43:
31:
945:
742:
731:
747:
653:
193:
623:
483:
550:
448:
1129:
514:
292:
940:{\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}}}
228:
315:
263:
1163:
968:
579:
412:
370:
350:
146:
126:
103:
648:
1266:
1225:
1207:
1249:
Lapaine, Miljenko (2017). "Basics of
Geodesy for Map Projections". In Lapaine, Miljenko; Usery, E. Lynn (eds.).
1039:. U.S. Geological Survey Professional Paper. Vol. 1395. Washington, D.C.: U.S. Government Printing Office.
988:
106:
1369:
154:
24:
585:
459:
971:
520:
418:
42:
1351:
1379:
1089:
1347:
1003:
8:
453:
205:
1109:
494:
297:
272:
245:
16:
Measure of compression between circle to ellipse or sphere to an ellipsoid of revolution
1339:
1316:
Uber die
Berechnung der geographischen Langen und Breiten aus geodatischen Vermessungen
1297:
1279:
1148:
1081:
1013:
953:
726:{\displaystyle {\begin{aligned}f={\frac {2n}{1+n}},\\n={\frac {f}{2-f}}.\end{aligned}}}
564:
397:
355:
335:
131:
111:
88:
1301:
1262:
1203:
1085:
20:
1384:
1374:
1323:
1289:
1254:
1176:
1073:
1040:
993:
983:
30:
1293:
1062:"Transformation of the Geodetic Horizontal Control to Another Reference Ellipsoid"
1258:
1180:
1138:
1093:
1199:
1077:
1034:
1363:
1221:(Technical report). Ohio State Univ. Dept. of Geodetic Science and Surveying.
1061:
736:
The flattenings are related to other parameters of the ellipse. For example,
19:"Ellipticity" redirects here. For ellipticity in differential calculus, see
1327:
1232:
231:
372:
is the smaller (semiminor axis). All flattenings are zero for a circle (
1008:
1253:. Lecture Notes in Geoinformation and Cartography. pp. 327–343.
70:
1332:
The calculation of longitude and latitude from geodesic measurements
1284:
1216:
1045:
74:
1343:
1330:, translated into English by C. F. F. Karney and R. E. Deakin as
998:
66:
62:
58:
627:
Used in geodetic calculations as a small expansion parameter.
321:, sometimes only given a symbol, or sometimes called the
1274:
Karney, Charles F.F. (2023). "On auxiliary latitudes".
352:
is the larger dimension (e.g. semimajor axis), whereas
1151:
1112:
956:
745:
651:
588:
567:
523:
497:
462:
421:
400:
358:
338:
300:
275:
248:
208:
157:
134:
114:
91:
1157:
1123:
962:
939:
725:
617:
573:
544:
508:
477:
442:
406:
364:
344:
309:
286:
257:
222:
187:
140:
120:
97:
474:
1361:
1143:(Technical report). MIT Lincoln Lab. p. 84.
230:in each case; for the ellipse, this is also its
50:compressed to an oblate ellipsoid of revolution.
1032:
642:The flattenings can be related to each-other:
989:Eccentricity (mathematics) § Ellipses
1283:
1170:
1044:
473:
242:There are three variants: the flattening
148:of the resulting ellipse or ellipsoid is
41:
29:
1248:
1223:
85:. The usual notation for flattening is
1362:
1273:
1196:Coordinate Systems and Map Projections
1193:
1059:
1308:
1026:
269:, as well as two other "flattenings"
77:) respectively. Other terms used are
57:is a measure of the compression of a
1214:
1136:
188:{\displaystyle f={\frac {a-b}{a}}.}
105:and its definition in terms of the
13:
1198:(2nd ed.). Oxford; New York:
1173:Practical Geodesy: Using Computers
618:{\displaystyle {\frac {a-b}{a+b}}}
478:{\displaystyle {\frac {1}{f}}\,\!}
14:
1396:
1036:Map Projections: A Working Manual
1338:331(8), 852–861 (2010), E-print
545:{\displaystyle {\frac {a-b}{b}}}
443:{\displaystyle {\frac {a-b}{a}}}
388:
1066:Studia Geophysica et Geodaetica
1242:
1187:
1100:
1053:
875:
862:
237:
1:
1294:10.1080/00396265.2023.2217604
1194:Maling, Derek Hylton (1992).
1019:
637:
1259:10.1007/978-3-319-51835-0_13
65:along a diameter to form an
7:
1181:10.1007/978-3-642-60584-0_3
1171:Hooijberg, Maarten (1997).
977:
25:Flattening (disambiguation)
10:
1401:
1226:"The Mercator Projections"
1137:Taff, Laurence G. (1980).
317:each sometimes called the
18:
1251:Choosing a Map Projection
1218:Geometric Geodesy, Part I
1215:Rapp, Richard H. (1991).
559:Third flattening
558:
488:
392:(First) flattening
391:
38:compressed to an ellipse.
1175:. Springer. p. 41.
1140:An Astronomical Glossary
1033:Snyder, John P. (1987).
456:are specified by giving
1078:10.1023/A:1019881431482
1060:Tenzer, Róbert (2002).
1328:10.1002/asna.201011352
1231:. §5.2. Archived from
1159:
1125:
964:
941:
727:
619:
575:
546:
510:
479:
452:Fundamental. Geodetic
444:
408:
366:
346:
311:
288:
259:
224:
189:
142:
122:
99:
51:
39:
23:. For other uses, see
1160:
1126:
965:
942:
728:
620:
576:
547:
511:
480:
445:
409:
367:
347:
312:
289:
265:sometimes called the
260:
225:
190:
143:
123:
100:
45:
33:
1314:F. W. Bessel, 1825,
1224:Osborne, P. (2008).
1149:
1110:
1004:Planetary flattening
954:
743:
649:
586:
565:
521:
495:
460:
454:reference ellipsoids
419:
398:
356:
336:
298:
273:
246:
206:
155:
132:
112:
89:
1370:Celestial mechanics
1352:1825AN......4..241B
223:{\displaystyle b/a}
46:A sphere of radius
34:A circle of radius
1322:, 4(86), 241–254,
1155:
1124:{\displaystyle f'}
1121:
1014:Roundness (object)
960:
937:
935:
723:
721:
615:
571:
542:
509:{\displaystyle f'}
506:
489:Second flattening
475:
440:
404:
362:
342:
332:In the following,
310:{\displaystyle n,}
307:
287:{\displaystyle f'}
284:
258:{\displaystyle f,}
255:
220:
200:compression factor
185:
138:
118:
95:
52:
40:
1268:978-3-319-51834-3
1167:second flattening
1158:{\displaystyle n}
1133:second flattening
963:{\displaystyle e}
928:
885:
803:
758:
714:
683:
631:
630:
613:
574:{\displaystyle n}
540:
471:
438:
407:{\displaystyle f}
365:{\displaystyle b}
345:{\displaystyle a}
323:second flattening
319:second flattening
180:
141:{\displaystyle b}
121:{\displaystyle a}
98:{\displaystyle f}
21:elliptic operator
1392:
1354:
1312:
1306:
1305:
1287:
1272:
1246:
1240:
1239:
1237:
1230:
1222:
1213:
1191:
1185:
1184:
1164:
1162:
1161:
1156:
1144:
1130:
1128:
1127:
1122:
1120:
1104:
1098:
1097:
1057:
1051:
1050:
1048:
1030:
994:Equatorial bulge
984:Earth flattening
969:
967:
966:
961:
946:
944:
943:
938:
936:
929:
927:
926:
911:
886:
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882:
860:
852:
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821:
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791:
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759:
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724:
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621:
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612:
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551:
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543:
541:
536:
525:
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484:
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449:
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423:
413:
411:
410:
405:
389:
381:
371:
369:
368:
363:
351:
349:
348:
343:
329:, respectively.
327:third flattening
316:
314:
313:
308:
293:
291:
290:
285:
283:
267:first flattening
264:
262:
261:
256:
229:
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226:
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194:
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176:
165:
147:
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127:
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124:
119:
104:
102:
101:
96:
49:
37:
1400:
1399:
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1391:
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1389:
1360:
1359:
1358:
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1313:
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1188:
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1054:
1031:
1027:
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897:
891:
890:
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689:
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587:
584:
583:
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563:
562:
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463:
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457:
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354:
353:
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299:
296:
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276:
274:
271:
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247:
244:
243:
240:
212:
207:
204:
203:
166:
164:
156:
153:
152:
133:
130:
129:
113:
110:
109:
90:
87:
86:
73:of revolution (
47:
35:
28:
17:
12:
11:
5:
1398:
1388:
1387:
1382:
1377:
1372:
1356:
1355:
1336:Astron. Nachr.
1307:
1267:
1241:
1238:on 2012-01-18.
1208:
1202:. p. 65.
1200:Pergamon Press
1186:
1165:is called the
1154:
1131:is called the
1119:
1116:
1099:
1052:
1046:10.3133/pp1395
1024:
1023:
1021:
1018:
1017:
1016:
1011:
1006:
1001:
996:
991:
986:
979:
976:
959:
948:
947:
932:
925:
921:
917:
914:
909:
906:
903:
900:
898:
896:
893:
892:
889:
881:
877:
873:
870:
867:
864:
859:
856:
850:
845:
841:
837:
834:
831:
828:
825:
823:
819:
815:
811:
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807:
801:
798:
795:
790:
787:
784:
778:
775:
772:
769:
766:
763:
761:
757:
754:
749:
748:
734:
733:
718:
712:
709:
706:
702:
697:
694:
691:
690:
687:
681:
678:
675:
670:
667:
661:
658:
655:
654:
639:
636:
635:
634:
633:
632:
629:
628:
625:
611:
608:
605:
600:
597:
594:
581:
570:
560:
556:
555:
552:
539:
535:
532:
529:
516:
504:
501:
490:
486:
485:
470:
467:
450:
437:
433:
430:
427:
414:
403:
393:
361:
341:
306:
303:
282:
279:
254:
251:
239:
236:
219:
215:
211:
196:
195:
184:
179:
175:
172:
169:
163:
160:
137:
117:
94:
15:
9:
6:
4:
3:
2:
1397:
1386:
1383:
1381:
1378:
1376:
1373:
1371:
1368:
1367:
1365:
1353:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1320:Astron.Nachr.
1317:
1311:
1303:
1299:
1295:
1291:
1286:
1281:
1277:
1276:Survey Review
1270:
1264:
1260:
1256:
1252:
1245:
1234:
1227:
1220:
1219:
1211:
1209:0-08-037233-3
1205:
1201:
1197:
1190:
1182:
1178:
1174:
1168:
1152:
1142:
1141:
1134:
1117:
1114:
1106:For example,
1103:
1095:
1091:
1087:
1083:
1079:
1075:
1071:
1067:
1063:
1056:
1047:
1042:
1038:
1037:
1029:
1025:
1015:
1012:
1010:
1007:
1005:
1002:
1000:
997:
995:
992:
990:
987:
985:
982:
981:
975:
973:
957:
930:
923:
919:
915:
912:
907:
904:
901:
899:
894:
887:
879:
871:
868:
865:
857:
854:
848:
843:
839:
835:
832:
829:
826:
824:
817:
813:
805:
799:
796:
793:
788:
785:
782:
776:
773:
770:
767:
764:
762:
755:
752:
739:
738:
737:
716:
710:
707:
704:
700:
695:
692:
685:
679:
676:
673:
668:
665:
659:
656:
645:
644:
643:
626:
609:
606:
603:
598:
595:
592:
582:
568:
561:
557:
554:Rarely used.
553:
537:
533:
530:
527:
517:
502:
499:
491:
487:
468:
465:
455:
451:
435:
431:
428:
425:
415:
401:
394:
390:
387:
386:
385:
384:
383:
380:
376:
359:
339:
330:
328:
324:
320:
304:
301:
280:
277:
268:
252:
249:
235:
233:
217:
213:
209:
201:
182:
177:
173:
170:
167:
161:
158:
151:
150:
149:
135:
115:
108:
92:
84:
80:
76:
72:
68:
64:
60:
56:
44:
32:
26:
22:
1380:Trigonometry
1335:
1331:
1319:
1315:
1310:
1275:
1250:
1244:
1233:the original
1217:
1195:
1189:
1172:
1166:
1139:
1132:
1102:
1072:(1): 27–32.
1069:
1065:
1055:
1035:
1028:
972:eccentricity
949:
735:
641:
378:
374:
331:
326:
322:
318:
266:
241:
232:aspect ratio
199:
197:
82:
78:
54:
53:
238:Definitions
79:ellipticity
1364:Categories
1285:2212.05818
1020:References
1009:Sphericity
638:Identities
83:oblateness
55:Flattening
1344:0908.1824
1302:254564050
1145:However,
1094:750849329
1086:117114346
916:−
908:−
836:−
786:−
771:−
708:−
596:−
531:−
429:−
171:−
107:semi-axes
71:ellipsoid
1278:: 1–16.
1118:′
1090:ProQuest
978:See also
503:′
281:′
75:spheroid
1385:Circles
1375:Geodesy
1348:Bibcode
999:Ovality
970:is the
67:ellipse
1300:
1265:
1206:
1092:
1084:
950:where
69:or an
63:sphere
59:circle
1340:arXiv
1298:S2CID
1280:arXiv
1236:(PDF)
1229:(PDF)
1082:S2CID
81:, or
1263:ISBN
1204:ISBN
1169:in:
1135:in:
325:and
294:and
198:The
128:and
1346:,
1324:doi
1290:doi
1255:doi
1177:doi
1074:doi
1041:doi
382:).
202:is
61:or
1366::
1334:,
1318:,
1296:.
1288:.
1261:.
1088:.
1080:.
1070:46
1068:.
1064:.
974:.
377:=
234:.
1350::
1342::
1326::
1304:.
1292::
1282::
1271:.
1257::
1212:.
1183:.
1179::
1153:n
1115:f
1096:.
1076::
1049:.
1043::
958:e
931:,
924:2
920:e
913:1
905:1
902:=
895:f
888:,
880:2
876:)
872:n
869:+
866:1
863:(
858:n
855:4
849:=
844:2
840:f
833:f
830:2
827:=
818:2
814:e
806:,
800:n
797:+
794:1
789:n
783:1
777:=
774:f
768:1
765:=
756:a
753:b
717:.
711:f
705:2
701:f
696:=
693:n
686:,
680:n
677:+
674:1
669:n
666:2
660:=
657:f
610:b
607:+
604:a
599:b
593:a
569:n
538:b
534:b
528:a
500:f
469:f
466:1
436:a
432:b
426:a
402:f
379:b
375:a
360:b
340:a
305:,
302:n
278:f
253:,
250:f
218:a
214:/
210:b
183:.
178:a
174:b
168:a
162:=
159:f
136:b
116:a
93:f
48:a
36:a
27:.
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