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Sphere

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sphere

A sphere is defined as the set of all points in three-dimensional Euclidean space R^3 that are located at a distance r (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.

In the standard convention, an n-sphere has intrinsic dimension n and lies in (n+1)-dimensional Euclidean space (Hocking and Young 1988, p. 17; Maunder 1997, p. 21). The ordinary sphere is therefore a 2-sphere, denoted S^2 when its radius is 1. A circle is a 1-sphere, and a glome is a 3-sphere. Coxeter (1973, p. 125) instead indexes spheres by the dimension of the ambient Euclidean space, calling a circle a 2-sphere and an ordinary sphere a 3-sphere. This alternative convention must be considered when consulting older literature.

The term "sphere" refers to the surface only, so the ordinary 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 R are given by

S=4piR^2
(1)
V=4/3piR^3
(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 pi in terms of the sphere's circular cross section). The fact that

(V_(sphere))/(V_(circumscribed cylinder)-V_(sphere))=2
(3)

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

Any nonempty 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 R centered at the origin is given in Cartesian coordinates by

x^2+y^2+z^2=R^2,
(4)

which is a special case of the ellipsoid

(x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=1
(5)

and spheroid

(x^2+y^2)/(a^2)+(z^2)/(c^2)=1.
(6)

The Cartesian equation of a sphere centered at the point (x_0,y_0,z_0) with radius R is given by

(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=R^2.
(7)

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

x=rhocosthetasinphi
(8)
y=rhosinthetasinphi
(9)
z=rhocosphi,
(10)

where theta is an azimuthal coordinate running from 0 to 2pi (longitude), phi is a polar coordinate running from 0 to pi (colatitude), and rho is the radius. Note that there are several other notations sometimes used in which the symbols for theta and phi are interchanged or where r is used instead of rho. If rho is allowed to run from 0 to a given radius r, then a solid ball is obtained.

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

x=sqrt(r^2-u^2)costheta
(11)
y=sqrt(r^2-u^2)sintheta
(12)
z=u,
(13)

where theta runs from 0 to 2pi and u runs from -r to r.

The generalization of a sphere to n-dimensional Euclidean space is a hypersphere. It is an (n-1)-sphere, and when centered at the origin it is specified by

x_1^2+x_2^2+...+x_n^2=r^2.
(14)

The volume of the enclosed ball, V=4piR^3/3, can be found in Cartesian, cylindrical, and spherical coordinates, respectively, using the integrals

V=int_(-R)^Rint_(-sqrt(R^2-x^2))^(sqrt(R^2-x^2))int_(-sqrt(R^2-x^2-y^2))^(sqrt(R^2-x^2-y^2))dzdydx
(15)
Image=int_0^(2pi)int_0^Rint_(-sqrt(R^2-r^2))^(sqrt(R^2-r^2))rdzdrdtheta
(16)
Image=int_0^(2pi)int_0^piint_0^Rrho^2sinphidrhodphidtheta.
(17)

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

I=[2/5MR^2 0 0; 0 2/5MR^2 0; 0 0 2/5MR^2].
(18)

Converting to "standard" parametric variables a=rho, u=theta, and v=phi gives the coefficients of the first fundamental form

E=a^2sin^2v
(19)
F=0
(20)
G=a^2,
(21)

second fundamental form coefficients

e=asin^2v
(22)
f=0
(23)
g=a,
(24)

area element

dA=a^2sinvdu ^ dv,
(25)

Gaussian curvature

K=1/(a^2),
(26)

and mean curvature

H=1/a.
(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 (x_1,y_1,z_1) and (x_2,y_2,z_2) lying on a diameter is given by

(x-x_1)(x-x_2)+(y-y_1)(y-y_2)+(z-z_1)(z-z_2)=0.
(28)

Four noncoplanar points uniquely define a sphere. Given the points (x_i,y_i,z_i) with i=1, 2, 3, and 4, the sphere containing them is given by the beautiful determinant equation

|x^2+y^2+z^2 x y z 1; x_1^2+y_1^2+z_1^2 x_1 y_1 z_1 1; x_2^2+y_2^2+z_2^2 x_2 y_2 z_2 1; x_3^2+y_3^2+z_3^2 x_3 y_3 z_3 1; x_4^2+y_4^2+z_4^2 x_4 y_4 z_4 1|=0
(29)

(Beyer 1987, p. 210).

See also

Ball, Bing's Theorem, Bowl of Integers, Bubble, Circle, Cone-Sphere Intersection, Cylinder-Sphere Intersection, Dandelin Spheres, Diameter, Double Sphere, Ellipsoid, Exotic Sphere, Geodesic Dome, Glome, Hypersphere, Liebmann's Theorem, Liouville's Conformality Theorem, Mikusiński's Problem, Noise Sphere, Oblate Spheroid, Osculating Sphere, Parallelizable, Prolate Spheroid, Radius, Space Division by Spheres, Sphere Line Picking, Sphere Packing, Sphere Point Picking, Sphere-Sphere Intersection, Spherical Code, Spherical Lune, Spherical Wedge, Superegg, Supersphere, Tangent Spheres, Tennis Ball Theorem Explore this topic in the MathWorld classroom

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References

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 227, 1987.Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, 1971.Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.Eppstein, D. "Circles and Spheres." https://ics.uci.edu/~eppstein/junkyard/sphere.html.Fukagawa, H. and Pedoe, D. "Spheres," "Spheres and Ellipsoids," and "Spheres, Pyramids and Prisms". §2.2-2.6 and 9.1-9.3 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Canada: Charles Babbage Research Foundation, pp. 26-37, 69-76, 102-116, and 160-166, 1989.Geometry Center. "The Sphere." http://www.geom.uiuc.edu/zoo/toptype/sphere/.Harris, J. W. and Stocker, H. "Sphere." §4.8 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 106-108, 1998.Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, p. 10, 1999.Hocking, J. G. and Young, G. S. Topology. New York: Dover, 1988.JavaView. "Classic Surfaces from Differential Geometry: Sphere." http://www.javaview.de/demo/surface/common/PaSurface_Sphere.html.Kenison, E. and Bradley, H. C. "The Intersection of a Sphere with Another Surface." §198 in Descriptive Geometry. New York: Macmillan, 1935.Kern, W. F. and Bland, J. R. "Sphere." §33 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 87-93, 1948.Kiang, T. "An Old Chinese Way of Finding the Volume of a Sphere." Math. Gaz. 56, 88-91, 1972.Maunder, C. R. F. Algebraic Topology. New York: Dover, 1997.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, 1991.

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Sphere

Cite this as:

Weisstein, Eric W. "Sphere." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Sphere.html

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