Pair of diametrically opposite points on a circle, sphere, or hypersphere
For the geographical antipodal point of a place on Earth, see antipodes.
The two points P and P' (red) are antipodal because they are ends of a diameter PP', a segment of the axisa (purple) passing through the sphere's center O (black). P and P' are the poles of a great circle g (green) whose points are equidistant from each (with a central right angle). Any great circle s (blue) passing through the poles is secondary to 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.[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.
^Chisholm, Hugh, ed. (1911). "Antipodes" . Encyclopædia Britannica. Vol. 2 (11th ed.). Cambridge University Press. pp. 133–34.
sphere and passing through its center. Given any point on a sphere, its antipodalpoint is the unique point at greatest distance, whether measured intrinsically...
ænˈtɪpədi/) of any spot on Earth is the point on Earth's surface diametrically opposite to it. A pair of points antipodal (/ænˈtɪpədəl/) to each other are situated...
any point also passes through its antipodalpoint, so there are infinitely many great circles through two antipodal points.) The shorter of the two great-circle...
northwest across the Pacific Ocean, near Easter Island, passing the antipodalpoint near Emlilia island, through the South Bering Sea and ending somewhere...
about a point whose antipodalpoint lies on land and the map is extended to the maximum distance of 20,000 km (12,427 mi) the antipode point smears into...
opposite is the diametrically opposite point of a circle or sphere. It is mathematically known as an antipodalpoint, or antipode when referring to the Earth...
even-dimensional sphere into itself has either a fixed point or a point that maps onto its own antipodalpoint. This can be seen by transforming the function...
over northern South America and the south Atlantic, near Singapore's antipodalpoint, the magnetic field is relatively weak, resulting in a lower repulsion...
meridian – Line of longitude, at which longitude is defined to be 0° "Geo mid point: Calculation Methods". 2022. Retrieved 16 September 2022. Woods, Andrew...
volcanism can be triggered by converging seismic energy focused at the antipodalpoint opposite major impact sites. Impact-induced volcanism has not been...
example, the cut locus of every point on the regular 2-sphere consists of exactly one point, namely the antipodalpoint. Let X {\displaystyle X} be a metric...
the northeast of Campbell Island. They are very close to being the antipodalpoint to Normandy in France, meaning that the city farthest away is Cherbourg-en-Cotentin...
= 1, the point reflection group is the full isometry group of the line. Point reflection across the center of a sphere yields the antipodal map. A symmetric...
meet at a pair of antipodal points; and any two distinct pairs of antipodal points lie on a single great circle. If we identify each point on the sphere with...
great circle passes through a point on the sphere, it also passes through the antipodalpoint (the unique furthest other point on the sphere). For any pair...
days: four times with the wave travelling away from the volcano to its antipodalpoint and three times travelling back to the volcano.: 63 Hence, the wave...
a geographical point that describes a centerpoint of the region's population. There are several ways of defining such a "center point", leading to different...
the year, the Sun passes through the nadir (the antipodal zenith) of the Kaaba, casting shadows that point in the opposite direction, and thus also determine...
days: four times with the wave travelling away from the volcano to its antipodalpoint, and three times travelling back to the volcano;: 63 the wave rounded...
called a pentagonal hosohedron with Schläfli symbol {2,5}. It has 2 (antipodalpoint) vertices, 5 edges, and 5 digonal faces. Weisstein, Eric W. "Pentahedron"...
sufficiently energetic it might cause mantle plume (volcanism) at the antipodalpoint (the opposite side of the world). This volcanism could alone therefore...
in inverse relief, formed by valleys rather than raised ground. The antipodalpoint of Medicine Hat is near Port-aux-Francais (Kerguelen Island) in the...
places observe the opposite phenomenon when the Sun passes above the antipodalpoint of the Kaaba (in other words, the Sun passes directly underneath the...
whose great-circle distances to two foci is constant. By taking the antipodalpoint to one focus, every spherical ellipse is also a spherical hyperbola...
centered on the pointantipodal to A, λ12 = π, and this means that meridional geodesics stop being shortest paths before the antipodalpoint is reached. Various...
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