Sphere

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.

Definitions/Postulates

Great circle - The intersection of the sphere and a plane that contains the center of the sphere

A great circle is finite and returns to its origional starting point

There is a unique circle passing through any pair of nonpolar points

Polar points - The intersection of a sphere and a line passing through the origin of the sphere

Polar points are opposite each other on a sphere

Arc of a Great circle - The shortest distance between two points on a sphere (while not going through the center)

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.

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

<math>(x - x_0 )^2 + (y - y_0 )^2 + ( z - z_0 )^2 = r^2 \,<math>

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>

(see also trigonometric functions and spherical coordinates).

A sphere of any radius centered at the origin is described by the following differential equation:

<math> x \, dx + y \, dy + z \, dz = 0. <math>

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:

<math>A = 4 \pi r^2 \,<math>

and its enclosed volume is:

<math>V = \frac{4 \pi r^3}{3}<math>

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 <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.

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 mentionned 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.

See also

• Ball (mathematics)
• Circle, 3-sphere, hypersphere
• Metric space
• Riemann sphere
• Smale's paradox

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 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.

See also

• Alexander horned sphere
• Ball (mathematics)
• Homology sphere
• Homotopy sphere
• Riemann sphere
• 3-sphere, hypersphere

External links

• Mathworld website (http://mathworld.wolfram.com/Hypersphere.html)de:Kugel

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