle b}
132:{\displaystyle a}
109:{\displaystyle f}
32:elliptic operator
16:(Redirected from
1403:
1365:
1323:
1317:
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1298:
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1115:
1109:
1108:
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1059:
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1005:Equatorial bulge
995:Earth flattening
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338:third flattening
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278:first flattening
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97:
84:of revolution (
58:
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39:
28:
23:
22:
15:
12:
11:
5:
1409:
1407:
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1366:
1347:Astron. Nachr.
1318:
1278:
1252:
1249:on 2012-01-18.
1219:
1213:. p. 65.
1211:Pergamon Press
1197:
1176:is called the
1165:
1142:is called the
1130:
1127:
1110:
1063:
1057:10.3133/pp1395
1035:
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26:
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14:
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4:
3:
2:
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1332:
1331:Astron.Nachr.
1328:
1322:
1319:
1314:
1310:
1306:
1302:
1297:
1292:
1288:
1287:Survey Review
1281:
1275:
1271:
1267:
1263:
1256:
1253:
1245:
1238:
1231:
1230:
1222:
1220:0-08-037233-3
1216:
1212:
1208:
1201:
1198:
1193:
1189:
1185:
1179:
1163:
1153:
1152:
1145:
1128:
1125:
1117:For example,
1114:
1111:
1106:
1102:
1098:
1094:
1090:
1086:
1082:
1078:
1074:
1067:
1064:
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1049:
1048:
1040:
1037:
1030:
1026:
1023:
1021:
1018:
1016:
1013:
1011:
1008:
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1003:
1001:
998:
996:
993:
992:
988:
986:
984:
968:
941:
934:
930:
926:
923:
918:
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910:
905:
898:
890:
882:
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876:
868:
865:
859:
854:
850:
846:
843:
840:
837:
835:
828:
824:
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810:
807:
804:
799:
796:
793:
787:
784:
781:
778:
775:
773:
766:
763:
750:
749:
748:
727:
721:
718:
715:
711:
706:
703:
696:
690:
687:
684:
679:
676:
670:
667:
656:
655:
654:
648:
637:
620:
617:
614:
609:
606:
603:
593:
579:
572:
568:
565:Rarely used.
564:
548:
544:
541:
538:
528:
513:
510:
502:
498:
479:
476:
466:
462:
446:
442:
439:
436:
426:
412:
405:
401:
398:
397:
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394:
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387:
370:
350:
341:
339:
335:
331:
315:
312:
291:
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279:
263:
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248:
246:
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228:
224:
220:
212:
193:
188:
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181:
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172:
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162:
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160:
146:
126:
119:
103:
95:
91:
87:
83:
79:
75:
71:
67:
55:
43:
37:
33:
19:
1391:Trigonometry
1346:
1342:
1330:
1326:
1321:
1286:
1261:
1255:
1244:the original
1228:
1206:
1200:
1183:
1177:
1150:
1143:
1113:
1083:(1): 27–32.
1080:
1076:
1066:
1046:
1039:
983:eccentricity
960:
746:
652:
389:
385:
342:
337:
333:
329:
277:
252:
243:aspect ratio
210:
208:
93:
89:
65:
64:
249:Definitions
90:ellipticity
1375:Categories
1296:2212.05818
1031:References
1020:Sphericity
649:Identities
94:oblateness
66:Flattening
18:Oblateness
1355:0908.1824
1313:254564050
1156:However,
1105:750849329
1097:117114346
927:−
919:−
847:−
797:−
782:−
719:−
607:−
542:−
440:−
182:−
118:semi-axes
82:ellipsoid
1289:: 1–16.
1129:′
1101:ProQuest
989:See also
514:′
292:′
86:spheroid
1396:Circles
1386:Geodesy
1359:Bibcode
1010:Ovality
981:is the
78:ellipse
1311:
1276:
1217:
1103:
1095:
961:where
80:or an
74:sphere
70:circle
1351:arXiv
1309:S2CID
1291:arXiv
1247:(PDF)
1240:(PDF)
1093:S2CID
92:, or
1274:ISBN
1215:ISBN
1180:in:
1146:in:
336:and
305:and
209:The
139:and
1357:,
1335:doi
1301:doi
1266:doi
1188:doi
1085:doi
1052:doi
393:).
213:is
72:or
1377::
1345:,
1329:,
1307:.
1299:.
1272:.
1099:.
1091:.
1081:46
1079:.
1075:.
985:.
388:=
245:.
1361::
1353::
1337::
1315:.
1303::
1293::
1282:.
1268::
1223:.
1194:.
1190::
1164:n
1126:f
1107:.
1087::
1060:.
1054::
969:e
942:,
935:2
931:e
924:1
916:1
913:=
906:f
899:,
891:2
887:)
883:n
880:+
877:1
874:(
869:n
866:4
860:=
855:2
851:f
844:f
841:2
838:=
829:2
825:e
817:,
811:n
808:+
805:1
800:n
794:1
788:=
785:f
779:1
776:=
767:a
764:b
728:.
722:f
716:2
712:f
707:=
704:n
697:,
691:n
688:+
685:1
680:n
677:2
671:=
668:f
621:b
618:+
615:a
610:b
604:a
580:n
549:b
545:b
539:a
511:f
480:f
477:1
447:a
443:b
437:a
413:f
390:b
386:a
371:b
351:a
316:,
313:n
289:f
264:,
261:f
229:a
225:/
221:b
194:.
189:a
185:b
179:a
173:=
170:f
147:b
127:a
104:f
59:a
47:a
38:.
20:)
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