Sphere

:"Globose" redirects here. See also Globose nucleus."A sphere (from Greek "σφαίρα" - "sphaira", "globe, ball," [ [http://www.perseus.tufts.edu/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3D%23101561 Sphaira, Henry George Liddell, Robert Scott, "A Greek-English Lexicon", at Perseus] ] ) 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 (R³) 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.

Equations in R^"3"

In analytic geometry, a sphere with center ("x"₀, "y"₀, "z"₀) and radius "r" is the locus 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

:$,\; x\; =\; x\_0\; +\; r\; cos\; varphi\; ;\; sin\; heta$:$,\; y\; =\; y\_0\; +\; r\; sin\; varphi\; ;\; sin\; heta\; qquad\; (0\; leq\; varphi\; leq\; 2pi\; mbox\{\; and\; \}\; 0\; leq\; heta\; leq\; pi\; )\; ,$:$,\; z\; =\; z\_0\; +\; r\; cos\; heta\; ,$

(see also trigonometric functions 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

:$A\; =\; 4\; pi\; r^2\; ,$

so the radius from surface area is

:$r\; =\; left(frac\{A\}\{4pi\}\; ight)^frac\{1\}\{2\}.$

Its volume is

:$V\; =\; frac\{4\}\{3\}pi\; r^3.$

so the radius from volume is

:$r\; =\; left(V\; frac\{3\}\{4pi\}\; ight)^frac\{1\}\{3\}.$

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 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:

:$SSA\; =\; frac\{A\}\{V\; ho\}\; =\; frac\{3\}\{r\; ho\}.$

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.

Terminology

Pairs of points on a sphere that lie on a straight line through its center are called antipodal points.A great circle is a circle on the sphere that has the same center and radius as the sphere, and consequently divides it into two equal parts. The shortest distance between two distinct non-antipodal points on the surface and measured along the surface, is on the unique great circle passing through the two points. Equipped with the great-circle distance, a great circle becomes the Riemannian circle.

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.

Generalization to other dimensions

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:

* a 0-sphere is a pair of endpoints of an interval (−"r", "r") of the real line
* 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 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

:$2\; frac\{pi^\{n/2\{Gamma(n/2)\}$

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

:

Generalization to metric spaces

More generally, in a metric space ("E","d"), the sphere of center "x" and radius (computer animation showing how the inside of a sphere can turn outside.)