39:
509:
391:
31:
106:
508:
280:
471:, the stereographic projection is infinitely large, and showing the South Pole (for a map centered on the North Pole) is impossible. However, it is possible to show points arbitrarily close to the South Pole as long as the boundaries of the map are extended far enough.
138:
The stereographic projection was exclusively used for star charts until 1507, when
Walther Ludd of St. Dié, Lorraine created the first known instance of a stereographic projection of the Earth's surface. Its popularity in cartography increased after
382:
passing through its center point. As a conformal projection, it faithfully represents angles everywhere. In addition, in its spherical form, the stereographic projection is the only map projection that renders all
188:
193:
398:
The spherical form of the stereographic projection is equivalent to a perspective projection where the point of perspective is on the point on the globe opposite the center point of the map.
991:
363:. There are various forms of transverse or oblique stereographic projections of ellipsoids. One method uses double projection via a conformal sphere, while other methods do not.
469:
352:
is normally chosen to model the Earth when the extent of the mapped region exceeds a few hundred kilometers in length in both dimensions. For maps of smaller regions, an
143:
used its equatorial aspect for his 1595 atlas. It subsequently saw frequent use throughout the seventeenth century with its equatorial aspect being used for maps of the
439:
343:
323:
94:
93:, the perspective definition of the stereographic projection is not conformal, and adjustments must be made to preserve its azimuthal and conformal properties. The
419:
303:
367:
1733:
1265:
771:
648:
515:
1351:
1147:
1137:
1057:
67:
1142:
723:
275:{\displaystyle {\begin{aligned}r&=2R\tan \left({\frac {\pi }{4}}-{\frac {\varphi }{2}}\right)\\\theta &=\lambda \end{aligned}}}
1152:
953:
603:
Sprinsky, William H.; Snyder, John P. (1986). "The Miller
Oblated Stereographic Projection for Africa, Europe, Asia and Australasia".
1285:
1275:
1270:
1245:
1237:
898:
824:
781:
776:
751:
743:
581:
557:
38:
1676:
1473:
1400:
1356:
1052:
1521:
1468:
1758:
1629:
1598:
1172:
1021:
799:
728:
83:
522:
and some azimuthal projections centred on 90° N at the same scale, ordered by projection altitude in Earth radii.
1713:
1681:
1531:
1162:
986:
819:
809:
641:
480:
1671:
1385:
948:
1661:
1611:
1574:
1341:
1034:
883:
733:
1255:
761:
1546:
1390:
981:
814:
804:
1526:
911:
1260:
766:
1616:
1556:
1536:
1167:
1129:
1094:
634:
75:
63:
378:
As an azimuthal projection, the stereographic projection faithfully represents the relative directions of all
593:
Timothy Feeman. 2002. "Portraits of the Earth: A Mathematician Looks at Maps". American
Mathematical Society.
366:
Examples of transverse or oblique stereographic projections include the Miller
Oblated Stereographic and the
829:
673:
523:
1728:
1361:
1336:
878:
668:
43:
1651:
1441:
1395:
1222:
1199:
1182:
893:
444:
109:
World map made by Rumold
Mercator in 1587, using two equatorial aspects of the stereographic projection.
17:
1656:
1551:
1331:
1326:
1321:
1298:
1293:
1214:
976:
916:
888:
873:
868:
863:
858:
1606:
1541:
1446:
1423:
1250:
1157:
1029:
756:
714:
353:
349:
122:
71:
390:
1478:
1089:
794:
384:
360:
148:
144:
1405:
1346:
1316:
1311:
1227:
1204:
1084:
1079:
998:
943:
921:
577:
553:
424:
328:
308:
182:
The spherical form of the stereographic projection is usually expressed in polar coordinates:
30:
1191:
971:
612:
491:
140:
121:, who was the first Greek to use it. Its oblique aspect was used by Greek Mathematician
1643:
1589:
1566:
1513:
1501:
1456:
1433:
1415:
1375:
1117:
1071:
1008:
963:
935:
843:
705:
693:
657:
495:
484:
404:
288:
131:
79:
1752:
499:
163:
155:
114:
1666:
379:
171:
105:
678:
616:
483:
are distributed with the same spacing as those on the central meridian of the
394:
3D illustration of the geometric construction of the stereographic projection.
159:
126:
118:
129:
in the eleventh century. The earliest written description of it is
Ptolemy's
125:
in the fourth century, and its equatorial aspect was used by Arab astronomer
1723:
90:
1718:
167:
113:
The stereographic projection was likely known in its polar aspect to the
1579:
34:
Stereographic projection of the world north of 30°S. 15° graticule.
626:
502:
and then transforming points on it via a tenth-order polynomial.
1702:
1499:
1115:
691:
630:
359:
The ellipsoidal form of the polar ellipsoidal projection uses
576:
p.~169. Chicago and London: The
University of Chicago Press.
548:
Snyder, John P. 1987. "Map
Projections---A Working Manual".
574:
Flattening the Earth: Two
Thousand Years of Map Projections
162:, published the first mathematical proof that this map is
447:
427:
407:
331:
311:
291:
191:
1642:
1597:
1588:
1565:
1512:
1455:
1432:
1414:
1374:
1284:
1236:
1213:
1190:
1181:
1128:
1070:
1020:
1007:
962:
934:
851:
842:
742:
713:
704:
552:. United States Geological Survey. 1395: 154--163.
463:
433:
413:
337:
317:
297:
274:
544:
542:
540:
356:must be chosen if greater accuracy is required.
135:, which calls it the "planisphere projection".
95:universal polar stereographic coordinate system
345:are the latitude and longitude, respectively.
642:
8:
568:
566:
166:. He used the recently established tools of
117:, though its invention is often credited to
66:whose use dates back to antiquity. Like the
1699:
1594:
1509:
1496:
1187:
1125:
1112:
1017:
848:
710:
701:
688:
649:
635:
627:
368:Roussilhe oblique stereographic projection
97:uses one such ellipsoidal implementation.
1734:Map projection of the tri-axial ellipsoid
451:
446:
426:
406:
330:
310:
290:
236:
223:
192:
190:
389:
104:
37:
29:
536:
18:Stereographic projection in cartography
7:
464:{\displaystyle -{\frac {\pi }{2}}}
42:The stereographic projection with
25:
305:is the radius of the sphere, and
1677:Quadrilateralized spherical cube
1357:Quadrilateralized spherical cube
507:
498:stereographic projection to the
27:Type of conformal map projection
158:, motivated by his interest in
82:, and when on a sphere, also a
1266:Lambert cylindrical equal-area
60:azimuthal conformal projection
1:
1714:Interruption (map projection)
481:Gall stereographic projection
1352:Lambert azimuthal equal-area
1148:Guyou hemisphere-in-a-square
1138:Adams hemisphere-in-a-square
524:(click for detail)
520:Stereographic map projection
401:Because the expression for
1775:
617:10.1559/152304086783899908
487:stereographic projection.
1709:
1698:
1625:
1508:
1495:
1307:
1124:
1111:
1048:
907:
790:
700:
687:
664:
605:The American Cartographer
494:is formed by mapping the
170:, invented by his friend
572:Snyder, John P. (1993).
434:{\displaystyle \varphi }
338:{\displaystyle \lambda }
318:{\displaystyle \varphi }
76:stereographic projection
64:conformal map projection
52:stereographic projection
1153:Lambert conformal conic
68:orthographic projection
1286:Tobler hyperelliptical
899:Tobler hyperelliptical
825:Space-oblique Mercator
465:
435:
415:
395:
339:
319:
299:
276:
110:
84:perspective projection
56:planisphere projection
47:
35:
1759:Conformal projections
479:The parallels on the
466:
436:
416:
393:
340:
320:
300:
277:
108:
41:
33:
1662:Cahill–Keyes M-shape
1522:Chamberlin trimetric
445:
425:
405:
329:
309:
289:
189:
80:azimuthal projection
54:, also known as the
1729:Tissot's indicatrix
1630:Central cylindrical
1271:Smyth equal-surface
1173:Transverse Mercator
1022:General perspective
777:Smyth equal-surface
729:Transverse Mercator
475:Derived projections
149:Western hemispheres
123:Theon of Alexandria
72:gnomonic projection
44:Tissot's indicatrix
1682:Waterman butterfly
1532:Miller cylindrical
1163:Peirce quincuncial
1058:Lambert equal-area
810:Gall stereographic
550:Professional Paper
518:Comparison of the
461:
431:
411:
396:
361:conformal latitude
335:
315:
295:
272:
270:
111:
48:
36:
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1486:
1370:
1369:
1107:
1106:
1103:
1102:
1066:
1065:
954:Lambert conformal
930:
929:
844:Pseudocylindrical
838:
837:
459:
414:{\displaystyle r}
354:ellipsoidal model
298:{\displaystyle R}
244:
231:
115:ancient Egyptians
16:(Redirected from
1766:
1700:
1657:Cahill Butterfly
1595:
1575:Goode homolosine
1510:
1497:
1462:
1461:(Mecca or Qibla)
1342:Goode homolosine
1188:
1126:
1113:
1018:
1013:
884:Goode homolosine
849:
734:Oblique Mercator
711:
702:
689:
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1584:
1561:
1547:Van der Grinten
1504:
1502:By construction
1483:
1460:
1459:
1451:
1428:
1410:
1391:Equirectangular
1377:
1366:
1303:
1280:
1276:Trystan Edwards
1232:
1209:
1177:
1120:
1099:
1072:Pseudoazimuthal
1062:
1044:
1011:
1010:
1003:
958:
926:
922:Winkel I and II
903:
834:
815:Gall isographic
805:Equirectangular
786:
782:Trystan Edwards
738:
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601:
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492:GS50 projection
477:
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327:
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307:
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287:
286:
269:
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141:Rumold Mercator
103:
46:of deformation.
28:
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15:
12:
11:
5:
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1549:
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1539:
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1529:
1527:Kavrayskiy VII
1524:
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1506:
1505:
1500:
1493:
1492:
1489:
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1457:Retroazimuthal
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1376:Equidistant in
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928:
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912:Kavrayskiy VII
908:
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658:Map projection
656:
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631:
623:
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611:(3): 253–261.
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132:Planisphaerium
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26:
24:
14:
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10:
9:
6:
4:
3:
2:
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1653:
1650:
1649:
1647:
1645:
1641:
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1628:
1627:
1624:
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1617:Stereographic
1615:
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1610:
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1596:
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1591:
1587:
1581:
1578:
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1573:
1572:
1570:
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1564:
1558:
1557:Winkel tripel
1555:
1553:
1550:
1548:
1545:
1543:
1540:
1538:
1537:Natural Earth
1535:
1533:
1530:
1528:
1525:
1523:
1520:
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1511:
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1503:
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1343:
1340:
1338:
1335:
1333:
1330:
1328:
1325:
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1320:
1318:
1317:Briesemeister
1315:
1313:
1310:
1309:
1306:
1300:
1297:
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1292:
1291:
1289:
1287:
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1277:
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1269:
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1257:
1254:
1252:
1249:
1247:
1244:
1243:
1241:
1239:
1235:
1229:
1226:
1224:
1221:
1220:
1218:
1216:
1212:
1206:
1203:
1201:
1198:
1197:
1195:
1193:
1189:
1186:
1184:
1180:
1174:
1171:
1169:
1168:Stereographic
1166:
1164:
1161:
1159:
1156:
1154:
1151:
1149:
1146:
1144:
1141:
1139:
1136:
1135:
1133:
1131:
1127:
1123:
1119:
1114:
1110:
1096:
1095:Winkel tripel
1093:
1091:
1088:
1086:
1083:
1081:
1078:
1077:
1075:
1073:
1069:
1059:
1056:
1054:
1051:
1050:
1047:
1041:
1040:Stereographic
1038:
1036:
1033:
1031:
1028:
1027:
1025:
1023:
1019:
1016:
1014:
1006:
1000:
997:
993:
990:
988:
985:
984:
983:
980:
978:
975:
973:
970:
969:
967:
965:
964:Pseudoconical
961:
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952:
950:
947:
945:
942:
941:
939:
937:
933:
923:
920:
918:
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703:
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629:
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596:
590:
587:
583:
582:0-226-76746-9
579:
575:
569:
567:
563:
559:
558:0-226-76746-9
555:
551:
545:
543:
541:
537:
530:
525:
521:
516:
510:
503:
501:
500:complex plane
497:
493:
488:
486:
482:
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453:
448:
428:
408:
399:
392:
388:
386:
385:small circles
381:
380:great circles
373:
371:
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364:
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292:
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196:
185:
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177:
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165:
161:
157:
156:Edmond Halley
152:
150:
146:
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136:
134:
133:
128:
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116:
107:
100:
98:
96:
92:
87:
85:
81:
77:
73:
69:
65:
61:
57:
53:
45:
40:
32:
19:
1612:Orthographic
1143:Gauss–Krüger
1039:
1035:Orthographic
830:Web Mercator
724:Gauss–Krüger
608:
604:
598:
589:
573:
549:
519:
489:
478:
421:diverges as
400:
397:
387:as circles.
377:
365:
358:
347:
284:
181:
172:Isaac Newton
153:
137:
130:
112:
88:
59:
55:
51:
49:
1590:Perspective
1378:some aspect
1362:Strebe 1995
1337:Equal Earth
1256:Gall–Peters
1238:Cylindrical
1053:Equidistant
949:Equidistant
879:Equal Earth
762:Gall–Peters
706:Cylindrical
441:approaches
160:star charts
1652:AuthaGraph
1644:Polyhedral
1514:Compromise
1442:Loximuthal
1434:Loxodromic
1396:Sinusoidal
1246:Balthasart
1223:Sinusoidal
1200:Sinusoidal
1183:Equal-area
894:Sinusoidal
852:Equal-area
752:Balthasart
744:Equal-area
717:-conformal
694:By surface
531:References
485:transverse
374:Properties
127:Al-Zarkali
119:Hipparchus
1724:Longitude
1552:Wagner VI
1401:Two-point
1332:Eckert VI
1327:Eckert IV
1322:Eckert II
1299:Mollweide
1294:Collignon
1261:Hobo–Dyer
1215:Bottomley
1130:Conformal
1118:By metric
1009:Azimuthal
982:Polyconic
977:Bottomley
917:Wagner VI
889:Mollweide
874:Eckert VI
869:Eckert IV
864:Eckert II
859:Collignon
767:Hobo–Dyer
454:π
449:−
429:φ
333:λ
313:φ
266:λ
256:θ
239:φ
234:−
226:π
216:
164:conformal
154:In 1695,
91:ellipsoid
1753:Category
1719:Latitude
1704:See also
1667:Dymaxion
1607:Gnomonic
1542:Robinson
1447:Mercator
1424:Gnomonic
1416:Gnomonic
1251:Behrmann
1158:Mercator
1030:Gnomonic
1012:(planar)
987:American
757:Behrmann
715:Mercator
178:Formulae
168:calculus
1580:HEALPix
1479:Littrow
1090:Wiechel
992:Chinese
936:Conical
800:Central
795:Cassini
772:Lambert
669:History
496:oblique
145:Eastern
101:History
62:, is a
58:or the
1599:Planar
1567:Hybrid
1474:Hammer
1406:Werner
1347:Hammer
1312:Albers
1228:Werner
1205:Werner
1085:Hammer
1080:Aitoff
999:Werner
944:Albers
820:Miller
679:Portal
580:
556:
350:sphere
285:where
89:On an
78:is an
74:, the
1469:Craig
1386:Conic
1192:Bonne
972:Bonne
1672:ISEA
674:List
578:ISBN
554:ISBN
490:The
348:The
325:and
147:and
70:and
50:The
613:doi
213:tan
1755::
609:13
607:.
565:^
539:^
370:.
174:.
151:.
86:.
650:e
643:t
636:v
619:.
615::
584:.
560:.
457:2
409:r
293:R
263:=
248:)
242:2
229:4
220:(
210:R
207:2
204:=
197:r
20:)
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