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The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. A sphere with a spherical triangle on it.
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, ... A sphere has 2/3 the volume and surface area of its circumscribing ...
In geometry, many uniform tilings on sphere, euclidean plane, and hyperbolic plane can be made by Wythoff construction within a fundamental triangle, (p q r), defined by internal angles as π/p, π/q, and π/r. Special cases are right triangles (p q 2).
The above -sphere exists in (+) -dimensional Euclidean space and is an example of an -manifold. The volume form ω {\displaystyle \omega } of an n {\displaystyle n} -sphere of radius r {\displaystyle r} is given by
The key observation that leads to Lie sphere geometry is that theorems of Euclidean geometry in the plane (resp. in space) which only depend on the concepts of circles (resp. spheres) and their tangential contact have a more natural formulation in a more general context in which circles, lines and points (resp. spheres, planes and points) are treated on an equal footing.
Great circle on a sphere. The basic elements of Euclidean plane geometry are points and lines. On the sphere, points are defined in the usual sense. The analogue of the "line" is the geodesic, which is a great circle; the defining characteristic of a great circle is that the plane containing all its points also passes through the center of the ...
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 ...
In Euclidean geometry, ... The hypersphere in 2 dimensions is a circle, sometimes called a 1-sphere (S 1) because it is a one-dimensional manifold.