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866:{\displaystyle {\frac {\partial y}{\partial \varphi }}\cdot {\frac {\partial x}{\partial \lambda }}-{\frac {\partial y}{\partial \lambda }}\cdot {\frac {\partial x}{\partial \varphi }}=R\cdot R\cdot \cos \varphi -0\cdot (-R\cdot \lambda \cdot \sin \varphi )=R^{2}\cdot \cos \varphi =s\cdot \cos \varphi }
74:
measure between any and all map regions. Equivalent projections are widely used for thematic maps showing scenario distribution such as population, farmland distribution, forested areas, and so forth, because an equal-area map does not change apparent density of the phenomenon being mapped.
89:. This implies that an equal-area projection inevitably distorts shapes. Even though a point or points or a path or paths on a map might have no distortion, the greater the area of the region being mapped, the greater and more obvious the distortion of shapes inevitably becomes.
115:
1128:{\displaystyle {\frac {\partial y}{\partial \varphi }}\cdot {\frac {\partial x}{\partial \lambda }}-{\frac {\partial y}{\partial \lambda }}\cdot {\frac {\partial x}{\partial \varphi }}=s\cdot \cos \varphi \cdot {\frac {(1-e^{2})}{(1-e^{2}\sin ^{2}\varphi )^{2}}}}
648:{\displaystyle {\frac {\partial x}{\partial \varphi }}=-R\cdot \lambda \cdot \sin \varphi ,\quad R\cdot {\frac {\partial x}{\partial \lambda }}=R\cdot \cos \varphi ,\quad {\frac {\partial y}{\partial \varphi }}=R,\quad {\frac {\partial y}{\partial \lambda }}=0}
455:
229:{\displaystyle {\frac {\partial y}{\partial \varphi }}\cdot {\frac {\partial x}{\partial \lambda }}-{\frac {\partial y}{\partial \lambda }}\cdot {\frac {\partial x}{\partial \varphi }}=s\cdot \cos \varphi }
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1497:. USGS Professional Paper. Vol. 1395. Washington: United States Government Printing Office. p. 28.
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450:{\displaystyle {\begin{aligned}x&=R\cdot \lambda \cos \varphi \\y&=R\cdot \varphi \end{aligned}}}
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Brodzik, Mary J.; Billingsley, Brendan; Haran, Terry; Raup, Bruce; Savoie, Matthew H. (13 March 2012).
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In order for a map projection of the sphere to be equal-area, its generating formulae must meet this
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1601:"EASE-Grid 2.0: Incremental but Significant Improvements for Earth-Gridded Data Sets"
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Lambert azimuthal equal-area projection of the world centered on 0° N 0° E.
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32:"Area-preserving maps" redirects here. For the mathematical concept, see
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Albers projection of the world with standard parallels 20° N and 50° N.
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378:
is a very simple equal-area projection. Its generating formulae are:
1209:(global or local) of an equal-area surface representation, used for
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1351:
Equal Earth projection, an equal-area pseudocylindrical projection
1279:
Bottomley projection of the world with standard parallel at 30° N.
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1660:
1214:
1542:
http://scorus.org/wp-content/uploads/2012/10/2010JurmalaP4.5.pdf
930:, the corresponding differential condition that must be met is:
480:
is the radius of the globe. Computing the partial derivatives,
1580:. In Annoni, Alessandro; Luzet, Claude; Gubler, Erich (eds.).
1578:"An Equal Area Projection for Statistical Mapping in the EU"
1598:
71:
1641:"McBryde-Thomas Flat-Polar Quartic Projection - MATLAB"
1309:
Lambert cylindrical equal-area projection of the world
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339:are the projected (planar) coordinates for a given
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1229:These are some projections that preserve area:
1605:ISPRS International Journal of Geo-Information
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1417:McBryde-Thomas Flat-Polar Quartic Projection
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1669:
1552:IBGE (2016), "Grade EstatĂstica". Arquivo
2768:Map projection of the tri-axial ellipsoid
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1205:The term "statistical grid" refers to a
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38:
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1588:, European Commission. pp. 50–55.
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259:is constant throughout the map. Here,
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85:, an equal-area projection cannot be
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1465:Measure-preserving dynamical system
1268:Lambert equal-area conic projection
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364:{\displaystyle (\varphi ,\lambda )}
34:Measure-preserving dynamical system
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1494:Map projections — A working manual
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1517:"INSPIRE helpdesk | INSPIRE"
1317:(with latitude of no distortion)
896:taking the value of the constant
2711:Quadrilateralized spherical cube
2391:Quadrilateralized spherical cube
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1162:of the ellipsoid of revolution.
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2300:Lambert cylindrical equal-area
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1321:Lambert cylindrical equal-area
1225:List of equal-area projections
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2748:Interruption (map projection)
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926:For an equal-area map of the
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2386:Lambert azimuthal equal-area
2182:Guyou hemisphere-in-a-square
2172:Adams hemisphere-in-a-square
1431:Snyder equal-area projection
1237:Lambert azimuthal equal-area
1219:statistical spatial analysis
7:
1576:Tsoulos, Lysandros (2003).
1460:Equiareal map (mathematics)
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10:
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1582:Map projections for Europe
299:represents longitude; and
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1491:Snyder, John P. (1987).
292:{\displaystyle \lambda }
272:{\displaystyle \varphi }
70:that preserves relative
2187:Lambert conformal conic
1563:2 December 2019 at the
2793:Equal-area projections
2320:Tobler hyperelliptical
1933:Tobler hyperelliptical
1859:Space-oblique Mercator
1403:Tobler hyperelliptical
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27:Type of map projection
1611:(1). MDPI AG: 32–45.
1586:Joint Research Centre
1554:grade_estatistica.pdf
1470:Geodesic polygon area
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916:{\displaystyle R^{2}}
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376:sinusoidal projection
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279:represents latitude;
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64:equal-area projection
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2696:Cahill–Keyes M-shape
2556:Chamberlin trimetric
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45:Mollweide projection
2763:Tissot's indicatrix
2664:Central cylindrical
2305:Smyth equal-surface
2207:Transverse Mercator
2056:General perspective
1811:Smyth equal-surface
1763:Transverse Mercator
1618:10.3390/ijgi1010032
1413:Eckert-Greifendorff
2716:Waterman butterfly
2566:Miller cylindrical
2197:Peirce quincuncial
2092:Lambert equal-area
1844:Gall stereographic
1523:on 22 January 2021
1388:Goode's homolosine
1356:Pseudocylindrical
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1211:data visualization
1185:. You can help by
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1988:Lambert conformal
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1878:Pseudocylindrical
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1645:www.mathworks.com
1450:Authalic latitude
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473:{\displaystyle R}
374:For example, the
371:coordinate pair.
332:{\displaystyle y}
312:{\displaystyle x}
252:{\displaystyle s}
206:
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109:-like condition:
83:Theorema Egregium
16:(Redirected from
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2691:Cahill Butterfly
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2609:Goode homolosine
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2376:Goode homolosine
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1918:Goode homolosine
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1519:. Archived from
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1406:
1405:
1400:
1395:
1390:
1385:
1380:
1367:
1362:
1345:
1344:
1343:
1342:
1336:
1330:
1324:
1303:
1302:
1301:
1300:
1295:
1290:
1284:Pseudoconical
1273:
1272:
1271:
1270:
1265:
1248:
1247:
1246:
1245:
1239:
1226:
1223:
1201:
1200:
1180:
1178:
1167:
1164:
1147:
1136:
1135:
1119:
1115:
1111:
1108:
1103:
1099:
1093:
1089:
1085:
1082:
1079:
1074:
1069:
1065:
1061:
1058:
1055:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1025:
1022:
1017:
1014:
1008:
1002:
999:
994:
991:
985:
979:
976:
971:
968:
962:
956:
953:
948:
945:
910:
906:
885:
874:
873:
862:
859:
856:
853:
850:
847:
844:
841:
838:
835:
830:
826:
822:
819:
816:
813:
810:
807:
804:
801:
798:
795:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
762:
759:
753:
750:
745:
742:
736:
730:
727:
722:
719:
713:
707:
704:
699:
696:
690:
684:
681:
676:
673:
656:
655:
644:
641:
635:
632:
627:
624:
617:
614:
611:
605:
602:
597:
594:
587:
584:
581:
578:
575:
572:
569:
563:
560:
555:
552:
546:
543:
539:
536:
533:
530:
527:
524:
521:
518:
515:
512:
506:
503:
498:
495:
469:
458:
457:
442:
439:
436:
433:
430:
428:
426:
423:
422:
419:
416:
413:
410:
407:
404:
401:
398:
396:
394:
391:
390:
360:
357:
354:
351:
348:
328:
308:
288:
268:
248:
237:
236:
225:
222:
219:
216:
213:
210:
204:
201:
196:
193:
187:
181:
178:
173:
170:
164:
158:
155:
150:
147:
141:
135:
132:
127:
124:
107:Cauchy-Riemann
102:
99:
68:map projection
26:
18:Equal-area map
9:
6:
4:
3:
2:
2810:
2799:
2796:
2794:
2791:
2790:
2788:
2769:
2766:
2764:
2761:
2759:
2756:
2754:
2751:
2749:
2746:
2745:
2742:
2735:
2731:
2717:
2714:
2712:
2709:
2707:
2704:
2702:
2699:
2697:
2694:
2692:
2689:
2687:
2684:
2683:
2681:
2679:
2675:
2665:
2662:
2661:
2658:
2652:
2651:Stereographic
2649:
2647:
2644:
2642:
2639:
2638:
2636:
2634:
2630:
2627:
2625:
2621:
2615:
2612:
2610:
2607:
2606:
2604:
2602:
2598:
2592:
2591:Winkel tripel
2589:
2587:
2584:
2582:
2579:
2577:
2574:
2572:
2571:Natural Earth
2569:
2567:
2564:
2562:
2559:
2557:
2554:
2553:
2551:
2549:
2545:
2541:
2537:
2532:
2528:
2514:
2511:
2509:
2506:
2504:
2501:
2500:
2498:
2492:
2488:
2482:
2479:
2477:
2474:
2473:
2471:
2469:
2465:
2459:
2456:
2455:
2453:
2451:
2447:
2441:
2438:
2436:
2433:
2431:
2428:
2426:
2423:
2421:
2418:
2417:
2415:
2413:
2407:
2397:
2394:
2392:
2389:
2387:
2384:
2382:
2379:
2377:
2374:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2351:Briesemeister
2349:
2347:
2344:
2343:
2340:
2334:
2331:
2329:
2326:
2325:
2323:
2321:
2317:
2311:
2308:
2306:
2303:
2301:
2298:
2296:
2293:
2291:
2288:
2286:
2283:
2281:
2278:
2277:
2275:
2273:
2269:
2263:
2260:
2258:
2255:
2254:
2252:
2250:
2246:
2240:
2237:
2235:
2232:
2231:
2229:
2227:
2223:
2220:
2218:
2214:
2208:
2205:
2203:
2202:Stereographic
2200:
2198:
2195:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2173:
2170:
2169:
2167:
2165:
2161:
2157:
2153:
2148:
2144:
2130:
2129:Winkel tripel
2127:
2125:
2122:
2120:
2117:
2115:
2112:
2111:
2109:
2107:
2103:
2093:
2090:
2088:
2085:
2084:
2081:
2075:
2074:Stereographic
2072:
2070:
2067:
2065:
2062:
2061:
2059:
2057:
2053:
2050:
2048:
2040:
2034:
2031:
2027:
2024:
2022:
2019:
2018:
2017:
2014:
2012:
2009:
2007:
2004:
2003:
2001:
1999:
1998:Pseudoconical
1995:
1989:
1986:
1984:
1981:
1979:
1976:
1975:
1973:
1971:
1967:
1957:
1954:
1952:
1949:
1947:
1944:
1943:
1940:
1934:
1931:
1929:
1926:
1924:
1921:
1919:
1916:
1914:
1911:
1909:
1906:
1904:
1901:
1899:
1896:
1894:
1891:
1890:
1888:
1884:
1881:
1879:
1875:
1865:
1862:
1860:
1857:
1855:
1852:
1850:
1847:
1845:
1842:
1840:
1837:
1835:
1832:
1830:
1827:
1826:
1823:
1817:
1814:
1812:
1809:
1807:
1804:
1802:
1799:
1797:
1794:
1792:
1789:
1787:
1784:
1783:
1781:
1779:
1775:
1769:
1766:
1764:
1761:
1759:
1756:
1755:
1753:
1750:
1746:
1743:
1741:
1737:
1733:
1729:
1724:
1720:
1714:
1711:
1709:
1706:
1704:
1701:
1700:
1697:
1693:
1686:
1681:
1679:
1674:
1672:
1667:
1666:
1663:
1646:
1642:
1636:
1628:
1624:
1619:
1614:
1610:
1606:
1602:
1595:
1587:
1583:
1579:
1572:
1566:
1562:
1559:
1549:
1543:
1538:
1522:
1518:
1512:
1504:
1500:
1496:
1495:
1487:
1485:
1480:
1471:
1468:
1466:
1463:
1461:
1458:
1456:
1453:
1451:
1448:
1447:
1436:
1432:
1429:
1427:
1424:
1422:
1419:
1416:
1414:
1411:
1410:
1408:
1404:
1401:
1399:
1396:
1394:
1391:
1389:
1386:
1384:
1381:
1379:
1375:
1371:
1368:
1366:
1363:
1361:
1358:
1357:
1355:
1354:
1349:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1318:
1316:
1313:
1312:
1307:
1299:
1296:
1294:
1291:
1289:
1286:
1285:
1283:
1282:
1277:
1269:
1266:
1264:
1261:
1260:
1258:
1257:
1252:
1243:
1240:
1238:
1235:
1234:
1232:
1231:
1230:
1222:
1220:
1216:
1212:
1208:
1207:discrete grid
1197:
1188:
1184:
1181:This section
1179:
1176:
1172:
1171:
1163:
1161:
1145:
1117:
1109:
1106:
1101:
1097:
1091:
1087:
1083:
1080:
1067:
1063:
1059:
1056:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1023:
1015:
1006:
1000:
992:
983:
977:
969:
960:
954:
946:
933:
932:
931:
929:
924:
908:
904:
883:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
828:
824:
820:
814:
811:
808:
805:
802:
799:
796:
793:
787:
784:
781:
778:
775:
772:
769:
766:
763:
760:
757:
751:
743:
734:
728:
720:
711:
705:
697:
688:
682:
674:
661:
660:
659:
642:
639:
633:
625:
615:
612:
609:
603:
595:
585:
582:
579:
576:
573:
570:
567:
561:
553:
544:
541:
537:
534:
531:
528:
525:
522:
519:
516:
513:
510:
504:
496:
483:
482:
481:
467:
440:
437:
434:
431:
429:
424:
417:
414:
411:
408:
405:
402:
399:
397:
392:
381:
380:
379:
377:
372:
355:
352:
349:
326:
306:
286:
266:
246:
223:
220:
217:
214:
211:
208:
202:
194:
185:
179:
171:
162:
156:
148:
139:
133:
125:
112:
111:
110:
108:
94:
90:
88:
84:
81:
76:
73:
69:
65:
61:
57:
53:
46:
41:
35:
30:
19:
2646:Orthographic
2216:
2177:Gauss–Krüger
2069:Orthographic
1864:Web Mercator
1758:Gauss–Krüger
1648:. Retrieved
1644:
1635:
1608:
1604:
1594:
1581:
1571:
1548:
1537:
1525:. Retrieved
1521:the original
1511:
1493:
1228:
1204:
1191:
1187:adding to it
1182:
1160:eccentricity
1137:
925:
875:
657:
459:
373:
238:
104:
77:
63:
59:
55:
49:
29:
2624:Perspective
2412:some aspect
2396:Strebe 1995
2371:Equal Earth
2290:Gall–Peters
2272:Cylindrical
2087:Equidistant
1983:Equidistant
1913:Equal Earth
1796:Gall–Peters
1740:Cylindrical
1433:, used for
1426:Strebe 1995
1383:Equal Earth
1339:Gall–Peters
1315:Cylindrical
101:Description
52:cartography
2787:Categories
2686:AuthaGraph
2678:Polyhedral
2548:Compromise
2476:Loximuthal
2468:Loxodromic
2430:Sinusoidal
2280:Balthasart
2257:Sinusoidal
2234:Sinusoidal
2217:Equal-area
1928:Sinusoidal
1886:Equal-area
1786:Balthasart
1778:Equal-area
1751:-conformal
1728:By surface
1558:Censo 2010
1527:1 December
1476:References
1398:Sinusoidal
1233:Azimuthal
1194:April 2020
56:equivalent
2758:Longitude
2586:Wagner VI
2435:Two-point
2366:Eckert VI
2361:Eckert IV
2356:Eckert II
2333:Mollweide
2328:Collignon
2295:Hobo–Dyer
2249:Bottomley
2164:Conformal
2152:By metric
2043:Azimuthal
2016:Polyconic
2011:Bottomley
1951:Wagner VI
1923:Mollweide
1908:Eckert VI
1903:Eckert IV
1898:Eckert II
1893:Collignon
1801:Hobo–Dyer
1650:3 January
1627:2220-9964
1393:Mollweide
1370:Eckert II
1365:Collignon
1333:Hobo–Dyer
1293:Bottomley
1110:φ
1107:
1084:−
1060:−
1048:⋅
1045:φ
1042:
1036:⋅
1024:φ
1021:∂
1013:∂
1007:⋅
1001:λ
998:∂
990:∂
984:−
978:λ
975:∂
967:∂
961:⋅
955:φ
952:∂
944:∂
928:ellipsoid
861:φ
858:
852:⋅
843:φ
840:
834:⋅
815:φ
812:
806:⋅
803:λ
800:⋅
794:−
788:⋅
782:−
779:φ
776:
770:⋅
764:⋅
752:φ
749:∂
741:∂
735:⋅
729:λ
726:∂
718:∂
712:−
706:λ
703:∂
695:∂
689:⋅
683:φ
680:∂
672:∂
634:λ
631:∂
623:∂
604:φ
601:∂
593:∂
583:φ
580:
574:⋅
562:λ
559:∂
551:∂
545:⋅
535:φ
532:
526:⋅
523:λ
520:⋅
514:−
505:φ
502:∂
494:∂
441:φ
438:⋅
418:φ
415:
409:λ
406:⋅
356:λ
350:φ
287:λ
267:φ
224:φ
221:
215:⋅
203:φ
200:∂
192:∂
186:⋅
180:λ
177:∂
169:∂
163:−
157:λ
154:∂
146:∂
140:⋅
134:φ
131:∂
123:∂
87:conformal
2753:Latitude
2738:See also
2701:Dymaxion
2641:Gnomonic
2576:Robinson
2481:Mercator
2458:Gnomonic
2450:Gnomonic
2285:Behrmann
2192:Mercator
2064:Gnomonic
2046:(planar)
2021:American
1791:Behrmann
1749:Mercator
1561:Archived
1444:See also
1335:(37°30′)
1327:Behrmann
60:authalic
2614:HEALPix
2513:Littrow
2124:Wiechel
2026:Chinese
1970:Conical
1834:Central
1829:Cassini
1806:Lambert
1703:History
1242:Wiechel
1215:geocode
1158:is the
658:and so
80:Gauss's
2633:Planar
2601:Hybrid
2508:Hammer
2440:Werner
2381:Hammer
2346:Albers
2262:Werner
2239:Werner
2119:Hammer
2114:Aitoff
2033:Werner
1978:Albers
1854:Miller
1713:Portal
1625:
1421:Hammer
1409:Other
1298:Werner
1263:Albers
1259:Conic
1138:where
460:where
239:where
2503:Craig
2420:Conic
2226:Bonne
2006:Bonne
1341:(45°)
1329:(30°)
1288:Bonne
876:with
66:is a
62:, or
54:, an
2706:ISEA
1708:List
1652:2024
1623:ISSN
1529:2019
1376:and
1323:(0°)
1217:and
319:and
72:area
1613:doi
1499:doi
1189:.
1098:sin
1039:cos
855:cos
837:cos
809:sin
773:cos
577:cos
529:sin
412:cos
218:cos
78:By
50:In
2789::
1643:.
1621:.
1607:.
1603:.
1584:.
1483:^
1378:VI
1374:IV
1372:,
1221:.
1213:,
923:.
58:,
1684:e
1677:t
1670:v
1654:.
1629:.
1615::
1609:1
1531:.
1505:.
1501::
1437:.
1196:)
1192:(
1146:e
1118:2
1114:)
1102:2
1092:2
1088:e
1081:1
1078:(
1073:)
1068:2
1064:e
1057:1
1054:(
1033:s
1030:=
1016:x
993:y
970:x
947:y
909:2
905:R
884:s
849:s
846:=
829:2
825:R
821:=
818:)
797:R
791:(
785:0
767:R
761:R
758:=
744:x
721:y
698:x
675:y
643:0
640:=
626:y
616:,
613:R
610:=
596:y
586:,
571:R
568:=
554:x
542:R
538:,
517:R
511:=
497:x
468:R
435:R
432:=
425:y
403:R
400:=
393:x
359:)
353:,
347:(
327:y
307:x
247:s
212:s
209:=
195:x
172:y
149:x
126:y
36:.
20:)
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