# Antipodal point

__: Pair of diametrically opposite points on a circle, sphere, or hypersphere__

**Short description**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.^{[1]}

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 (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 degenerate if antipodal points are allowed; for example, a spherical triangle degenerates to an underspecified lune if two of the vertices are antipodal.

The point antipodal to a given point is called its **antipodes**, from the Greek ἀντίποδες (*antípodes*) meaning "opposite feet"; see 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 spheres 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 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]\displaystyle{ S^n }[/math] to [math]\displaystyle{ \R^n }[/math] maps some pair of antipodal points in [math]\displaystyle{ S^n }[/math] to the same point in [math]\displaystyle{ \R^n. }[/math] Here, [math]\displaystyle{ S^n }[/math] denotes the [math]\displaystyle{ n }[/math]-dimensional sphere and [math]\displaystyle{ \R^n }[/math] is [math]\displaystyle{ n }[/math]-dimensional real coordinate space.

The **antipodal map** [math]\displaystyle{ A : S^n \to S^n }[/math] sends every point on the sphere to its antipodal point. If points on the [math]\displaystyle{ n }[/math]-sphere are represented as displacement vectors from the sphere's center in Euclidean [math]\displaystyle{ (n+1) }[/math]-space, then two antipodal points are represented by additive inverses [math]\displaystyle{ \bold{v} }[/math] and [math]\displaystyle{ -\bold{v}, }[/math] and the antipodal map can be defined as [math]\displaystyle{ A(\bold{x}) = -\bold{x}. }[/math] The antipodal map preserves orientation (is homotopic to the identity map)^{[2]} when [math]\displaystyle{ n }[/math] is odd, and reverses it when [math]\displaystyle{ n }[/math] is even. Its degree is [math]\displaystyle{ (-1)^{n+1}. }[/math]

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

## See also

## References

- ↑ Chisholm, Hugh, ed (1911). "Antipodes".
*Encyclopædia Britannica*.**2**(11th ed.). Cambridge University Press. pp. 133–34. - ↑ V. Guillemin; A. Pollack (1974).
*Differential topology*. Prentice-Hall.

## External links

- Hazewinkel, Michiel, ed. (2001), "Antipodes",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/a012720 - "antipodal". http://planetmath.org/?op=getobj&from=objects&id={{{id}}}.

Original source: https://en.wikipedia.org/wiki/Antipodal point.
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