In international politics, a state's claim to exclusive or predominant control over a foreign area or territory. Beginning in the late 1880s, European colonial powers undertook legal agreements consisting of promises not to interfere with each other's actions in mutually recognized spheres of influence in Africa and Asia. After colonial expansion ceased, geopolitical rather than legal claims to spheres of influence became common, examples being the U.S. claim to dominance in the Western Hemisphere under the much-earlier Monroe Doctrine and the Soviet Union's expansion of its sphere of influence to eastern Europe following World War II. Seealso Iron Curtain.
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Apparent surface of the heavens, on which the stars seem to be fixed. For the purpose of establishing celestial coordinate systems to mark the positions of heavenly bodies, it can be thought of as a real sphere at an infinite distance from Earth. Earth's rotational axis, extended to infinity, touches this sphere at the northern and southern celestial poles, around which the heavens seem to turn. The intersection of the plane of Earth's Equator with the sphere marks the celestial equator.
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A sphere (from Greek σφαίρα - sphaira, "globe, ball,) is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface. In mathematics, a sphere is the set of all points in three-dimensional space (R^{3}) which are at distance r from a fixed point of that space, where r is a positive real number called the radius of the sphere. Thus, in three dimensions, a mathematical sphere is considered to be a two-dimensional spherical surface embedded in three-dimensional space, rather than the volume contained within it (which mathematicians would instead describe as a ball). The fixed point is called the center, and is not part of the sphere itself. The special case of r = 1 is called a unit sphere.
This article deals with the mathematical concept of a sphere. In physics, a sphere is an object (usually idealized for the sake of simplicity) capable of colliding or stacking with other objects which occupy space.
The points on the sphere with radius r can be parametrized via
(see also trigonometric functions and spherical coordinates).
A sphere of any radius centered at the origin is described by the following differential equation:
This equation reflects the fact that the position and velocity vectors of a point travelling on the sphere are always orthogonal to each other.
The surface area of a sphere of radius r is
so the radius from surface area is
Its volume is
so the radius from volume is
The sphere has the smallest surface area among all surfaces enclosing a given volume and it encloses the largest volume among all closed surfaces with a given surface area. For this reason, the sphere appears in nature: for instance bubbles and small water drops are roughly spherical, because the surface tension locally minimizes surface area. The surface area in relation to the mass of a sphere is called the specific surface area. From the above stated equations it can be expressed as follows:
The circumscribed cylinder for a given sphere has a volume which is 3/2 times the volume of the sphere, and also the curved portion has a surface area which is equal to the surface area of the sphere. This fact, along with the volume and surface formulas given above were known as far back as Archimedes.
A sphere can also be defined as the surface formed by rotating a circle about any diameter. If the circle is replaced by an ellipse, and rotated about the major axis, the shape becomes a prolate spheroid, rotated about the minor axis, an oblate spheroid.
If a particular point on a sphere is (arbitrarily) designated as its north pole, then the corresponding antipodal point is called the south pole and the equator is the great circle that is equidistant to them. Great circles through the two poles are called lines (or meridians) of longitude, and the line connecting the two poles is called the axis of rotation. Circles on the sphere that are parallel to the equator are lines of latitude. This terminology is also used for astronomical bodies such as the planet Earth, even though it is neither spherical nor even spheroidal (see geoid).
A sphere is divided into two equal hemispheres by any plane that passes through its center. If two intersecting planes pass through its center, then they will subdivide the sphere into four lunes or biangles, the vertices of which all coincide with the antipodal points lying on the line of intersection of the planes.
The antipodal quotient of the sphere is the surface called the real projective plane, which can also be thought of as the northern hemisphere with antipodal points of the equator identified.
Spheres can be generalized to spaces of any dimension. For any natural number n, an n-sphere, often written as S^{n}, is the set of points in (n+1)-dimensional Euclidean space which are at a fixed distance r from a central point of that space, where r is, as before, a positive real number. In particular:
Spheres for n > 2 are sometimes called hyperspheres.
The n-sphere of unit radius centred at the origin is denoted S^{n} and is often referred to as "the" n-sphere. Note that the ordinary sphere is a 2-sphere, because it is a 2-dimensional surface (which is embedded in 3-dimensional space).
The surface area of the (n−1)-sphere of radius 1 is
where Γ(z) is Euler's Gamma function.
Another formula for surface area is
and the volume within is the surface area times $\{r\; over\; n\}$ or
More generally, in a metric space (E,d), the sphere of center x and radius is the set of points y such that d(x,y) = r.
If the center is a distinguished point considered as origin of E, as in a normed space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken equal to one, as in the case of a unit sphere.
In contrast to a ball, a sphere may be an empty set, even for a large radius. For example, in Z^{n} with Euclidean metric, a sphere of radius r is nonempty only if r^{2} can be written as sum of n squares of integers.
In topology, an n-sphere is defined as a space homeomorphic to the boundary of an (n+1)-ball; thus, it is homeomorphic to the Euclidean n-sphere, but perhaps lacking its metric.
The n-sphere is denoted S^{n}. It is an example of a compact topological manifold without boundary. A sphere need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere.
The Heine-Borel theorem implies that a Euclidean n-sphere is compact. The sphere is the inverse image of a one-point set under the continuous function ||x||. Therefore the sphere is closed. S^{n} is also bounded, therefore it is compact.
The basic elements of plane geometry are points and lines. On the sphere, points are defined in the usual sense, but the analogue of "line" may not be immediately apparent. If one measures by arc length one finds that the shortest path connecting two points lying entirely in the sphere is a segment of the great circle containing the points; see geodesic. Many theorems from classical geometry hold true for this spherical geometry as well, but many do not (see parallel postulate). In spherical trigonometry, angles are defined between great circles. Thus spherical trigonometry is different from ordinary trigonometry in many respects. For example, the sum of the interior angles of a spherical triangle exceeds 180 degrees. Also, any two similar spherical triangles are congruent.