# Dictionary Definition

spherical adj

1 of or relating to spheres or resembling a
sphere; "spherical geometry" [ant: nonspherical]

2 having the shape of a sphere or ball; "a
spherical object"; "nearly orbicular in shape"; "little globular
houses like mud-wasp nests"- Zane Grey [syn: ball-shaped,
global, globose, globular, orbicular, spheric]

# User Contributed Dictionary

## English

### Etymology

spherical < Latin spheric-us (< Greek σφαιρικός) + -al### Adjective

- shaped like a sphere.
- (no comparative or superlative) Of, or pertaining to, spheres.

#### Derived terms

- spherical aberration
- spherical angle
- spherical cap
- spherical distance
- spherical geometry
- spherical lune
- spherical sector
- spherical segment
- spherical triangle
- spherical trigonometry
- spherical wedge

#### Translations

shaped like a sphere

of or relating to a sphere or spheres

# Extensive Definition

- "Globose" redirects here. See also Globose nucleus.

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.

## Equations in R3

In analytic geometry, a sphere with center (x0, y0, z0) and radius r is the locus 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 \cos \varphi \; \sin \theta
- \, y = y_0 + r \sin \varphi \; \sin \theta \qquad (0 \leq \varphi \leq 2\pi \mbox 0 \leq \theta \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 \,

so the radius from surface area
is

- r = \left(\frac \right)^\frac.

Its volume is

- V = \frac\pi r^3.

so the radius from volume is

- r = \left(V \frac\right)^\frac.

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 surface area in
relation to the mass of a sphere is called the specific
surface area. From the above stated equations it can be
expressed as follows:

- SSA = \frac = \frac.

The circumscribed cylinder
for a given sphere has a volume which is 3/2 times the volume of
the sphere, and also the curved portion has a surface area which is
equal to 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.

## 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 distinct non-antipodal points on the surface and measured along the surface, is on the unique 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 of
rotation. 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 is 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 spaces of any dimension. 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 a fixed distance r from a central point of that space, where r is, as before, a positive real number. In particular:Spheres for n > 2 are sometimes called
hyperspheres.

The n-sphere of unit radius centred at the origin
is denoted S'n 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.

The surface area of the (n−1)-sphere of radius 1
is

- 2 \frac

where Γ(z) is Euler's Gamma
function.

Another formula for surface area is

\begin \displaystyle \frac , & \text n \text;
\\ \\ \displaystyle \frac , & \text n \text. \end

and the volume within is the surface area times
or

\begin \displaystyle \frac , & \text n \text;
\\ \\ \displaystyle \frac , & \text n \text. \end

## Generalization to metric spaces

More generally, in a metric space
(E,d), the sphere of center x and radius is the set of points y
such that d(x,y) = r.

If the center is a distinguished point considered
as origin of E, as 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, as in the case
of a unit
sphere.

In contrast to a ball,
a sphere may be an empty set, even for a large radius. 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.

## 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.

- 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
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 implies 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.

## 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.## Eleven properties of the sphere

In their book Geometry and the imagination David Hilbert and Stephan Cohn-Vossen describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere. Several properties hold for the plane which can be thought of as a sphere with infinite radius. These properties are:- The points on the sphere are all the same distance from a fixed
point. Also, the ratio of the distance of its points from two fixed
points is constant.
- The first part is the usual definition of the sphere and determines it uniquely. The second part can be easily deduced and follows a similar result of Apollonius of Perga for the circle. This second part also holds for the plane.

- The contours and plane sections of the sphere are circles.
- This property defines the sphere uniquely.

- The sphere has constant width and constant girth.
- The width of a surface is the distance between pairs of parallel tangent planes. There are numerous other closed convex surfaces which have constant width, for example Meissner's tetrahedron. The girth of a surface is the circumference of the boundary of its orthogonal projection on to a plane. It can be proved that each of these properties implies the other.#All points of a sphere are umbilics.
- At any point on a surface we can find a normal direction which is at right angles to the surface, for the sphere these on the lines radiating out from the center of the sphere. The intersection of a plane containing the normal with the surface will form a curve called a normal section and the curvature of this curve is the sectional curvature. For most points on a surfaces different sections will have different curvatures, the maximum and minimum values of these are called the principal curvatures. It can be proved that any closed surface will have at least four points called umbilical points. At an umbilic all the sectional curvatures are equal, in particular the principal curvature's are equal. Umbilical points can be thought of as the points where the surface is closely approximated by a sphere.
- For the sphere the curvatures of all normal sections are equal, so every point is an umbilic. The sphere and plane are the only surfaces with this property.

- The sphere does not have a surface of centers.
- For a given normal section there is a circle whose curvature is the same as the sectional curvature, is tangent to the surface and whose center lines along on the normal line. Take the two center corresponding to the maximum and minimum sectional curvatures these are called the focal points, and the set of all such centers forms the focal surface.
- For most surfaces the focal surface forms two sheets each of which is a surface and which come together at umbilical points. There are a number of special cases. For channel surfaces one sheet forms a curve and the other sheet is a surface; For cones, cylinders, toruses and cyclides both sheets form curves. For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This is a unique property of the sphere.

- All geodesics of the sphere are closed curves.
- Geodesics are curves on a surface which give the shortest distance between two points. They are generalisation of the concept of a straight line in the plane. For the sphere the geodesics are great circles. There are many other surfaces with this property.

- Of all the solids having a given volume, the sphere is the one
with the smallest surface area; of all solids having a given
surface area, the sphere is the one having the greatest volume.
- These properties define the sphere uniquely. These properties can be seen by observing soap bubbles. A soap bubble will enclose a fixed volume and due to surface tension it will try to minimize its surface area. Therefore a free floating soap bubble will be approximately a sphere, factors like gravity will cause a slight distortion.

- The sphere has the smallest total mean curvature among all
convex solids with a given surface area.
- The mean curvature is the average of the two principal curvatures and as these are constant at all points of the sphere then so is the mean curvature.

- The sphere has constant positive mean curvature.
- The sphere is the only surface without boundary or singularities with constant positive mean curvature. There are other surfaces with constant mean curvature, the minimal surfaces have zero mean curvature.

- The sphere has constant positive Gaussian curvature.
- Gaussian curvature is the product of the two principle curvatures. It is an intrinsic property which can be determined by measuring length and angles and does not depend on the way the surface is embedded in space. Hence, bending a surface will not alter the Gaussian curvature and other surfaces with constant positive Gaussian curvature can be obtained by cutting a small slit in the sphere and bending it. All these other surfaces would have boundaries and the sphere is the only surface without boundary with constant positive Gaussian curvature. The pseudosphere is an example of a surface with constant negative Gaussian curvature.

- The sphere is transformed into itself by a three-parameter
family of rigid motions.
- Consider a unit sphere place at the origin, a rotation around the x, y or z axis will map the sphere onto itself, indeed any rotation about a line through the origin can be expressed as a combination of rotations around the three coordinate axis, see Euler angles. Thus there is a three parameter family of rotations which transform the sphere onto itself, this is the rotation group, SO(3). The plane is the only other surface with a three parameter family of transformations (translations along the x and y axis and rotations around the origin). Circular cylinders are the only surfaces with two parameter families of rigid motions and the surfaces of revolution and helicoids are the only surfaces with a one parameter family.

## References

## See also

- 3-sphere
- Alexander horned sphere
- Ball (mathematics)
- Banach-Tarski Paradox
- Circle
- Curvature
- Directional statistics
- Dome (mathematics)
- Dyson sphere
- Homology sphere
- Homotopy groups of spheres
- Homotopy sphere
- Hypersphere
- Metric space
- Pseudosphere
- Riemann sphere
- Smale's paradox
- Solid angle
- Spherical cap
- Spherical coordinates
- Spherical Earth

## External links

- Surface Area MATHguide
- Volume MATHguide
- Mathworld website
- calculate area and volume with your own radius-values to understand the equations
- MathAce » Spheres MathAce's basic article on spheres - good step by step explanation of equation transformation.
- (computer animation showing how the inside of a sphere can turn outside.)

spherical in Arabic: كرة

spherical in Bulgarian: Сфера

spherical in Catalan: Esfera

spherical in Czech: Koule

spherical in Chuvash: Сфера

spherical in Danish: Kugle

spherical in German: Kugel

spherical in Spanish: Esfera

spherical in Esperanto: Sfero

spherical in Persian: گوی

spherical in French: Sphère

spherical in Scottish Gaelic: Cruinne

spherical in Croatian: Sfera

spherical in Interlingua (International
Auxiliary Language Association): Sphera

spherical in Italian: Sfera

spherical in Hebrew: ספירה (גאומטריה)

spherical in Latin: Sphaera

spherical in Latvian: Sfēra

spherical in Hungarian: Gömb

spherical in Dutch: Bol (lichaam)

spherical in Japanese: 球面

spherical in Norwegian: Kule (geometri)

spherical in Norwegian Nynorsk: Sfære

spherical in Polish: Sfera

spherical in Portuguese: Esfera
(geometria)

spherical in Russian: Сфера

spherical in Simple English: Sphere

spherical in Slovenian: Sfera

spherical in Serbian: Сфера

spherical in Finnish: Pallo (geometria)

spherical in Swedish: Sfär

spherical in Thai: ทรงกลม

spherical in Turkish: Küre (geometri)

spherical in Ukrainian: Сфера

spherical in Chinese: 球面

# Synonyms, Antonyms and Related Words

3-D, bulblike, bulbous, cubic, dimensional, ellipsoid, flat, fourth-dimensional, global, globate, globe-shaped, globed, globelike, globoid, globose, globular, hemispheric, obovoid, orb, orbed, orbic, orbicular, orbiculate, orblike, orby, ovoid, proportional, round, space, space-time, spatial, spatiotemporal,
sphere-shaped, spherelike, spheric, spheriform, spheroid, spheroidal, stereoscopic, superficial, surface, three-dimensional,
two-dimensional, volumetric