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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 ...
There is a natural unit of angle measurement (based on a revolution), a natural unit of length (based on the circumference of a great circle) and a natural unit of area (based on the area of the sphere). Each great circle is associated with a pair of antipodal points, called its poles which are the common intersections of the set of great ...
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).
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 ...
The dimensionless quantities often represent the degree of deviation from an ideal shape, such as a circle, sphere or equilateral polyhedron. [1] Shape factors are often normalized, that is, the value ranges from zero to one. A shape factor equal to one usually represents an ideal case or maximum symmetry, such as a circle, sphere, square or cube.
Figure 1: Coordinate isosurfaces for a point P (shown as a black sphere) in oblate spheroidal coordinates (μ, ν, φ). The z-axis is vertical, and the foci are at ±2. The red oblate spheroid (flattened sphere) corresponds to μ = 1, whereas the blue half-hyperboloid corresponds to ν = 45°.
In two dimensions, the chordal metric on the sphere is not intrinsic, and the induced intrinsic metric is given by the great-circle distance. Every connected Riemannian manifold can be turned into a path metric space by defining the distance of two points as the infimum of the lengths of continuously differentiable curves connecting the two points.
In mathematics, a circle bundle is a fiber bundle where the fiber is the circle. Oriented circle bundles are also known as principal U(1)-bundles, or equivalently, as principal SO(2)-bundles. In physics, circle bundles are the natural geometric setting for electromagnetism. A circle bundle is a special case of a sphere bundle.