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A sphere (from Greek σφαῖρα, sphaîra) [1] is a geometrical object that is a three-dimensional analogue to a two-dimensional circle.Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. [2]
Since spherical geometry violates the parallel postulate, there exists no such triangle on the surface of a sphere. The sum of the angles of a triangle on a sphere is 180°(1 + 4f), where f is the fraction of the sphere's surface that is enclosed by the triangle.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry are the natural analog of straight lines in Euclidean space. For any pair of distinct non-antipodal points on the sphere, there is a unique great circle passing through both. (Every great circle through any point also passes through its antipodal ...
A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two antipodal points, u and v are also shown. The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them. This arc is the shortest path ...
If the sphere is isometrically embedded in Euclidean space, the sphere's intersection with a plane is a circle, which can be interpreted extrinsically to the sphere as a Euclidean circle: a locus of points in the plane at a constant Euclidean distance (the extrinsic radius) from a point in the plane (the extrinsic center). A great circle lies ...
The n dimensional real projective space is the quotient of the n sphere by the antipodal map. It follows that its Euler characteristic is exactly half that of the corresponding sphere – either 0 or 1. The n dimensional torus is the product space of n circles. Its Euler characteristic is 0, by the product property.
Euclidean geometry is a mathematical system ... and hydrofoils where geometric shape directly impacts lift and drag characteristics. ... A sphere has 2/3 the volume ...
A sphere is a surface that can be ... Congruence and similarity are concepts that describe when two shapes have similar characteristics. [68] In Euclidean geometry, ...