134:
1417:
122:
1051:
513:
25:
1278:
699:
820:
305:
1416:
153:. When the Earth is photographed from space, the camera records the view as a perspective projection. When the camera is aimed toward the center of the Earth, the resulting projection is called Vertical Perspective. When aimed in other directions, the resulting projection is called a Tilted Perspective.
214:
Space exploration led to a renewed interest in the perspective projection. Now the concern was for a pictorial view from space, not for minimal distortion. A picture taken with a hand-held camera from the window of a spacecraft has a tilted vertical perspective, so the crewed Gemini and Apollo space
188:
When tilted, the
General Perspective projection, also called the tilted perspective projection, is not azimuthal (see second figure below); directions are not true from the central point, and the projection plane is not tangent to the sphere. Tilted perspectives are common from aerial and low orbit
210:
Some forms of the projection were known to the Greeks and
Egyptians 2,000 years ago. It was studied by several French and British scientists in the 18th and 19th centuries. However, the projection had little practical value at that time; computationally simpler nonperspective azimuthal projections
1062:
1046:{\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}}}
524:
201:
show the globe as it appears from space. These applications permit a wide variety of interactive pan and zoom operations, including fly-through simulations, mimicking pictures or movies taken with a hand-held camera from an airplane or spacecraft.
508:{\displaystyle {\begin{aligned}x&=R\,k'\cos \varphi \sin \left(\lambda -\lambda _{0}\right)\\y&=R\,k'{\big (}\cos \varphi _{0}\sin \varphi -\sin \varphi _{0}\cos \varphi \cos \left(\lambda -\lambda _{0}\right){\big )}\\\end{aligned}}}
185:, or vantage point, for the General Perspective Projection is at a finite distance. It depicts the earth as it appears from some relatively short distance above the surface, typically a few hundred to a few tens of thousands of kilometers.
1439:
747:
denotes the distance from the perspective point to the center of earth. It is positive in the direction of the center of the projection (for the “view from space”) and negative in the opposite direction. For the
1273:{\displaystyle {\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\c&=\arcsin {\frac {P-{\sqrt {1-{\frac {\rho ^{2}(P+1)}{R^{2}(P-1)}}}}}{{\frac {R(P-1)}{\rho }}+{\frac {\rho }{R(P-1)}}}}\end{aligned}}}
1067:
825:
529:
310:
694:{\displaystyle {\begin{aligned}k'&={\frac {d-R}{d-R\cos c}}\\\cos c&=\sin \varphi _{0}\sin \varphi +\cos \varphi _{0}\cos \varphi \cos \left(\lambda -\lambda _{0}\right)\end{aligned}}}
189:
photography, generally taken from at a height measured in kilometers to hundreds of kilometers, rather than the hundreds or thousands of kilometers typical of a vertical perspective. However,
193:
pioneered the use of this projection on strategic maps showing military theaters during WWII. Some prominent
Internet mapping tools also use the tilted perspective projection. For example,
1876:
1364:
1321:
42:
779:
809:
1384:
1301:
745:
721:
181:
projections, meaning that the projection surface is a plane tangent to the sphere. This results in correct directions from the center to all other points. The
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68:
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108:
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1937:
75:
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1399:
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2057:
1684:
1613:
46:
1430:
and some azimuthal projections centred on 90° N at the same scale, ordered by projection altitude in Earth radii.
57:
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1866:
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1796:
2145:
1651:
2643:
2501:
2441:
2421:
2052:
2014:
1979:
1519:
749:
162:
1476:
PROJ contributors (2020). PROJ coordinate transformation software library. Open Source
Geospatial Foundation.
1714:
1558:
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82:
35:
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138:
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125:
Vertical perspective from an altitude of 35,786 km over (0°, 90°W), corresponding to a view from
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2135:
2042:
1914:
1641:
1599:
782:
166:
126:
2363:
1974:
1679:
1403:
1442:
Geometric projection of the parallels of the polar
Perspective projections, Vertical and Tilted.
2290:
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2196:
2112:
2089:
1969:
1964:
1883:
1828:
1806:
1326:
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1497:
724:
266:
1306:
137:
The vertical perspective projection showing exactly one third of the Earth's surface, with
1395:
755:
198:
788:
177:, meaning that they result from viewing the globe from some vantage point. They are also
121:
2528:
2474:
2451:
2398:
2386:
2341:
2318:
2300:
2260:
2002:
1956:
1893:
1848:
1820:
1728:
1590:
1578:
1542:
1457:
1369:
1286:
730:
706:
150:
2637:
1477:
2551:
228:
194:
1563:
1387:
24:
1493:
Map
Projections—A Working Manual (US Geologic Survey Professional Paper 1395)
2608:
288:
232:
1491:
2603:
240:
224:
1438:
2464:
1496:. Washington, D.C.: US Government Printing Office. pp. 169–181.
1386:
must be subtracted from 180° to place it in the proper quadrant. For
256:
252:
248:
1501:
1391:
1511:
2587:
2384:
2000:
1576:
1515:
18:
1406:
of the orthographic projection as written is correct in all
227:
for the general perspective projection are derived using
1372:
1329:
1309:
1289:
1065:
823:
791:
758:
733:
709:
527:
308:
1390:
of the inverse formulas the use of the two-argument
2527:
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2397:
2340:
2317:
2299:
2259:
2169:
2121:
2098:
2075:
2066:
2013:
1955:
1905:
1892:
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1819:
1736:
1727:
1627:
1598:
1589:
49:. Unsourced material may be challenged and removed.
1378:
1358:
1315:
1295:
1272:
1045:
803:
773:
739:
715:
693:
507:
215:missions sparked interest in this projection.
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496:
404:
8:
161:The Vertical Perspective is related to the
2584:
2479:
2394:
2381:
2072:
2010:
1997:
1902:
1733:
1595:
1586:
1573:
1534:
1520:
1512:
291:for the orthographic projection onto the (
2619:Map projection of the tri-axial ellipsoid
1402:) is recommended. This ensures that the
1371:
1348:
1328:
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1236:
1206:
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1151:
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1127:
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494:
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419:
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365:
326:
309:
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299:) tangent plane reduce to the following:
109:Learn how and when to remove this message
1437:
132:
120:
1469:
7:
47:adding citations to reliable sources
16:Azimuthal perspective map projection
814:The inverse formulas are given by:
14:
2562:Quadrilateralized spherical cube
2242:Quadrilateralized spherical cube
1415:
231:. They are written in terms of
58:"General Perspective projection"
23:
34:needs additional citations for
2151:Lambert cylindrical equal-area
1428:General Perspective projection
1345:
1333:
1257:
1245:
1224:
1212:
1196:
1184:
1169:
1157:
147:General Perspective projection
1:
2599:Interruption (map projection)
2237:Lambert azimuthal equal-area
2033:Guyou hemisphere-in-a-square
2023:Adams hemisphere-in-a-square
1432:(click for detail)
2660:
2594:
2583:
2510:
2393:
2380:
2192:
2009:
1996:
1933:
1792:
1675:
1585:
1572:
1549:
1398:function (as opposed to
1359:{\displaystyle R(P-1)/P}
750:stereographic projection
163:stereographic projection
2038:Lambert conformal conic
1453:List of map projections
211:could be used instead.
175:perspective projections
171:orthographic projection
2171:Tobler hyperelliptical
1784:Tobler hyperelliptical
1710:Space-oblique Mercator
1490:Snyder, J. P. (1987).
1443:
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1318:
1316:{\displaystyle \rho }
1298:
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806:
776:
742:
718:
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510:
273:) of the projection (
191:Richard Edes Harrison
173:. These are all true
157:Perspective and usage
136:
124:
2547:Cahill–Keyes M-shape
2407:Chamberlin trimetric
1478:"Tilted perspective"
1370:
1327:
1307:
1287:
1063:
821:
789:
774:{\displaystyle d=-R}
756:
731:
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183:point of perspective
43:improve this article
2614:Tissot's indicatrix
2515:Central cylindrical
2156:Smyth equal-surface
2058:Transverse Mercator
1907:General perspective
1662:Smyth equal-surface
1614:Transverse Mercator
804:{\displaystyle d=0}
783:gnomonic projection
167:gnomonic projection
139:Tissot's indicatrix
127:geostationary orbit
2567:Waterman butterfly
2417:Miller cylindrical
2048:Peirce quincuncial
1943:Lambert equal-area
1695:Gall stereographic
1444:
1426:Comparison of the
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1992:
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1988:
1987:
1951:
1950:
1839:Lambert conformal
1815:
1814:
1729:Pseudocylindrical
1723:
1722:
1379:{\displaystyle c}
1296:{\displaystyle P}
1264:
1261:
1231:
1202:
1200:
1105:
1033:
912:
740:{\displaystyle d}
716:{\displaystyle c}
580:
119:
118:
111:
93:
2651:
2585:
2542:Cahill Butterfly
2480:
2460:Goode homolosine
2395:
2382:
2347:
2346:(Mecca or Qibla)
2227:Goode homolosine
2073:
2011:
1998:
1903:
1898:
1769:Goode homolosine
1734:
1619:Oblique Mercator
1596:
1587:
1574:
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1529:
1522:
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1506:
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1481:
1474:
1419:
1385:
1383:
1382:
1377:
1365:
1363:
1362:
1357:
1352:
1323:is greater than
1322:
1320:
1319:
1314:
1303:is negative and
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1300:
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725:angular distance
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129:. 10° graticule.
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27:
19:
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2644:Map projections
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2590:
2571:
2519:
2506:
2469:
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2432:Van der Grinten
2389:
2387:By construction
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2344:
2336:
2313:
2295:
2276:Equirectangular
2262:
2251:
2188:
2165:
2161:Trystan Edwards
2117:
2094:
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2005:
1984:
1957:Pseudoazimuthal
1947:
1929:
1896:
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1843:
1811:
1807:Winkel I and II
1788:
1719:
1700:Gall isographic
1690:Equirectangular
1671:
1667:Trystan Edwards
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1396:inverse tangent
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199:NASA World Wind
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141:of deformation.
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2412:Kavrayskiy VII
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1797:Kavrayskiy VII
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1543:Map projection
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1502:10.3133/pp1395
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60: –
59:
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54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
2497:Orthographic
2483:
2028:Gauss–Krüger
1920:Orthographic
1906:
1715:Web Mercator
1609:Gauss–Krüger
1492:
1485:
1472:
1427:
1394:form of the
1282:
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517:
296:
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263:
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244:
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229:trigonometry
222:
213:
209:
195:Google Earth
187:
182:
160:
146:
144:
105:
99:January 2021
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
2475:Perspective
2263:some aspect
2247:Strebe 1995
2222:Equal Earth
2141:Gall–Peters
2123:Cylindrical
1938:Equidistant
1834:Equidistant
1764:Equal Earth
1647:Gall–Peters
1591:Cylindrical
1388:computation
219:Mathematics
2537:AuthaGraph
2529:Polyhedral
2399:Compromise
2327:Loximuthal
2319:Loxodromic
2281:Sinusoidal
2131:Balthasart
2108:Sinusoidal
2085:Sinusoidal
2068:Equal-area
1779:Sinusoidal
1737:Equal-area
1637:Balthasart
1629:Equal-area
1602:-conformal
1579:By surface
1464:References
69:newspapers
2609:Longitude
2437:Wagner VI
2286:Two-point
2217:Eckert VI
2212:Eckert IV
2207:Eckert II
2184:Mollweide
2179:Collignon
2146:Hobo–Dyer
2100:Bottomley
2015:Conformal
2003:By metric
1894:Azimuthal
1867:Polyconic
1862:Bottomley
1802:Wagner VI
1774:Mollweide
1759:Eckert VI
1754:Eckert IV
1749:Eckert II
1744:Collignon
1652:Hobo–Dyer
1408:quadrants
1340:−
1311:ρ
1252:−
1239:ρ
1229:ρ
1219:−
1191:−
1149:ρ
1142:−
1134:−
1125:
1071:ρ
1024:φ
1020:
1011:
1002:−
993:φ
989:
980:
974:ρ
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950:
935:λ
924:λ
910:ρ
900:φ
896:
887:
866:φ
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829:φ
766:−
674:λ
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667:λ
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653:φ
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638:φ
634:
625:φ
622:
610:φ
606:
590:
574:
565:−
554:−
481:λ
477:−
474:λ
466:
460:φ
457:
445:φ
441:
435:−
432:φ
429:
417:φ
413:
363:λ
359:−
356:λ
348:
342:φ
339:
289:equations
247:) on the
233:longitude
179:azimuthal
2638:Category
2604:Latitude
2589:See also
2552:Dymaxion
2492:Gnomonic
2427:Robinson
2332:Mercator
2309:Gnomonic
2301:Gnomonic
2136:Behrmann
2043:Mercator
1915:Gnomonic
1897:(planar)
1872:American
1642:Behrmann
1600:Mercator
1447:See also
537:′
399:′
332:′
287:). The
262:and the
241:latitude
225:formulas
2465:HEALPix
2364:Littrow
1975:Wiechel
1877:Chinese
1821:Conical
1685:Central
1680:Cassini
1657:Lambert
1554:History
723:is the
255:of the
206:History
83:scholar
2484:Planar
2452:Hybrid
2359:Hammer
2291:Werner
2232:Hammer
2197:Albers
2113:Werner
2090:Werner
1970:Hammer
1965:Aitoff
1884:Werner
1829:Albers
1705:Miller
1564:Portal
1122:arcsin
1056:where
947:arctan
839:arcsin
518:where
282:φ
275:λ
271:origin
264:center
257:sphere
253:radius
249:sphere
245:φ
239:) and
237:λ
169:, and
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2271:Conic
2077:Bonne
1857:Bonne
1392:atan2
269:(and
267:point
149:is a
90:JSTOR
76:books
2557:ISEA
1559:List
1404:sign
1400:atan
727:and
223:The
197:and
145:The
62:news
1498:doi
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1008:sin
986:cos
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963:sin
893:cos
884:sin
859:sin
850:cos
656:cos
647:cos
631:cos
619:sin
603:sin
587:cos
571:cos
463:cos
454:cos
438:sin
426:sin
410:cos
345:sin
336:cos
45:by
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87:·
80:·
73:·
66:·
39:.
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