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SPHERE

For other senses of this word, see sphere (disambiguation).

A sphere is a perfectly 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. 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.

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.

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[edit] 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

(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 \theta \; \cos \phi
y = y_0 + r \sin \theta \; \sin \phi \qquad (0 \leq \theta \leq \pi \mbox{ and } -\pi < \phi \leq \pi) \,
z = z_0 + r \cos \theta \,

(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 \,

and its enclosed volume is

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

An image of one of the most accurate spheres ever created by humans, as it refracts 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.
An image of one of the most accurate spheres ever created by humans, as it refracts 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 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.

[edit] 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 points on the surface and measured along the surface, is on a great circle passing through the two points.

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 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.

[edit] Generalization to other dimensions

Spheres can be generalized to other dimensions. For any natural number n, an n-sphere, often written as Sn, 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. For n> 0, the n-sphere is the simply connected n-dimensional manifold of constant, positive curvature, and can also be thought of embedded in an n+1-dimensional manifold, as the surface or boundary of a ball in the n+1-dimensional manifold.

  • a 0-sphere is a pair of points on the line at ( − 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. Note that the ordinary sphere is a 2-sphere, because it is a 2-dimensional surface, though it is also a 3-dimensional object because it can be embedded in ordinary 3-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.

[edit] 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, even for a large radius. For example, in Zn with Euclidean metric, a sphere of radius r is nonempty only if r 2 can be written as sum of n squares of integers.

[edit] 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, but perhaps lacking its metric.

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 a closed. Sn is also bounded. Therefore it is compact.

[edit] Spherical geometry

Main article: Spherical geometry

The basic elements of plane geometry are points and lines. On the sphere, points are defined in the usual sense, but the analogue of "line" may not be immediately apparent. If one measures by arc length one finds that the shortest path connecting two points lying entirely in the sphere is a segment of the great circle containing the points; see geodesic. Many theorems from classical geometry hold true for this spherical geometry as well, but many do not (see parallel postulate). In spherical trigonometry, angles are defined between great circles. Thus spherical trigonometry is different from ordinary trigonometry in many respects. For example, the sum of the interior angles of a spherical triangle exceeds 180 degrees. Also, any two similar spherical triangles are congruent.

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