sphere - Geometry

In mathematics, the locus in space of all points equidistant from a fixed point (the centre). The distance of each point from the centre is the radius r of the sphere. The surface area of a sphere is 4?r²; the volume of a sphere is . Archimedes discovered that the surface area of a sphere is equal to the curved surface area of the cylinder circumscribing the sphere, and asked that this should be commemorated on his tombstone. When Cicero was serving in Syracuse 200 years after the death of Archimedes, he discovered a tomb which he recognized as that of Archimedes, because it displayed the diagram.

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.

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

(see also trigonometric functions and spherical coordinates).

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 locally 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.

A sphere can also be defined as the surface formed by rotating a circle about any diameter.

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.

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. Circles on the sphere that are parallel to the equator are lines of latitude.

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.

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 ( − 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 > 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

where Γ(z) is Euler's Gamma function.

Generalization to metric spaces

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

In contrast to a ball, a sphere may be empty, even for a large radius. thus, for example, (the image of) any knot is a 1-sphere a 2-sphere is an ordinary sphere (up to homeomorphism);

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; In spherical trigonometry, angles are defined between great circles.