39:
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31:
904:
469:
673:
278:
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1118:
In a wide sense, all projections with the point of perspective at infinity (and therefore parallel projecting lines) are considered as orthographic, regardless of the surface onto which they are projected. Such projections distort angles and areas close to the poles.
899:{\displaystyle {\begin{aligned}\varphi &=\arcsin \left(\cos c\sin \varphi _{0}+{\frac {y\sin c\cos \varphi _{0}}{\rho }}\right)\\\lambda &=\lambda _{0}+\arctan \left({\frac {x\sin c}{\rho \cos c\cos \varphi _{0}-y\sin c\sin \varphi _{0}}}\right)\end{aligned}}}
601:
1002:
464:{\displaystyle {\begin{aligned}x&=R\,\cos \varphi \sin \left(\lambda -\lambda _{0}\right)\\y&=R{\big (}\cos \varphi _{0}\sin \varphi -\sin \varphi _{0}\cos \varphi \cos \left(\lambda -\lambda _{0}\right){\big )}\end{aligned}}}
157:
The earliest surviving maps on the projection appear as crude woodcut drawings of terrestrial globes of 1509 (anonymous), 1533 and 1551 (Johannes Schöner), and 1524 and 1551 (Apian). A highly-refined map, designed by
Renaissance
661:
491:
920:
678:
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1969:
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1199:
38:
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2018:
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2139:
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2216:
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1346:
596:{\displaystyle \cos c=\sin \varphi _{0}\sin \varphi +\cos \varphi _{0}\cos \varphi \cos \left(\lambda -\lambda _{0}\right)\,}
66:
1105:
and some azimuthal projections centred on 90° N at the same scale, ordered by projection altitude in Earth radii.
129:
used the projection in the 2nd century BC to determine the places of star-rise and star-set. In about 14 BC, Roman engineer
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2003:
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2008:
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997:{\displaystyle {\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\c&=\arcsin {\frac {\rho }{R}}\end{aligned}}}
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62:
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43:
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2059:
2013:
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485:
of the orthographic projection. This ensures that points on the opposite hemisphere are not plotted:
17:
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The inverse formulas are particularly useful when trying to project a variable defined on a (
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1275:
146:, which also meant a sundial showing latitude and longitude, was the common name until
2370:
74:
2284:
201:
106:
78:
1050:). Direct application of the orthographic projection yields scattered points in (
125:
has been known since antiquity, with its cartographic uses being well documented.
180:
from spacecraft have inspired renewed interest in the orthographic projection in
1296:
1008:
98:
34:
Orthographic projection (equatorial aspect) of eastern hemisphere 30W–150E
140:(= “straight”) and graphē (= “drawing”)) for the projection. However, the name
1085:
See
References for an ellipsoidal version of the orthographic map projection.
126:
110:
1166:
Map
Projections—A Working Manual (US Geologic Survey Professional Paper 1395)
611:) is negative. That is, all points that are included in the mapping satisfy:
2341:
261:
205:
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130:
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Latitudes beyond the range of the map should be clipped by calculating the
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Vitruvius also seems to have devised the term orthographic (from the Greek
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213:
197:
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used the projection to construct sundials and to compute sun positions.
2197:
1074:) projection plane and construct the image from the values defined in (
177:
151:
102:
229:
225:
221:
90:
70:
1214:"Ellipsoidal Orthographic Projection via ECEF and Topocentric (ENU)"
1012:
173:
94:
37:
29:
1244:
1194:
pp. 16–18. Chicago and London: The
University of Chicago Press.
1122:
An example of an orthographic projection onto a cylinder is the
1082:) by using the inverse formulas of the orthographic projection.
2320:
2117:
1733:
1309:
1248:
1169:. Washington, D.C.: US Government Printing Office. pp.
1027:
of the orthographic projection as written is correct in all
656:{\displaystyle -{\frac {\pi }{2}}<c<{\frac {\pi }{2}}}
200:
for the spherical orthographic projection are derived using
1192:
Flattening the Earth: Two
Thousand Years of Map Projections
1158:
1156:
918:
676:
620:
494:
281:
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of the inverse formulas the use of the two-argument
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1992:
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607:The point should be clipped from the map if cos(
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1186:
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264:for the orthographic projection onto the (
2352:Map projection of the tri-axial ellipsoid
1124:Lambert cylindrical equal-area projection
1023:) is recommended. This ensures that the
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272:) tangent plane reduce to the following:
53:has been used since antiquity. Like the
1152:
1141:Stereographic projection in cartography
1114:Orthographic projections onto cylinders
1240:Orthographic Projection—from MathWorld
1066:. One solution is to start from the (
85:for the orthographic projection is at
51:Orthographic projection in cartography
18:Orthographic projection in cartography
67:perspective (or azimuthal) projection
7:
27:Azimuthal perspective map projection
1042:) grid onto a rectilinear grid in (
667:The inverse formulas are given by:
154:promoted its present name in 1613.
25:
42:The orthographic projection with
2295:Quadrilateralized spherical cube
1975:Quadrilateralized spherical cube
1090:
204:. They are written in terms of
113:, particularly near the edges.
1884:Lambert cylindrical equal-area
1058:), which creates problems for
1:
2332:Interruption (map projection)
1970:Lambert azimuthal equal-area
1766:Guyou hemisphere-in-a-square
1756:Adams hemisphere-in-a-square
1107:(click for detail)
109:. The shapes and areas are
1103:Orthographic map projection
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2113:
1925:
1742:
1729:
1666:
1525:
1408:
1318:
1305:
1282:
1212:Zinn, Noel (June 2011).
1190:Snyder, John P. (1993).
1019:function (as opposed to
89:distance. It depicts a
55:stereographic projection
1771:Lambert conformal conic
1136:List of map projections
131:Marcus Vitruvius Pollio
123:orthographic projection
63:orthographic projection
1904:Tobler hyperelliptical
1517:Tobler hyperelliptical
1443:Space-oblique Mercator
1163:Snyder, J. P. (1987).
998:
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47:
35:
1064:numerical integration
999:
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246:) of the projection (
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2280:Cahill–Keyes M-shape
2140:Chamberlin trimetric
916:
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492:
279:
169:, appeared in 1515.
83:point of perspective
73:is projected onto a
2347:Tissot's indicatrix
2248:Central cylindrical
1889:Smyth equal-surface
1791:Transverse Mercator
1640:General perspective
1395:Smyth equal-surface
1347:Transverse Mercator
172:Photographs of the
97:as it appears from
59:gnomonic projection
44:Tissot's indicatrix
2300:Waterman butterfly
2150:Miller cylindrical
1781:Peirce quincuncial
1676:Lambert equal-area
1428:Gall stereographic
1101:Comparison of the
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186:planetary science
16:(Redirected from
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2275:Cahill Butterfly
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2193:Goode homolosine
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2079:(Mecca or Qibla)
1960:Goode homolosine
1806:
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1352:Oblique Mercator
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476:angular distance
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167:Johannes Stabius
165:and executed by
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2165:Van der Grinten
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2009:Equirectangular
1995:
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1894:Trystan Edwards
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1690:Pseudoazimuthal
1680:
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1540:Winkel I and II
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1433:Gall isographic
1423:Equirectangular
1404:
1400:Trystan Edwards
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1994:Equidistant in
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75:tangent plane
72:
69:in which the
68:
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45:
40:
32:
19:
2230:Orthographic
2229:
1761:Gauss–Krüger
1653:Orthographic
1652:
1448:Web Mercator
1342:Gauss–Krüger
1220:. Retrieved
1207:
1191:
1165:
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1117:
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269:
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236:
232:
217:
209:
202:trigonometry
195:
171:
156:
141:
137:
135:
120:
107:great circle
101:, where the
82:
79:secant plane
50:
49:
2208:Perspective
1996:some aspect
1980:Strebe 1995
1955:Equal Earth
1874:Gall–Peters
1856:Cylindrical
1671:Equidistant
1567:Equidistant
1497:Equal Earth
1380:Gall–Peters
1324:Cylindrical
1009:computation
192:Mathematics
99:outer space
2270:AuthaGraph
2262:Polyhedral
2132:Compromise
2060:Loximuthal
2052:Loxodromic
2014:Sinusoidal
1864:Balthasart
1841:Sinusoidal
1818:Sinusoidal
1801:Equal-area
1512:Sinusoidal
1470:Equal-area
1370:Balthasart
1362:Equal-area
1335:-conformal
1312:By surface
1222:2011-11-11
1147:References
176:and other
127:Hipparchus
91:hemisphere
2342:Longitude
2170:Wagner VI
2019:Two-point
1950:Eckert VI
1945:Eckert IV
1940:Eckert II
1917:Mollweide
1912:Collignon
1879:Hobo–Dyer
1833:Bottomley
1748:Conformal
1736:By metric
1627:Azimuthal
1600:Polyconic
1595:Bottomley
1535:Wagner VI
1507:Mollweide
1492:Eckert VI
1487:Eckert IV
1482:Eckert II
1477:Collignon
1385:Hobo–Dyer
1029:quadrants
983:ρ
978:
924:ρ
877:φ
873:
864:
855:−
846:φ
842:
833:
827:ρ
819:
803:
788:λ
777:λ
763:ρ
753:φ
749:
740:
719:φ
715:
706:
695:
682:φ
646:π
627:π
622:−
579:λ
575:−
572:λ
564:
558:φ
555:
543:φ
539:
530:φ
527:
515:φ
511:
499:
481:from the
437:λ
433:−
430:λ
422:
416:φ
413:
401:φ
397:
391:−
388:φ
385:
373:φ
369:
328:λ
324:−
321:λ
313:
307:φ
304:
262:equations
220:) on the
206:longitude
182:astronomy
111:distorted
2371:Category
2337:Latitude
2322:See also
2285:Dymaxion
2225:Gnomonic
2160:Robinson
2065:Mercator
2042:Gnomonic
2034:Gnomonic
1869:Behrmann
1776:Mercator
1648:Gnomonic
1630:(planar)
1605:American
1375:Behrmann
1333:Mercator
1130:See also
1060:plotting
260:). The
235:and the
214:latitude
198:formulas
160:polymath
143:analemma
87:infinite
2198:HEALPix
2097:Littrow
1708:Wiechel
1610:Chinese
1554:Conical
1418:Central
1413:Cassini
1390:Lambert
1287:History
228:of the
178:planets
152:Antwerp
117:History
103:horizon
93:of the
2217:Planar
2185:Hybrid
2092:Hammer
2024:Werner
1965:Hammer
1930:Albers
1846:Werner
1823:Werner
1703:Hammer
1698:Aitoff
1617:Werner
1562:Albers
1438:Miller
1297:Portal
1198:
1080:φ
1076:λ
1040:φ
1036:λ
975:arcsin
909:where
800:arctan
692:arcsin
483:center
255:φ
248:λ
244:origin
237:center
230:sphere
226:radius
222:sphere
218:φ
212:) and
210:λ
138:orthos
81:. The
71:sphere
2087:Craig
2004:Conic
1810:Bonne
1590:Bonne
1217:(PDF)
1173:–153.
1013:atan2
242:(and
240:point
174:Earth
105:is a
95:globe
65:is a
2290:ISEA
1292:List
1196:ISBN
1062:and
1025:sign
1021:atan
1007:For
641:<
635:<
196:The
184:and
121:The
57:and
1171:145
870:sin
861:sin
839:cos
830:cos
816:sin
746:cos
737:sin
712:sin
703:cos
561:cos
552:cos
536:cos
524:sin
508:sin
496:cos
419:cos
410:cos
394:sin
382:sin
366:cos
310:sin
301:cos
150:of
77:or
2373::
1179:^
1155:^
1126:.
1078:,
1070:,
1054:,
1046:,
1038:,
1031:.
268:,
253:,
188:.
61:,
1268:e
1261:t
1254:v
1225:.
1202:.
1072:y
1068:x
1056:y
1052:x
1048:y
1044:x
986:R
972:=
965:c
954:2
950:y
946:+
941:2
937:x
931:=
889:)
881:0
867:c
858:y
850:0
836:c
822:c
813:x
807:(
797:+
792:0
784:=
769:)
757:0
743:c
734:y
728:+
723:0
709:c
699:(
689:=
663:.
649:2
638:c
630:2
609:c
603:.
589:)
583:0
568:(
547:0
533:+
519:0
505:=
502:c
479:c
453:)
447:)
441:0
426:(
405:0
377:0
361:(
356:R
353:=
346:y
338:)
332:0
317:(
297:R
294:=
287:x
270:y
266:x
258:0
251:0
233:R
216:(
208:(
20:)
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