View source for Antipodal point

{{Short description|Pair of diametrically opposite points on a circle, sphere, or hypersphere}}
{{Hatnote|For the geographical antipodal point of a place on Earth, see [[antipodes]].}}

[[File:Great circle, axis, and poles.svg|thumb|upright=1.25|The two points {{mvar|P}} and {{math|''P''{{'}}}} (red) are ''antipodal'' because they are ends of a diameter {{math|''PP''{{'}}}}, a segment of the ''axis'' {{mvar|a}} (purple) passing through the sphere's center {{mvar|O}} (black). {{mvar|P}} and {{math|''P''{{'}}}} are the ''poles'' of a great circle {{mvar|g}} (green) whose points are equidistant from each (with a central right angle). Any great circle {{mvar|s}} (blue) passing through the poles is ''secondary'' to {{mvar|g}}.]]

In [[mathematics]], two points of a [[sphere]] (or [[n-sphere]], including a [[circle]]) are called '''antipodal''' or '''diametrically opposite''' if they are the endpoints of a [[diameter]], a straight [[line segment]] between two points on a sphere and passing through its [[center (geometry)|center]].<ref name="EB1911">{{Cite EB1911|wstitle=Antipodes|volume=2|pages=133–34}}</ref>

Given any point on a sphere, its antipodal point is the unique point at greatest [[distance]], whether measured intrinsically ([[great-circle distance]] on the surface of the sphere) or extrinsically ([[Chord (geometry)|chordal]] distance through the sphere's interior). Every [[great circle]] 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 spherical geometry depend on choosing non-antipodal points, and [[degeneracy (mathematics)|degenerate]] if antipodal points are allowed; for example, a [[spherical triangle]] degenerates to an underspecified [[spherical lune|lune]] if two of the vertices are antipodal.

The point antipodal to a given point is called its '''antipodes''', from the [[Ancient Greek|Greek]] {{lang|grc|ἀντίποδες}} ({{transliteration|grc|antípodes}}) meaning "opposite feet"; see {{slink|Antipodes#Etymology}}. Sometimes the ''s'' is dropped, and this is rendered '''antipode''', a [[back-formation]].

== Higher mathematics ==
The concept of ''antipodal points'' is generalized to [[sphere]]s of any dimension: two points on the sphere are antipodal if they are opposite ''through the centre''. Each line through the centre intersects the sphere in two points, one for each [[ray (geometry)|ray]] emanating from the centre, and these two points are antipodal.

The [[Borsuk–Ulam theorem]] is a result from [[algebraic topology]] dealing with such pairs of points. It says that any [[continuous function]] from <math>S^n</math> to <math>\R^n</math> maps some pair of antipodal points in <math>S^n</math> to the same point in <math>\R^n.</math> Here, <math>S^n</math> denotes the {{nobr|<math>n</math>-dimensional}} sphere and <math>\R^n</math> is {{nobr|<math>n</math>-dimensional}} [[real coordinate space]].

The '''antipodal map''' <math>A : S^n \to S^n</math> sends every point on the sphere to its antipodal point. If points on the {{nobr|<math>n</math>-sphere}} are represented as [[position (geometry)|displacement vectors]] from the sphere's center in Euclidean {{nobr|<math>(n+1)</math>-space,}} then two antipodal points are represented by additive inverses <math>\mathbf{v}</math> and <math>-\mathbf{v},</math> and the antipodal map can be defined as <math>A(\mathbf{x}) = -\mathbf{x}.</math> The antipodal map preserves [[Orientability|orientation]] (is [[homotopy|homotopic]] to the [[identity function|identity map]])<ref>{{cite book |author1=V. Guillemin |author2=A. Pollack |title=Differential topology |publisher=Prentice-Hall |year=1974}}</ref> when <math>n</math> is odd, and reverses it when <math>n</math> is even. Its [[Degree of a continuous mapping|degree]] is <math>(-1)^{n+1}.</math>

If antipodal points are identified (considered equivalent), the sphere becomes a model of [[real projective space]].

==See also==
*[[Cut locus]]

==References==
<references />

==External links==
* {{springer|title=Antipodes|id=p/a012720}}
* {{planetmath reference|urlname=Antipodal|title=antipodal}}

[[Category:Spherical geometry]]
[[Category:Point (geometry)]]