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Encyclopedia > Spherical
For other uses, see sphere (disambiguation).

A sphere is, roughly speaking, a ball-shaped object. In non-mathematical usage, the term sphere is often used for something "solid" (which mathematicians call ball). But in mathematics, sphere refers to the boundary of a ball, which is "hollow". This article deals with the mathematical concept of sphere.

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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.

### Equations

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

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.

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.

### Generalization to higher dimensions

Spheres can be generalized to higher 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 ( - 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 hyperspheres. The n-sphere of unit radius centred at the origin is denoted Sn and is often referred to as "the" n-sphere.

## Topology

In topology, an n-sphere is defined as 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, example, 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 n-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 an 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. Results from FactBites:

 Spherical trigonometry - Wikipedia, the free encyclopedia (603 words) Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. Hence, a spherical triangle is specified as usual by its corner angles and its sides, but the sides are given not by their length, but by their arc angle. Spherical Trigonometry — for the use of colleges and schools by I. Todhunter, M.A., F.R.S. Historical Math Monograph posted by Cornell University Library.
 Spherical aberration (1391 words) Spherical aberration (SA) is an image imperfection that is due to the spherical lens shape. Spherical aberration is uniform over the field, in the sense that the longitudinal focus difference between the lens margins and center does not depend on the obliquity of the incident light. Spherical aberration is responsible for the rugged blur character at negative distances, and the transition to a smoother appearance at positive distances.
More results at FactBites »

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