WordNet

an opposition that can be graded between two extremes or poles
the relation between opposed entities (同)oppositeness
a direction opposite to another
a contestant that you are matched against (同)opponent, opposite
a body of people united in opposing something
of or existing at or near a geographical pole or within the Arctic or Antarctic Circles; "polar regions"
having a pair of equal and opposite charges
located at or near or coming from the earths poles; "polar diameter"; "polar zone"; "a polar air mass"; "Antarctica is the only polar continent"
the major political party opposed to the party in office and prepared to replace it if elected; "Her Majestys loyal opposition"

PrepTutorEJDIC

〈U〉〈C〉(…に)『反対すること』,(…に対する)反対,対立,抵抗,敵対《+『to』+『名』(a person's do『ing』)》 / 《時にO-》《the~》野党,反対党
『極地の』;南(北)極の / 正反対の
《話》政治家

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2016/09/25 14:37:12」(JST)

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Antipodal points on a circle are 180 degrees apart.

In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite to it — so situated that a line drawn from the one to the other passes through the center of the sphere and forms a true diameter.

This term applies to opposite points on a circle or any n-sphere.

An antipodal point is sometimes called an antipode, a back-formation from the Greek loan word antipodes, which originally meant "opposite the feet."

Theory

In 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; for example, taking the centre as origin, they are points with related vectors v and −v. On a circle, such points are also called diametrically opposite. In other words, each line through the centre intersects the sphere in two points, one for each ray out 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 Sⁿ to Rⁿ maps some pair of antipodal points in Sⁿ to the same point in Rⁿ. Here, Sⁿ denotes the n-dimensional sphere in (n + 1)-dimensional space (so the "ordinary" sphere is S² and a circle is S¹).

The antipodal map A : Sⁿ → Sⁿ, defined by A(x) = −x, sends every point on the sphere to its antipodal point. It is homotopic to the identity map if n is odd, and its degree is (−1)ⁿ⁺¹.

If one wants to consider antipodal points as identified, one passes to projective space (see also projective Hilbert space, for this idea as applied in quantum mechanics).

Antipodal pair of points on a convex polygon

An antipodal pair of a convex polygon is a pair of 2 points admitting 2 infinite parallel lines being tangent to both points included in the antipodal without crossing any other line of the convex polygon.

References

This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Antipodes". Encyclopædia Britannica (11th ed.). Cambridge University Press.

External links

Hazewinkel, Michiel, ed. (2001), "Antipodes", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
‹See Tfm›

"antipodal". PlanetMath.

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