Sphere

A sphere is defined as the set of all points in three-dimensional Euclidean space that are located at a distance (the "radius") from a given point (the "center"). Twice the radius is called the diameter, and pairs of points on the sphere on opposite sides of a diameter are called antipodes.

Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of "-sphere," with geometers referring to the number of coordinates in the underlying space ("thus a two-dimensional sphere is a circle," Coxeter 1973, p. 125) and topologists referring to the dimension of the surface itself ("the -dimensional sphere is defined to be the set of all points in satisfying ," Hocking and Young 1988, p. 17; "the -sphere is ," Maunder 1997, p. 21). As a result, geometers call the surface of the usual sphere the 3-sphere, while topologists refer to it as the 2-sphere and denote it .

Regardless of the choice of convention for indexing the number of dimensions of a sphere, the term "sphere" refers to the surface only, so the usual sphere is a two-dimensional surface. The colloquial practice of using the term "sphere" to refer to the interior of a sphere is therefore discouraged, with the interior of the sphere (i.e., the "solid sphere") being more properly termed a "ball."

The sphere is implemented in the Wolfram Language as Sphere[x, y, z, r].

The surface area of a sphere and volume of the ball of radius are given by

			(1)
			(2)

(Beyer 1987, p. 130). In On the Sphere and Cylinder (ca. 225 BC), Archimedes became the first to derive these equations (although he expressed in terms of the sphere's circular cross section). The fact that

(3)

was also known to Archimedes (Steinhaus 1999, p. 223; Wells 1991, pp. 236-237).

Any cross section through a sphere is a circle (or, in the degenerate case where the slicing plane is tangent to the sphere, a point). The size of the circle is maximized when the plane defining the cross section passes through a diameter.

The equation of a sphere of radius centered at the origin is given in Cartesian coordinates by

(4)

which is a special case of the ellipsoid

(5)

and spheroid

(6)

The Cartesian equation of a sphere centered at the point with radius is given by

(7)

A sphere with center at the origin may also be specified in spherical coordinates by

			(8)
			(9)
			(10)

where is an azimuthal coordinate running from 0 to (longitude), is a polar coordinate running from 0 to (colatitude), and is the radius. Note that there are several other notations sometimes used in which the symbols for and are interchanged or where is used instead of . If is allowed to run from 0 to a given radius , then a solid ball is obtained.

A sphere with center at the origin may also be represented parametrically by letting , so

			(11)
			(12)
			(13)

where runs from 0 to and runs from to .

The generalization of a sphere in dimensions is called a hypersphere. An -dimensional hypersphere, also known as an -sphere (in a geometer's convention), that is centered at the origin can therefore be specified by the equation

(14)

Of course, topologists would regard this equation as instead describing an -sphere.

The volume of the sphere, , can be found in Cartesian, cylindrical, and spherical coordinates, respectively, using the integrals

			(15)
			(16)
			(17)

The interior of the sphere of radius and mass has moment of inertia tensor

(18)

Converting to "standard" parametric variables , , and gives the coefficients of the first fundamental form

			(19)
			(20)
			(21)

second fundamental form coefficients

			(22)
			(23)
			(24)

area element

(25)

Gaussian curvature

(26)

and mean curvature

(27)

Given two points on a sphere, the shortest path on the surface of the sphere which connects them (the geodesic) is an arc of a circle known as a great circle. The equation of the sphere with points and lying on a diameter is given by

(28)

Four points are sufficient to uniquely define a sphere. Given the points with , 2, 3, and 4, the sphere containing them is given by the beautiful determinant equation

(29)

(Beyer 1987, p. 210).

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