28:
20:
426:
206:
211:
804:
158:
573:
722:
752:
662:
602:
541:
515:
469:
449:
489:
178:
778:
688:
628:
198:
421:{\displaystyle {\begin{aligned}&x=\lambda \\\alpha &y=\sin \varphi +{\frac {\alpha -1}{\gamma }}\int _{0}^{y}(\gamma ^{k}-z^{k})^{1/k}dz\end{aligned}}}
1285:
2027:
1559:
1065:
942:
1645:
1441:
1431:
1351:
1569:
1564:
1539:
1411:
1075:
1070:
1045:
1436:
1017:
1446:
1247:
1531:
1118:
1037:
83:
27:
1970:
1767:
1694:
1650:
1346:
2052:
1815:
1762:
1923:
1892:
1466:
1315:
1093:
1022:
856:
2007:
1975:
1825:
1456:
1280:
1113:
1103:
935:
783:
110:
491:
is the relative weight given to the cylindrical equal-area projection. For a purely cylindrical equal-area,
1965:
1679:
1333:
1242:
1955:
1905:
1868:
1635:
1328:
1177:
1027:
843:
879:
Tobler, Waldo (1973). "The hyperelliptical and other new pseudocylindrical equal area map projections".
1883:
1840:
1684:
1549:
1275:
1108:
1098:
1055:
546:
1807:
987:
1820:
1205:
901:
693:
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1850:
1830:
1610:
1554:
1461:
1423:
1388:
1060:
928:
631:
1137:
1123:
967:
815:
47:
2022:
1655:
1630:
1365:
1172:
962:
896:
731:
641:
581:
520:
494:
32:
1795:
1750:
454:
434:
1945:
1735:
1689:
1516:
1493:
1476:
1257:
1187:
474:
163:
82:. Their spacing is calculated to provide the equal-area property. The projection blends the
1950:
1845:
1669:
1625:
1620:
1615:
1592:
1587:
1508:
1270:
1210:
1182:
1167:
1162:
1157:
1152:
999:
888:
757:
725:
635:
87:
1937:
1727:
44:
8:
1900:
1835:
1740:
1717:
1544:
1451:
1323:
1302:
1050:
1008:
667:
607:
1709:
892:
1772:
1383:
1229:
1088:
183:
71:
19:
1860:
1699:
1640:
1605:
1521:
1498:
1378:
1373:
1292:
1237:
1215:
1485:
1265:
906:
867:
70:
As with any pseudocylindrical projection, in the projection’s normal aspect, the
55:
728:. Tobler favored the parameterization shown with the top illustration; that is,
98:
951:
835:
79:
2046:
910:
1960:
92:
23:
Tobler hyperelliptical projection of the world; α = 0, γ = 1.18314, k = 2.5
972:
62:
projection, now usually known as the Tobler hyperelliptical projection.
200:
are free parameters. Tobler's hyperelliptical projection is given as:
2017:
51:
2012:
75:
1873:
920:
90:, with meridians that follow a particular kind of curve known as
517:; for a projection with pure hyperellipses for meridians,
786:
760:
734:
696:
670:
644:
610:
584:
549:
523:
497:
477:
457:
437:
209:
186:
166:
113:
840:
1314:
798:
772:
746:
716:
682:
656:
622:
596:
567:
535:
509:
483:
463:
443:
420:
192:
172:
152:
2044:
936:
58:introduced the construction in 1973 as the
31:The Tobler hyperelliptical projection with
16:Pseudocylindrical equal-area map projection
943:
929:
2028:Map projection of the tri-axial ellipsoid
1145:
900:
857:Mapthematics directory of map projections
868:"Superellipse" in MathWorld encyclopedia
26:
18:
828:
799:{\displaystyle \gamma \approx 1.183136}
153:{\displaystyle x^{k}+y^{k}=\gamma ^{k}}
2045:
878:
834:
1996:
1891:
1793:
1409:
985:
924:
1794:
13:
950:
14:
2064:
568:{\displaystyle 0<\alpha <1}
543:; and for weighted combinations,
107:. A hyperellipse is described by
84:cylindrical equal-area projection
50:projections that may be used for
41:Tobler hyperelliptical projection
1971:Quadrilateralized spherical cube
1651:Quadrilateralized spherical cube
881:Journal of Geophysical Research
86:, which has straight, vertical
1560:Lambert cylindrical equal-area
986:
872:
861:
850:
717:{\displaystyle \gamma =4/\pi }
391:
364:
301:
278:
251:
245:
233:
224:
1:
2008:Interruption (map projection)
1410:
821:
1997:
1646:Lambert azimuthal equal-area
1442:Guyou hemisphere-in-a-square
1432:Adams hemisphere-in-a-square
35:of deformation; α = 0, k = 3
7:
844:University of Chicago Press
809:
724:the projection becomes the
65:
10:
2069:
2003:
1992:
1936:
1919:
1882:
1859:
1806:
1802:
1789:
1749:
1726:
1708:
1668:
1601:
1578:
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1507:
1484:
1475:
1422:
1418:
1405:
1364:
1342:
1301:
1256:
1228:
1201:
1136:
1084:
1036:
1007:
998:
994:
981:
958:
747:{\displaystyle \alpha =0}
657:{\displaystyle \alpha =0}
597:{\displaystyle \alpha =0}
536:{\displaystyle \alpha =0}
510:{\displaystyle \alpha =1}
464:{\displaystyle \varphi }
444:{\displaystyle \lambda }
1447:Lambert conformal conic
911:10.1029/JB078i011p01753
816:List of map projections
484:{\displaystyle \alpha }
173:{\displaystyle \gamma }
2053:Equal-area projections
1580:Tobler hyperelliptical
1193:Tobler hyperelliptical
1119:Space-oblique Mercator
800:
774:
748:
718:
684:
658:
624:
598:
569:
537:
511:
485:
465:
445:
422:
194:
174:
154:
36:
24:
801:
775:
773:{\displaystyle k=2.5}
749:
719:
685:
659:
625:
599:
570:
538:
512:
486:
471:is the latitude, and
466:
446:
423:
195:
175:
155:
30:
22:
1956:Cahill–Keyes M-shape
1816:Chamberlin trimetric
784:
758:
732:
726:Mollweide projection
694:
668:
642:
636:Collignon projection
608:
582:
547:
521:
495:
475:
455:
435:
207:
184:
164:
111:
2023:Tissot's indicatrix
1924:Central cylindrical
1565:Smyth equal-surface
1467:Transverse Mercator
1316:General perspective
1071:Smyth equal-surface
1023:Transverse Mercator
893:1973JGR....78.1753T
683:{\displaystyle k=2}
623:{\displaystyle k=1}
363:
33:Tissot's indicatrix
1976:Waterman butterfly
1826:Miller cylindrical
1457:Peirce quincuncial
1352:Lambert equal-area
1104:Gall stereographic
796:
770:
744:
714:
680:
654:
620:
594:
565:
533:
507:
481:
461:
451:is the longitude,
441:
418:
416:
349:
190:
170:
150:
37:
25:
2040:
2039:
2036:
2035:
1988:
1987:
1984:
1983:
1932:
1931:
1785:
1784:
1781:
1780:
1664:
1663:
1401:
1400:
1397:
1396:
1360:
1359:
1248:Lambert conformal
1224:
1223:
1138:Pseudocylindrical
1132:
1131:
887:(11): 1753–1759.
347:
299:
193:{\displaystyle k}
48:pseudocylindrical
2060:
1994:
1993:
1951:Cahill Butterfly
1889:
1888:
1869:Goode homolosine
1804:
1803:
1791:
1790:
1756:
1755:(Mecca or Qibla)
1636:Goode homolosine
1482:
1481:
1420:
1419:
1407:
1406:
1312:
1311:
1307:
1178:Goode homolosine
1143:
1142:
1028:Oblique Mercator
1005:
1004:
996:
995:
983:
982:
945:
938:
931:
922:
921:
915:
914:
904:
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865:
859:
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263:
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249:
213:
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191:
179:
177:
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171:
159:
157:
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149:
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136:
135:
123:
122:
103:or sometimes as
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2067:
2063:
2062:
2061:
2059:
2058:
2057:
2043:
2042:
2041:
2032:
1999:
1980:
1928:
1915:
1878:
1855:
1841:Van der Grinten
1798:
1796:By construction
1777:
1754:
1753:
1745:
1722:
1704:
1685:Equirectangular
1671:
1660:
1597:
1574:
1570:Trystan Edwards
1526:
1503:
1471:
1414:
1393:
1366:Pseudoazimuthal
1356:
1338:
1305:
1304:
1297:
1252:
1220:
1216:Winkel I and II
1197:
1128:
1109:Gall isographic
1099:Equirectangular
1080:
1076:Trystan Edwards
1032:
990:
977:
954:
949:
919:
918:
902:10.1.1.495.6424
877:
873:
866:
862:
855:
851:
836:Snyder, John P.
833:
829:
824:
812:
785:
782:
781:
759:
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755:
733:
730:
729:
706:
695:
692:
691:
669:
666:
665:
643:
640:
639:
630:the projection
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518:
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165:
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109:
108:
68:
60:hyperelliptical
56:Waldo R. Tobler
43:is a family of
17:
12:
11:
5:
2066:
2056:
2055:
2038:
2037:
2034:
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1865:
1863:
1857:
1856:
1854:
1853:
1848:
1843:
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1833:
1828:
1823:
1821:Kavrayskiy VII
1818:
1812:
1810:
1800:
1799:
1787:
1786:
1783:
1782:
1779:
1778:
1776:
1775:
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1759:
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1751:Retroazimuthal
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1712:
1706:
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1697:
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1676:
1674:
1670:Equidistant in
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1234:
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1226:
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1222:
1221:
1219:
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1213:
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1206:Kavrayskiy VII
1202:
1199:
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1190:
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979:
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952:Map projection
948:
947:
940:
933:
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917:
916:
871:
860:
849:
846:. p. 220.
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80:straight lines
78:are parallel,
67:
64:
15:
9:
6:
4:
3:
2:
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2029:
2026:
2024:
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2019:
2016:
2014:
2011:
2009:
2006:
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2002:
1995:
1991:
1977:
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1972:
1969:
1967:
1964:
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1952:
1949:
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1944:
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1925:
1922:
1921:
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1912:
1911:Stereographic
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1907:
1904:
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1898:
1896:
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1885:
1881:
1875:
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1852:
1851:Winkel tripel
1849:
1847:
1844:
1842:
1839:
1837:
1834:
1832:
1831:Natural Earth
1829:
1827:
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1634:
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1629:
1627:
1624:
1622:
1619:
1617:
1614:
1612:
1611:Briesemeister
1609:
1607:
1604:
1603:
1600:
1594:
1591:
1589:
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1497:
1495:
1492:
1491:
1489:
1487:
1483:
1480:
1478:
1474:
1468:
1465:
1463:
1462:Stereographic
1460:
1458:
1455:
1453:
1450:
1448:
1445:
1443:
1440:
1438:
1435:
1433:
1430:
1429:
1427:
1425:
1421:
1417:
1413:
1408:
1404:
1390:
1389:Winkel tripel
1387:
1385:
1382:
1380:
1377:
1375:
1372:
1371:
1369:
1367:
1363:
1353:
1350:
1348:
1345:
1344:
1341:
1335:
1334:Stereographic
1332:
1330:
1327:
1325:
1322:
1321:
1319:
1317:
1313:
1310:
1308:
1300:
1294:
1291:
1287:
1284:
1282:
1279:
1278:
1277:
1274:
1272:
1269:
1267:
1264:
1263:
1261:
1259:
1258:Pseudoconical
1255:
1249:
1246:
1244:
1241:
1239:
1236:
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1233:
1231:
1227:
1217:
1214:
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1191:
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1125:
1122:
1120:
1117:
1115:
1112:
1110:
1107:
1105:
1102:
1100:
1097:
1095:
1092:
1090:
1087:
1086:
1083:
1077:
1074:
1072:
1069:
1067:
1064:
1062:
1059:
1057:
1054:
1052:
1049:
1047:
1044:
1043:
1041:
1039:
1035:
1029:
1026:
1024:
1021:
1019:
1016:
1015:
1013:
1010:
1006:
1003:
1001:
997:
993:
989:
984:
980:
974:
971:
969:
966:
964:
961:
960:
957:
953:
946:
941:
939:
934:
932:
927:
926:
923:
912:
908:
903:
898:
894:
890:
886:
882:
875:
869:
864:
858:
853:
845:
841:
837:
831:
827:
817:
814:
813:
807:
793:
790:
787:
767:
764:
761:
741:
738:
735:
727:
711:
707:
703:
700:
697:
677:
674:
671:
651:
648:
645:
637:
633:
617:
614:
611:
591:
588:
585:
576:
562:
559:
556:
553:
550:
530:
527:
524:
504:
501:
498:
478:
458:
438:
411:
408:
403:
399:
395:
385:
381:
377:
372:
368:
359:
354:
350:
344:
340:
337:
334:
328:
325:
322:
319:
316:
313:
308:
296:
290:
286:
282:
272:
268:
264:
259:
255:
242:
239:
236:
230:
227:
221:
218:
215:
203:
202:
201:
187:
167:
145:
141:
137:
132:
128:
124:
119:
115:
106:
105:hyperellipses
102:
100:
95:
94:
93:superellipses
89:
85:
81:
77:
73:
63:
61:
57:
53:
49:
46:
42:
34:
29:
21:
1906:Orthographic
1579:
1437:Gauss–Krüger
1329:Orthographic
1192:
1124:Web Mercator
1018:Gauss–Krüger
884:
880:
874:
863:
852:
839:
830:
577:
430:
104:
97:
91:
69:
59:
40:
38:
1884:Perspective
1672:some aspect
1656:Strebe 1995
1631:Equal Earth
1550:Gall–Peters
1532:Cylindrical
1347:Equidistant
1243:Equidistant
1173:Equal Earth
1056:Gall–Peters
1000:Cylindrical
842:. Chicago:
632:degenerates
1946:AuthaGraph
1938:Polyhedral
1808:Compromise
1736:Loximuthal
1728:Loxodromic
1690:Sinusoidal
1540:Balthasart
1517:Sinusoidal
1494:Sinusoidal
1477:Equal-area
1188:Sinusoidal
1146:Equal-area
1046:Balthasart
1038:Equal-area
1011:-conformal
988:By surface
822:References
52:world maps
45:equal-area
2018:Longitude
1846:Wagner VI
1695:Two-point
1626:Eckert VI
1621:Eckert IV
1616:Eckert II
1593:Mollweide
1588:Collignon
1555:Hobo–Dyer
1509:Bottomley
1424:Conformal
1412:By metric
1303:Azimuthal
1276:Polyconic
1271:Bottomley
1211:Wagner VI
1183:Mollweide
1168:Eckert VI
1163:Eckert IV
1158:Eckert II
1153:Collignon
1061:Hobo–Dyer
897:CiteSeerX
791:≈
788:γ
736:α
712:π
698:γ
646:α
586:α
557:α
525:α
499:α
479:α
459:φ
439:λ
378:−
369:γ
351:∫
345:γ
338:−
335:α
326:φ
323:
309:α
297:γ
265:−
256:γ
243:α
240:−
228:α
222:λ
168:γ
142:γ
88:meridians
72:parallels
2047:Category
2013:Latitude
1998:See also
1961:Dymaxion
1901:Gnomonic
1836:Robinson
1741:Mercator
1718:Gnomonic
1710:Gnomonic
1545:Behrmann
1452:Mercator
1324:Gnomonic
1306:(planar)
1281:American
1051:Behrmann
1009:Mercator
838:(1993).
810:See also
794:1.183136
160:, where
76:latitude
66:Overview
1874:HEALPix
1773:Littrow
1384:Wiechel
1286:Chinese
1230:Conical
1094:Central
1089:Cassini
1066:Lambert
963:History
889:Bibcode
638:; when
634:to the
1893:Planar
1861:Hybrid
1768:Hammer
1700:Werner
1641:Hammer
1606:Albers
1522:Werner
1499:Werner
1379:Hammer
1374:Aitoff
1293:Werner
1238:Albers
1114:Miller
973:Portal
899:
780:, and
690:, and
431:where
101:curves
1763:Craig
1680:Conic
1486:Bonne
1266:Bonne
578:When
1966:ISEA
968:List
604:and
560:<
554:<
180:and
99:Lamé
39:The
907:doi
768:2.5
320:sin
96:or
74:of
2049::
905:.
895:.
885:78
883:.
806:.
754:,
664:,
575:.
54:.
944:e
937:t
930:v
913:.
909::
891::
765:=
762:k
742:0
739:=
708:/
704:4
701:=
678:2
675:=
672:k
652:0
649:=
618:1
615:=
612:k
592:0
589:=
563:1
551:0
531:0
528:=
505:1
502:=
412:z
409:d
404:k
400:/
396:1
392:)
386:k
382:z
373:k
365:(
360:y
355:0
341:1
329:+
317:=
314:y
302:]
291:k
287:/
283:1
279:)
273:k
269:y
260:k
252:(
246:)
237:1
234:(
231:+
225:[
219:=
216:x
188:k
146:k
138:=
133:k
129:y
125:+
120:k
116:x
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