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William A. Briesemeister presented a variant of the Hammer in 1953. In this version, the central meridian is set to 10°E, the coordinate system is rotated to bring the 45°N parallel to the center, and the resulting map is squashed horizontally and reciprocally stretched vertically to achieve a 7:4
81:
647:
414:{\displaystyle {\begin{aligned}x&={\frac {2{\sqrt {2}}\cos \varphi \sin {\frac {\lambda }{2}}}{\sqrt {1+\cos \varphi \cos {\frac {\lambda }{2}}}}}\\y&={\frac {{\sqrt {2}}\sin \varphi }{\sqrt {1+\cos \varphi \cos {\frac {\lambda }{2}}}}}\end{aligned}}}
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210:{\displaystyle {\begin{aligned}x&=\operatorname {laea} _{x}\left({\frac {\lambda }{2}},\varphi \right)\\y&={\tfrac {1}{2}}\operatorname {laea} _{y}\left({\frac {\lambda }{2}},\varphi \right)\end{aligned}}}
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rotated the coordinate system to bring the 45° north parallel to the center, leaving the prime meridian as the central meridian. He called this variant the "Nordic" projection.
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is another equal-area projection of similar aspect, though with straight parallels of latitude, unlike the Hammer's curved parallels.
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Visually, the Aitoff and Hammer projections are very similar. The Hammer has seen more use because of its equal-area property. The
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aspect ratio instead of the 2:1 of the Hammer. The purpose is to present the land masses more centrally and with lower distortion.
642:{\displaystyle {\begin{aligned}\lambda &=2\arctan {\frac {zx}{2\left(2z^{2}-1\right)}}\\\varphi &=\arcsin zy\end{aligned}}}
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55:, Hammer intended to reduce distortion in the regions of the outer meridians, where it is extreme in the Mollweide.
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520:{\displaystyle z\equiv {\sqrt {1-\left({\tfrac {1}{4}}x\right)^{2}-\left({\tfrac {1}{2}}y\right)^{2}}}}
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components of the equatorial
Lambert azimuthal equal-area projection. Written out explicitly:
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761:"Hammer–Aitoff Equal-Area Projection." From MathWorld—A Wolfram Web Resource
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Flattening the Earth: Two
Thousand Years of Map Projections
51:
in 1892. Using the same 2:1 elliptical outer shape as the
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The inverse is calculated with the intermediate variable
67:, Hammer suggested the use of the equatorial form of the
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The longitude and latitudes can then be calculated by
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742:, John P. Snyder, 1993, pp. 130–133,
1977:Map projection of the tri-axial ellipsoid
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16:Pseudoazimuthal equal-area map projection
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69:Lambert azimuthal equal-area projection
774:"A new oblique equal-area projection"
772:Briesemeister, William (April 1953).
7:
825:. Professional Paper 1453. Denver:
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14:
1920:Quadrilateralized spherical cube
1600:Quadrilateralized spherical cube
73:azimuthal equidistant projection
71:instead of Aitoff's use of the
1509:Lambert cylindrical equal-area
721:Eckert-Greifendorff projection
660:from the central meridian and
23:Hammer projection of the world
1:
1957:Interruption (map projection)
692:Before projecting to Hammer,
1595:Lambert azimuthal equal-area
1391:Guyou hemisphere-in-a-square
1381:Adams hemisphere-in-a-square
865:Table of common projections
822:An Album of Map Projections
31:The Hammer projection with
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63:Directly inspired by the
1396:Lambert conformal conic
706:List of map projections
2002:Equal-area projections
1529:Tobler hyperelliptical
1142:Tobler hyperelliptical
1068:Space-oblique Mercator
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1905:Cahill–Keyes M-shape
1765:Chamberlin trimetric
711:Mollweide projection
673:Mollweide projection
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53:Mollweide projection
1972:Tissot's indicatrix
1873:Central cylindrical
1514:Smyth equal-surface
1416:Transverse Mercator
1265:General perspective
1020:Smyth equal-surface
972:Transverse Mercator
778:Geographical Review
759:Weisstein, Eric W.
33:Tissot's indicatrix
1925:Waterman butterfly
1775:Miller cylindrical
1406:Peirce quincuncial
1301:Lambert equal-area
1053:Gall stereographic
817:Voxland, Philip M.
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65:Aitoff projection
43:is an equal-area
41:Hammer projection
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1818:Goode homolosine
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1704:(Mecca or Qibla)
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839:. Archived from
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1025:Trystan Edwards
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829:. p. 162.
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1770:Kavrayskiy VII
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859:External links
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836:978-0160033681
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881:
876:
875:
872:
866:
863:
862:
858:
843:on 2010-07-01
842:
838:
832:
828:
824:
823:
818:
814:
808:
806:
802:
791:
787:
783:
779:
775:
768:
765:
762:
756:
753:
749:
748:0-226-76747-7
745:
741:
736:
733:
726:
722:
719:
717:
714:
712:
709:
707:
704:
703:
699:
697:
695:
687:
685:
679:Briesemeister
678:
676:
674:
669:
667:
663:
659:
655:
632:
629:
626:
623:
620:
618:
613:
602:
598:
595:
590:
586:
582:
578:
574:
569:
566:
560:
557:
554:
551:
549:
544:
533:
532:
531:
510:
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501:
495:
492:
485:
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475:
470:
466:
460:
457:
450:
445:
442:
437:
434:
427:
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379:
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368:
365:
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357:
349:
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342:
330:
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297:
294:
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254:
243:
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239:
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199:
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169:
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158:
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134:
130:
127:
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104:
100:
96:
94:
89:
78:
77:
76:
74:
70:
66:
58:
56:
54:
50:
47:described by
46:
42:
34:
29:
21:
1855:Orthographic
1589:
1386:Gauss–Krüger
1327:
1278:Orthographic
1073:Web Mercator
967:Gauss–Krüger
845:. Retrieved
841:the original
821:
793:. Retrieved
781:
777:
767:
755:
739:
735:
691:
682:
670:
661:
653:
651:
529:
423:
237:
233:
228:
222:
219:
62:
49:Ernst Hammer
40:
38:
1833:Perspective
1621:some aspect
1605:Strebe 1995
1580:Equal Earth
1499:Gall–Peters
1481:Cylindrical
1296:Equidistant
1192:Equidistant
1122:Equal Earth
1005:Gall–Peters
949:Cylindrical
59:Development
1895:AuthaGraph
1887:Polyhedral
1757:Compromise
1685:Loximuthal
1677:Loxodromic
1639:Sinusoidal
1489:Balthasart
1466:Sinusoidal
1443:Sinusoidal
1426:Equal-area
1137:Sinusoidal
1095:Equal-area
995:Balthasart
987:Equal-area
960:-conformal
937:By surface
847:2018-03-29
795:2024-01-18
727:References
220:where laea
1967:Longitude
1795:Wagner VI
1644:Two-point
1575:Eckert VI
1570:Eckert IV
1565:Eckert II
1542:Mollweide
1537:Collignon
1504:Hobo–Dyer
1458:Bottomley
1373:Conformal
1361:By metric
1252:Azimuthal
1225:Polyconic
1220:Bottomley
1160:Wagner VI
1132:Mollweide
1117:Eckert VI
1112:Eckert IV
1107:Eckert II
1102:Collignon
1010:Hobo–Dyer
658:longitude
627:
614:φ
596:−
561:
545:λ
481:−
446:−
438:≡
397:λ
392:
386:φ
383:
369:φ
366:
328:λ
323:
317:φ
314:
295:λ
290:
284:φ
281:
196:φ
185:λ
175:
131:φ
120:λ
110:
1996:Category
1962:Latitude
1947:See also
1910:Dymaxion
1850:Gnomonic
1785:Robinson
1690:Mercator
1667:Gnomonic
1659:Gnomonic
1494:Behrmann
1401:Mercator
1273:Gnomonic
1255:(planar)
1230:American
1000:Behrmann
958:Mercator
819:(1989).
700:See also
666:latitude
232:are the
226:and laea
1823:HEALPix
1722:Littrow
1333:Wiechel
1235:Chinese
1179:Conical
1043:Central
1038:Cassini
1015:Lambert
912:History
664:is the
656:is the
1842:Planar
1810:Hybrid
1717:Hammer
1649:Werner
1590:Hammer
1555:Albers
1471:Werner
1448:Werner
1328:Hammer
1323:Aitoff
1242:Werner
1187:Albers
1063:Miller
922:Portal
833:
746:
688:Nordic
652:where
624:arcsin
558:arctan
1712:Craig
1629:Conic
1435:Bonne
1215:Bonne
1915:ISEA
917:List
831:ISBN
827:USGS
744:ISBN
236:and
166:laea
101:laea
39:The
786:doi
389:cos
380:cos
363:sin
320:cos
311:cos
287:sin
278:cos
1998::
815:;
804:^
782:43
780:.
776:.
668:.
75::
893:e
886:t
879:v
850:.
798:.
788::
750:.
662:φ
654:λ
633:y
630:z
621:=
603:)
599:1
591:2
587:z
583:2
579:(
575:2
570:x
567:z
555:2
552:=
511:2
506:)
502:y
496:2
493:1
486:(
476:2
471:)
467:x
461:4
458:1
451:(
443:1
435:z
400:2
377:+
374:1
358:2
350:=
343:y
331:2
308:+
305:1
298:2
273:2
268:2
262:=
255:x
238:y
234:x
229:y
223:x
200:)
193:,
188:2
179:(
170:y
159:2
156:1
150:=
143:y
135:)
128:,
123:2
114:(
105:x
97:=
90:x
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