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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.
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.
Lexell's theorem: Orange triangles ABC share a base AB and area. The locus of their apex C is a small circle (dashed green) passing through the points antipodal to A and B. In the half-plane model, antipodal points are reflections into the opposite half-plane (shaded gray). Hyperbolic triangles ABC (orange) share a base AB and area.
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.
Consider a sphere, and let the great circles of the sphere be "lines", and let pairs of antipodal points be "points". It is easy to check that this system obeys the axioms required of a projective plane: any pair of distinct great circles meet at a pair of antipodal points; and; any two distinct pairs of antipodal points lie on a single great ...
Antipodal point, the diametrically opposite point on a circle or n-sphere, also known as an antipode; Antipode, the convolution inverse of the identity on a Hopf algebra;
In geometry, a bigon, [1] digon, or a 2-gon, is a polygon with two sides and two vertices.Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visualised in elliptic space.