Sphere

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Sphere

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

The globose nucleus is one of the deep cerebellar nuclei. It is located medial to the emboliform nucleus and lateral to the fastigial nucleus. This nucleus contains primarily large and small multipolar neurons....
.

A sphere (from Greek

Greek language

Greek is an Indo-European languages native to the southern Balkan peninsula, the language of the Greek people. It forms an independent branch within Indo-European....
sfa??a — sphaira, "globe, ball,") is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball

Ball

A ball is a round object with various uses. It is usually sphere but can be ovoid. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players....
or to its two-dimensional surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
. In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a sphere is the locus

Locus (mathematics)

Solid geometry

In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in....
space (R³) which are at distance r from a fixed point of that space, where r is a positive real number

Real number

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Equations in R³

In analytic geometry

Analytic geometry

Analytic geometry, usually called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra; the modern development of analytic geometry is thus suggestively called algebraic geometry....
, a sphere with center (x₀, y₀, z₀) and radius r is the locus

Locus (mathematics)

In mathematics, a locus is a collection of point which share a property. The term locus is usually used of a condition which defines a continuous figure or figures, that is, a curve....
of all points (x, y, z) such that

The points on the sphere with radius r can be parametrized via

(see also trigonometric function

Trigonometric function

In mathematics, the trigonometric functions are function s of an angle. They are important in the trigonometry of Triangle and modeling Periodic function, among many other applications....
s and spherical coordinates).

A sphere of any radius centered at zero is an integral surface of the following differential form

Differential form

In the mathematics fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates....
:

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

Surface area

Surface area is how much exposed area an object has. It is expressed in square units. If an object has flat Face , its surface area can be calculated by adding together the areas of its faces....
of a sphere with radius r is

More generally, the area element on the sphere is given in spherical coordinates by

In particular, the total area can be obtained by integration

Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...

The volume

Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
of a sphere with radius r and diameter d = 2r 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

Surface tension

Surface tension is an attractive property of the surface of a liquid. It is what causes the surface portion of liquid to be attracted to another surface, such as that of another portion of liquid ....
locally minimizes surface area. The surface area in relation to the mass of a sphere is called the specific surface area

Specific surface area

Specific surface area is a material property of solids which measures the total surface area per unit of mass, solid or bulk volume, or cross-sectional area...
. From the above stated equations it can be expressed as follows:

The circumscribed cylinder

Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the 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 known as far back as Archimedes

Archimedes

Archimedes of Syracuse was a Greek mathematics, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity....
. The assertion about the volume can be proved by using Cavalieri's principle

Cavalieri's principle

File:Cavalieri's principle.jpgIn geometry, Cavalieri's principle, sometimes called the method of indivisibles, named after Bonaventura Cavalieri, is as follows:...
.

A sphere can also be defined as the surface formed by rotating a circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
about any diameter

Diameter

In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle....
. If the circle is replaced by an ellipse

Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
, and rotated about the major axis, the shape becomes a prolate spheroid

Spheroid

A spheroid is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters....
, 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 point

Antipodal point

In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diameter opposite it ? so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter....
s. A great circle

Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere....
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. Equipped with the great-circle distance

Great-circle distance

The great-circle distance is the shortest distance between any two Point s on the surface of a sphere measured along a path on the surface of the sphere ....
, a great circle becomes the Riemannian circle

Riemannian circle

In metric space theory and Riemannian geometry, the term Riemannian circle refers to a great circle equipped with its great-circle distance. In more detail, the term refers to the circle equipped with its intrinsic Riemannian metric of a compact 1-dimensional manifold of total length 2π, as opposed to the extrinsic metric obtaine...
.

If a particular point on a sphere is (arbitrarily) designated as its north pole, then the corresponding antipodal point is called the south pole and the equator

Equator

The equator is the intersection of the Earth's surface with the Plane perpendicular to the Earth's rotation and containing the Earth's center of mass....
is the great circle that is equidistant to them. Great circles through the two poles are called lines (or meridian

Meridian (geography)

A meridian is an imaginary arc on the Earth's surface from the North Pole to the South Pole that connects all locations running along it with a given longitude....
s) of longitude

Longitude

Longitude , symbolized by the Greek character lambda , is the geographic coordinate most commonly used in cartography and global navigation for east-west measurement....
, 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

Latitude

Latitude, usually denoted symbolically by the Greek letter phi gives the location of a place on Earth north or south of the equator. Lines of Latitude are the horizontal lines shown running east-to-west on maps ....
. This terminology is also used for astronomical bodies such as the planet Earth

Earth

Earth is the third planet from the Sun. Earth is the largest of the terrestrial planets in the Solar System in diameter, mass and density. It is also referred to as the World and Wiktionary:Terra.Note that by International Astronomical Union convention, the term "Terra" is used for naming extensive land masses, rather...
, even though it is neither spherical nor even spheroid

Spheroid

A spheroid is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters....
al (see geoid

Geoid

The geoid is that equipotential surface which would coincide exactly with the mean ocean surface of the Earth, if the oceans were in equilibrium, at rest, and extended through the continents ....
).

Hemisphere

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.

The antipodal quotient of the sphere is the surface called the real projective plane

Real projective plane

In mathematics, the real projective plane is the space of lines in R3 passing through the origin. It is a non-Orientability two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedding in our usual three-dimensional space without intersecting itself....
, which can also be thought of as the northern hemisphere with antipodal points of the equator identified.

Generalization to other dimensions

Spheres can be generalized to spaces of any dimension

Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
. For any natural number

Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
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:

a 0-sphere is a pair of endpoints of an interval (-r, r) of the real line
a 1-sphere is a circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
of radius r
a 2-sphere is an ordinary sphere
a 3-sphere
3-sphere

In mathematics, a '3-sphere' is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space....
is a sphere in 4-dimensional Euclidean space.

Spheres for n > 2 are sometimes called hypersphere

Hypersphere

In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real num...
s.

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 (which is embedded in 3-dimensional space).

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

where Γ(z) is Euler's Gamma function

Gamma function

In mathematics, the Gamma function is an extension of the factorial function to real number and complex number numbers. For a complex number z with positive real part the Gamma function is defined by...
.

Another formula for surface area is

and the volume within is the surface area times or

Generalization to metric spaces

More generally, in a metric space

Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean 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 norm

Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector....
ed 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

Ball (mathematics)

In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general....
, 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

Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
, an n-sphere is defined as a space homeomorphic to the boundary of an (n+1)-ball

Ball (mathematics)

In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general....
; thus, it is homeomorphic to the Euclidean n-sphere, but perhaps lacking its metric

Metric space

a 0-sphere is a pair of points with the discrete topology
a 1-sphere is a circle (up to
Up to

In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e....
homeomorphism
Homeomorphism

In the mathematics field of topology, a homeomorphism or topological isomorphism is a continuous function between two topological spaces....
); thus, for example, (the image of) any knot
Knot (mathematics)

In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations ....
is a 1-sphere
a 2-sphere is an ordinary sphere (up to
Up to

In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e....
homeomorphism
Homeomorphism

In the mathematics field of topology, a homeomorphism or topological isomorphism is a continuous function between two topological spaces....
); thus, for example, any spheroid
Spheroid

A spheroid is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters....
is a 2-sphere

The n-sphere is denoted Sⁿ. It is an example of a compact

Compact space

In mathematics, a topological space is called compact if each of its open covers has a finite set subcover.Note: Some authors such as Nicolas Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff spaces and "quasi-compact"....
topological manifold

Topological manifold

In mathematics, a topological manifold is a Hausdorff space topological space which looks locally like Euclidean space in a sense defined below....
without boundary

Boundary (topology)

In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S....
. A sphere need not be smooth

Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
; 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 closed. Sⁿ is also bounded, therefore it is compact.

Spherical geometry

The basic elements of plane geometry

Plane geometry

In mathematics, plane geometry may mean:*geometry of a plane ,*geometry of the Euclidean plane,or sometimes a plane is any flat surface that extends without end in all directions....
are point

Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
s and line

Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
s. 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

Arc length

Determining the length of an irregular arc segment ? also called rectification of a curve ? was historically difficult. Although many methods were used for specific curves, the advent of calculus led to a general formula that provides closed-form expression in some cases....
one finds that the shortest path connecting two points lying entirely in the sphere is a segment of the great circle

Great circle

Geodesic

In mathematics, a geodesic [jee-uh-des-ik, -dee-sik] is a generalization of the notion of a "Line " to "manifolds".In presence of a Metric , geodesics are defined to be the shortest path between points on the space....
. Many theorems from classical geometry hold true for this spherical geometry as well, but many do not (see parallel postulate

Parallel postulate

In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in what is now called Euclidean geometry....
). In spherical trigonometry

Spherical trigonometry

Spherical trigonometry is a part of spherical geometry that deals with polygons on the sphere and explains how to find relations between the involved angles....
, angle

Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
s are defined between great circles. Thus spherical trigonometry is different from ordinary trigonometry

Trigonometry

Trigonometry is a branch of mathematics that deals with triangle s, particularly those plane triangles in which one angle has 90 degrees . Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships....
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

David Hilbert

David Hilbert was a Germany mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries....
and Stephan Cohn-Vossen

Stephan Cohn-Vossen

Stefan or Stephan Cohn-Vossen was a mathematician, now best known for his collaboration with David Hilbert on the 1932 book Anschauliche Geometrie, translated into English language as Geometry and the Imagination....
describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere. Several properties hold for the plane

Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
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
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
of Apollonius of Perga
Apollonius of Perga

Apollonius of Perga [Pergaeus] was a Greeks geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and Ren? Descartes....
for the circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
. This second part also holds for the plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
.
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
Reuleaux triangle

A Reuleaux polygon is a curve of constant width - that is, a curve such that, if two parallel lines are drawn tangent to the curve in any orientation, the distance between them is fixed....
. 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
Surface normal

A surface normal, or simply normal, to a Flatness is a vector which is perpendicular to that surface. A normal to a non-flat surface at a Point P on the surface is a vector perpendicular to the Tangent space to that surface at P....
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 curvature
Principal curvature

In differential geometry, the two principal curvatures at a given point of a surface measure how the surface bends by different amounts in different directions at that point....
s. It can be proved that any closed surface will have at least four points called umbilical point
Umbilical point

In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points which are locally spherical. At such points both principal curvatures are equal, and every tangent vector is a principal direction....
s. At an umbilic all the sectional curvatures are equal, in particular the principal curvature
Principal curvature

In differential geometry, the two principal curvatures at a given point of a surface measure how the surface bends by different amounts in different directions at that point....
'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
Focal surface

For a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the tangential spheres whose radii are the reciprocals of one of the principal curvatures at the point of tangency....
.
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 surface
Channel surface

A channel or canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve. One sheet of the focal surface of a channel surface will be the generating curve....
s one sheet forms a curve and the other sheet is a surface; For cone
Cone (geometry)

A cone is a dimension geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base....
s, cylinders, torus
Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle....
es 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 bubble
Soap bubble

A soap bubble is a very thin film of soap water that forms a sphere with an iridescence surface. Soap bubbles usually last for only a few moments before bursting: either on their own or on contact with another object....
s. A soap bubble will enclose a fixed volume and due to surface tension
Surface tension

Surface tension is an attractive property of the surface of a liquid. It is what causes the surface portion of liquid to be attracted to another surface, such as that of another portion of liquid ....
it will try to minimize its surface area. This is why a free floating soap bubble approximates a sphere (though external forces such as gravity will distort the bubble's shape slightly).
The sphere has the smallest total mean curvature among all convex solids with a given surface area.
The mean curvature
Mean curvature

In mathematics, the mean curvature of a surface is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedding surface in some ambient space such as Euclidean space....
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 surface
Minimal surface

In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
s have zero mean curvature.
The sphere has constant positive Gaussian curvature.
Gaussian curvature
Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, ?1 and ?2, of the given point....
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
Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
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
Pseudosphere

In geometry, a pseudosphere of radius R is a surface of curvature −1/R2 , by analogy with the sphere of radius R, which is a surface of curvature 1/R2....
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
Euler angles

The Euler angles were developed by Leonhard Euler to describe the orientation of a rigid body in dimension Euclidean space. To give an object a specific orientation it may be subjected to a sequence of three rotations described by the Euler angles....
. Thus there is a three parameter family of rotations which transform the sphere onto itself, this is the rotation group
Rotation group

In classical mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of functional composition....
, 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
Surface of revolution

A surface of revolution is a surface created by rotating a curve lying on some plane around a straight line that lies on the same plane.Examples of surfaces generated by a straight line are the cylinder and conical surfaces....
and helicoid
Helicoid

The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776....
s are the only surfaces with a one parameter family.

External links

MATHguide
MATHguide
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.)

Sphere

Equations in R3

Terminology

Hemisphere

Generalization to other dimensions

Generalization to metric spaces

Topology

Spherical geometry

Eleven properties of the sphere

See also

External links

Equations in R³