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on the sphere, there is a unique great circle passing through both. (Every great circle through any point also passes through its antipodal point, so there are infinitely many great circles through two antipodal points.) The shorter of the two great-circle arcs between two distinct points on the
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1099:{\displaystyle {\frac {\sin \theta \cos \theta \phi '^{2}}{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}}={\frac {d}{dt}}{\frac {\theta '}{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}}.}
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of the idealized earth is a great circle and any meridian and its opposite meridian form a great circle. Another great circle is the one that divides the
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coincides with the north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by
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To prove that the minor arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply
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Loxodrome (Rhumb Line), Orthodrome (Great Circle), Great
Ellipse and Geodetic Line (Geodesic) in Navigation
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861:{\displaystyle {\frac {\sin ^{2}\theta \phi '}{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}}=C}
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is allowed to take on arbitrary real values. The infinitesimal arc length in these coordinates is
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of any great circle coincides with a diameter of the sphere, and therefore every great circle is
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by John Snyder with additional contributions by Jeff Bryant, Pratik Desai, and Carl Woll,
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Great Circle description, figures, and equations. Mathworld, Wolfram
Research, Inc. c1999
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and a plane passing through its center. In higher dimensions, the great circles on the
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Integrating both sides and considering the boundary condition, the real solution of
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A great circle is the largest circle that can be drawn on any given sphere. Any
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417:{\displaystyle \theta =\theta (t),\quad \phi =\phi (t),\quad a\leq t\leq b}
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Every circle in
Euclidean 3-space is a great circle of exactly one sphere.
535:{\displaystyle ds=r{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}\,dt.}
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and if a great circle passes through a point it must pass through its
30:"Great Circle" redirects here. For the novel by Maggie Shipstead, see
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which is a plane through the origin, i.e., the center of the sphere.
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1401:. Great circles are also used as rather accurate approximations of
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From the first equation of these two, it can be obtained that
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integrates a function along all great circles of the sphere.
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A great circle divides the sphere in two equal hemispheres.
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formed by the two points and the center of the sphere.
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Consider the class of all regular paths from a point
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1505:: CS1 maint: multiple names: authors list (
1432:. A great circle divides the earth into two
27:Spherical geometry analog of a straight line
57:(purple) through the center is called the
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1557:Great Circles on Mercator's Chart
53:(black). The perpendicular line
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167:. For any pair of distinct non-
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1551:Great Circle – from MathWorld
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264:Derivation of shortest paths
247:are the intersection of the
90:(blue) through the poles is
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1296:{\displaystyle \theta _{0}}
912:-independent constant, and
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1430:land and water hemispheres
1409:'s surface for air or sea
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159:are the natural analog of
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545:So the length of a curve
1249:{\displaystyle \phi '=0}
187:between the points (the
1526:. USA: CRC Press, Inc.
1415:is not a perfect sphere
1269:{\displaystyle \theta }
730:Euler–Lagrange equation
558:{\displaystyle \gamma }
151:of a great circle is a
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34:. For other uses, see
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1481:W., Weisstein, Eric.
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440:{\displaystyle \phi }
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270:Great-circle distance
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32:Great Circle (novel)
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1577:Elementary geometry
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86:. Any great circle
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1395:celestial equator
1391:celestial horizon
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754:{\displaystyle S}
728:According to the
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598:{\displaystyle q}
578:{\displaystyle p}
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338:{\displaystyle p}
314:{\displaystyle q}
301:to another point
294:{\displaystyle p}
45:The great circle
16:(Redirected from
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36:The Great Circle
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18:Great circles
1522:
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1490:. Retrieved
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1389:include the
1384:
1381:Applications
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321:. Introduce
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227:
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220:small circle
218:is called a
214:. Any other
201:
176:
146:
142:center point
127:intersection
119:
116:great circle
115:
109:
91:
79:
71:
58:
1434:hemispheres
112:mathematics
1571:Categories
1492:2022-09-30
1468:References
1462:Rhumb line
1411:navigation
1397:, and the
1307:, this is
607:functional
268:See also:
234:great disk
208:concentric
181:arc length
120:orthodrome
1403:geodesics
1350:ϕ
1346:
1337:−
1328:ϕ
1324:
1285:θ
1264:θ
1234:ϕ
1174:−
1171:θ
1168:
1153:θ
1150:
1138:θ
1121:ϕ
1088:θ
1085:
1061:ϕ
1043:θ
1034:θ
1007:θ
1004:
980:ϕ
962:θ
945:ϕ
941:θ
938:
932:θ
929:
847:θ
844:
820:ϕ
802:θ
792:ϕ
788:θ
785:
746:γ
701:θ
698:
674:ϕ
656:θ
639:∫
626:γ
553:γ
518:θ
515:
491:ϕ
473:θ
435:ϕ
427:provided
409:≤
403:≤
384:ϕ
378:ϕ
362:θ
356:θ
177:minor arc
169:antipodal
92:secondary
1501:cite web
1451:See also
1399:ecliptic
1237:′
1141:′
1124:′
1065:′
1047:′
1037:′
984:′
966:′
949:′
824:′
806:′
795:′
678:′
660:′
495:′
477:′
325:so that
204:diameter
153:geodesic
124:circular
1592:Circles
1426:equator
1405:on the
278:to it.
245:-sphere
195:of the
193:measure
183:is the
122:is the
1530:
1393:, the
872:where
212:radius
172:points
133:and a
131:sphere
1407:Earth
892:is a
605:is a
565:from
135:plane
129:of a
80:poles
75:'
1528:ISBN
1507:link
1443:The
1424:The
1256:and
238:ball
230:disk
228:The
147:Any
114:, a
69:and
59:axis
1343:cos
1321:sin
1159:sin
1147:sin
1076:sin
995:sin
935:cos
926:sin
835:sin
776:sin
689:sin
585:to
506:sin
163:in
149:arc
118:or
110:In
94:to
82:of
61:of
1573::
1503:}}
1499:{{
1485:.
1440:.
1421:.
732:,
260:.
144:.
1563:.
1536:.
1509:)
1495:.
1362:0
1359:=
1354:0
1340:y
1332:0
1318:x
1289:0
1244:0
1241:=
1213:C
1201:.
1182:2
1178:C
1163:2
1134:C
1128:=
1094:.
1080:2
1069:2
1057:+
1051:2
1025:t
1022:d
1018:d
1013:=
999:2
988:2
976:+
970:2
953:2
900:t
880:C
868:,
856:C
853:=
839:2
828:2
816:+
810:2
780:2
749:]
743:[
740:S
713:.
710:t
707:d
693:2
682:2
670:+
664:2
648:b
643:a
635:r
632:=
629:]
623:[
620:S
593:q
573:p
530:.
527:t
524:d
510:2
499:2
487:+
481:2
467:r
464:=
461:s
458:d
412:b
406:t
400:a
396:,
393:)
390:t
387:(
381:=
374:,
371:)
368:t
365:(
359:=
333:p
309:q
289:p
257:R
249:n
243:n
98:.
96:g
88:s
84:g
72:P
67:P
63:g
55:a
51:O
47:g
38:.
20:)
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