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on a sphere passing through a point also passes through its antipodal point, and there are infinitely many great circles passing through a pair of antipodal points (unlike the situation for any non-antipodal pair of points, which have a unique great circle passing through both). Many results in
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88:(green) whose points are equidistant from each (with a central right angle). Any great circle
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If antipodal points are identified (considered equivalent), the sphere becomes a model of
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230:. Each line through the centre intersects the sphere in two points, one for each
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Given any point on a sphere, its antipodal point is the unique point at greatest
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of any dimension: two points on the sphere are antipodal if they are opposite
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828:. Vol. 2 (11th ed.). Cambridge University Press. pp. 133–34.
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sends every point on the sphere to its antipodal point. If points on the
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Pair of diametrically opposite points on a circle, sphere, or hypersphere
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spherical geometry depend on choosing non-antipodal points, and
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then two antipodal points are represented by additive inverses
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emanating from the centre, and these two points are antipodal.
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For the geographical antipodal point of a place on Earth, see
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between two points on a sphere and passing through its
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dealing with such pairs of points. It says that any
175:The point antipodal to a given point is called its
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164:if antipodal points are allowed; for example, a
151:on the surface of the sphere) or extrinsically (
155:distance through the sphere's interior). Every
671:{\displaystyle A(\mathbf {x} )=-\mathbf {x} .}
63:(purple) passing through the sphere's center
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633:and the antipodal map can be defined as
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548:from the sphere's center in Euclidean
305:maps some pair of antipodal points in
172:if two of the vertices are antipodal.
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92:(blue) passing through the poles is
47:because they are ends of a diameter
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147:, whether measured intrinsically (
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839:V. Guillemin; A. Pollack (1974).
357:{\displaystyle \mathbb {R} ^{n}.}
202:is dropped, and this is rendered
168:degenerates to an underspecified
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647:
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591:
514:{\displaystyle A:S^{n}\to S^{n}}
436:{\displaystyle \mathbb {R} ^{n}}
298:{\displaystyle \mathbb {R} ^{n}}
194:) meaning "opposite feet"; see
128:if they are the endpoints of a
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626:{\displaystyle -\mathbf {v} ,}
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710:is odd, and reverses it when
678:The antipodal map preserves
598:{\displaystyle \mathbf {v} }
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863:Encyclopedia of Mathematics
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772:{\displaystyle (-1)^{n+1}.}
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196:Antipodes § Etymology
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825:Encyclopædia Britannica
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126:diametrically opposite
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841:Differential topology
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574:{\displaystyle (n+1)}
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467:real coordinate space
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384:{\displaystyle S^{n}}
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332:to the same point in
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325:{\displaystyle S^{n}}
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269:{\displaystyle S^{n}}
149:great-circle distance
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546:displacement vectors
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544:are represented as
247:continuous function
239:Borsuk–Ulam theorem
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243:algebraic topology
228:through the centre
222:is generalized to
214:Higher mathematics
166:spherical triangle
108:, two points of a
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84:of a great circle
819:"Antipodes"
723:{\displaystyle n}
703:{\displaystyle n}
535:{\displaystyle n}
457:{\displaystyle n}
405:{\displaystyle n}
241:is a result from
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904:Point (geometry)
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843:. Prentice-Hall.
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220:antipodal points
198:. Sometimes the
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30:The two points
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851:External links
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816:, ed. (1911).
814:Chisholm, Hugh
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208:back-formation
116:, including a
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474:antipodal map
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688:identity map
473:
471:
464:-dimensional
412:-dimensional
391:denotes the
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157:great circle
142:
134:line segment
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876:"antipodal"
858:"Antipodes"
680:orientation
414:sphere and
179:, from the
106:mathematics
893:Categories
881:PlanetMath
801:References
162:degenerate
43:(red) are
868:EMS Press
795:Cut locus
745:−
684:homotopic
658:−
613:−
499:→
191:antĂpodes
185:ἀντίποδες
177:antipodes
122:antipodal
94:secondary
67:(black).
45:antipodal
21:antipodes
789:See also
204:antipode
145:distance
130:diameter
114:n-sphere
80:are the
870:, 2001
690:) when
686:to the
581:-space,
542:-sphere
224:spheres
153:chordal
732:degree
364:Here,
138:center
118:circle
110:sphere
249:from
181:Greek
82:poles
77:'
53:'
40:'
682:(is
605:and
472:The
237:The
206:, a
170:lune
112:(or
71:and
58:axis
34:and
734:is
443:is
276:to
232:ray
124:or
104:In
96:to
895::
878:.
866:,
860:,
822:.
785:.
469:.
210:.
140:.
50:PP
884:.
767:.
762:1
759:+
756:n
752:)
748:1
742:(
718:n
698:n
666:.
662:x
655:=
652:)
648:x
644:(
641:A
621:,
617:v
592:v
569:)
566:1
563:+
560:n
557:(
530:n
507:n
503:S
494:n
490:S
486::
483:A
452:n
429:n
424:R
400:n
377:n
373:S
352:.
347:n
342:R
318:n
314:S
291:n
286:R
262:n
258:S
200:s
188:(
100:.
98:g
90:s
86:g
74:P
69:P
65:O
61:a
37:P
32:P
23:.
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