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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.
An area formula for spherical triangles analogous to the formula for planar triangles. Given a fixed base , an arc of a great circle on a sphere, and two apex points and on the same side of great circle , Lexell's theorem holds that the surface area of the spherical triangle is equal to that of if and only if lies on the small-circle arc , where and are the points antipodal to and , respectively.
In geography, the antipode (/ ˈ æ n t ɪ ˌ p oʊ d, æ n ˈ t ɪ p ə d i /) of any spot on Earth is the point on Earth's surface diametrically opposite to it. A pair of points antipodal (/ æ n ˈ t ɪ p ə d əl /) to each other are situated such that a straight line connecting the two would pass through Earth's center.
Antipodal. In mathematics, the Borsuk–Ulam theorem states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.
Elliptic geometry may be derived from spherical geometry by identifying antipodal points of the sphere to a single elliptic point. The elliptic lines correspond to great circles reduced by the identification of antipodal points. As any two great circles intersect, there are no parallel lines in elliptic geometry.
For example, the excentral triangle is the antipedal triangle of the incenter. Suppose that P does not lie on any of the extended sides BC, CA, AB, and let P −1 denote the isogonal conjugate of P. The pedal triangle of P is homothetic to the antipedal triangle of P −1.
An antipodal pair of vertex and their supporting parallel lines.. The rotating calipers method was first used in the dissertation of Michael Shamos in 1978. [2] Shamos used this method to generate all antipodal pairs of points on a convex polygon and to compute the diameter of a convex polygon in () time.
We can also look for antipodal points on a hypercycle (a "diameter" being a a geodesic orthogonal to the hypercycle) with the caveat that there's no well-defined "center"; or even on a horocycle (every point of which is is antipodal to the ideal point where the horocycle is tangent to the boundary). –jacobolus 16:00, 13 July 2023 (UTC)