Sphere

A sphere (< Greek ''σφαίρα'') is a perfectly Symmetrical Geometrical object. In Mathematics , the term refers to the Surface or Boundary of a ''' Ball ''', but in non-mathematical usage, the term is used to refer either to a three-dimensional ball or to its surface. This article deals with the mathematical concept of a sphere. GEOMETRY In Three-dimensional Euclidean Geometry , a sphere is the set of points in '''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. The fixed point is called the '''center''' or '''centre''', and is not part of the sphere itself. The special case of ''r'' = 1 is called a '''unit sphere'''. Equations In Analytic Geometry , a sphere with center (''x''₀, ''y''₀, ''z''₀) and radius ''r'' is the set of all points (''x'', ''y'', ''z'') such that : $(x - x_0 )^2 + (y - y_0 )^2 + ( z - z_0 )^2 = r^2 \,$ The points on the sphere with radius ''r'' can be parametrized via : $x = x_0 + r \sin heta \; \cos \phi$ : $y = y_0 + r \sin heta \; \sin \phi \qquad (0 \leq heta \leq \pi \mbox{ and } -\pi < \phi \leq \pi) \,$ : $z = z_0 + r \cos heta \,$ (see also Trigonometric Function s and Spherical Coordinates ). A sphere of any radius centered at the origin is described by the following Differential Equation : : $x \, dx + y \, dy + z \, dz = 0.$ 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: : $A = 4 \pi r^2 \,$ and its enclosed Volume is: : $V = rac{4 \pi r^3}{3}$ 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 minimizes surface area. the image of Einstein in the background. A Fused Quartz Gyroscope for the Gravity Probe B experiment which differs in shape from a perfect sphere by no more than a mere 40 atoms of thickness. It is thought that only Neutron Stars are smoother.]] The circumscribed Cylinder for a given sphere has a volume which is 3/2 times the volume of the sphere, and also a surface area which is 3/2 times the surface area of the sphere. This fact, along with the volume and surface formulas given above, was already known to Archimedes . A sphere can also be defined as the surface formed by rotating a Circle about its Diameter . If the circle is replaced by an Ellipse , the shape becomes a Spheroid . Terminology Pairs of points on a sphere that lie on a straight line through its center are called Antipodal Point s. A Great Circle is a circle on the sphere that has the same center as the sphere. If a particular point on a sphere is 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 Meridian s) of Longitude , and the line connecting the two poles is called the Axis . 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 may be divided into two equal hemispheres by any plane that passes through its center. If two intersecting planes pass through it's 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. Generalization to other dimensions Spheres can be generalized to other Dimension s. For any Natural Number ''n'', an ''n''-sphere is the set of points in (''n''+1)-dimensional Euclidean space which are at distance ''r'' from a fixed point of that space, where ''r'' is, as before, a positive real number: a 0-sphere is a pair of points $(-r, r)$ a 1-sphere is a Circle of radius ''r'' a 2-sphere is an ordinary sphere a 3-sphere is a sphere in 4-dimensional Euclidean space. Spheres for ''n'' > 2 are sometimes called Hypersphere s. The ''n''-sphere of unit radius centred at the origin is denoted ''S''^''n'' and is often referred to as "the" ''n''-sphere. Generalization to metric spaces More generally, in a Metric Space ''(E,d)'', the sphere of center ''x'' and radius ''r'' > 0 is the set
	The	"http://wwwinformationdelightinfo/encyclopedia/entry/Heine-Borel_theorem" class="copylinks">Heine-Borel Theorem is used in a short proof 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''<sup>''n''</sup> is also bounded Therefore it is compact

	CATEGORIES ABOUT SPHERE

	differential geometry

	elementary geometry

	surfaces

	topology

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Sphere