Sphere

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For other uses, see sphere (disambiguation).

Image:Sphere.jpg 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 sphere.

1 Geometry
2 Topology
- 2.1 Spherical Geometry
- 2.2 See also
3 External links

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Geometry

In three-dimensional Euclidean geometry, a sphere is the set of points in 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. 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.

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Equations

Image:Jade.png In analytic geometry, a sphere with center (x₀, y₀, z₀) and radius r is the set of all points (x, y, z) such that

The points on the sphere with radius r can be parametrized via

<math> x = x_0 + r \sin \theta \; \cos \phi </math>

<math> y = y_0 + r \sin \theta \; \sin \phi \qquad (0 \leq \theta \leq \pi \mbox{ and } -\pi < \phi \leq \pi) \,</math>

<math> z = z_0 + r \cos \theta \,</math>

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:

and its enclosed 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 minimizes surface area. Image:Einstein gyro gravity probe b.jpg 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.

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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 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 meridians) 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.

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Generalization to other dimensions

Spheres can be generalized to other dimensions. 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 <math>(-r, r)</math>
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 Sⁿ and is often referred to as "the" n-sphere.

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Generalization to metric spaces

More generally, in a metric space (E,d), the sphere of center x and radius r > 0 is the set

S(x;r) = { y ∈ E | d(x,y) = r } .

If the center is a distinguished point considered as origin of E, e.g. 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, i.e. in the case of a unit sphere.

In contrast to a ball, a sphere may be empty. For example, in Zⁿ with Euclidean metric, a sphere of radius r is nonempty only if r² can be written as sum of n squares of integers.

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Topology

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 described above under Geometry, but perhaps lacking its metric.

a 0-sphere is a pair of points with the discrete topology
a 1-sphere is a circle (up to homeomorphism); thus, for example, (the image of) any knot is a 1-sphere
a 2-sphere is an ordinary sphere (up to homeomorphism); thus, for example, any spheroid is a 2-sphere

The n-sphere is denoted Sⁿ. 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 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. Sⁿ is also bounded. Therefore it is compact.

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Spherical Geometry

main article Spherical geometry

In plane geometry the basic concepts are points and line. On the sphere, points are defined in the usual sense. The equivalents of lines are not defined in the usual sense of "straight line" but in the sense of "the shortest paths between points" which is called a geodesic. On the sphere the geodesics are the great circles, so the other geometric concepts are defined like in plane geometry but with lines replaced by great circles. Thus, in spherical geometry angles are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects (for example, the sum of the interior angles of a triangle exceeds 180 degrees).

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