Editing Antipodal point (section)

{{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]].