A sphere (from Greek —, "globe, ball"^{[1]}) is a geometrical object in threedimensional space that is the surface of a ball. Like a circle in a twodimensional space, a sphere is defined mathematically as the set of points that are all at the same distance from a given point in a threedimensional space.^{[2]} This distance is the radius of the ball, which is made up from all points with a distance less than (or, for a closed ball, less than or equal to) from the given point, which is the center of the mathematical ball. These are also referred to as the radius and center of the sphere, respectively. The longest straight line segment through the ball, connecting two points of the sphere, passes through the center and its length is thus twice the radius; it is a diameter of both the sphere and its ball.
While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in mathematics the above distinction is made between a sphere, which is a twodimensional closed surface embedded in a threedimensional Euclidean space, and a ball, which is a threedimensional shape that includes the sphere and everything inside the sphere (a closed ball), or, more often, just the points inside, but not on the sphere (an open ball). The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid. This is analogous to the situation in the plane, where the terms "circle" and "disk" can also be confounded.
See also: trigonometric function and spherical coordinates.
In analytic geometry, a sphere with center and radius is the locus of all points such that
(xx_{0})^{2}+(yy_{0})^{2}+(zz_{0})^{2}=r^{2.}
Let be real numbers with and put
x_{0}=
b  
a 
, y_{0}=
c  
a 
, z_{0}=
d  
a 
, \rho=
b^{2}+c^{2+d}^{2}ae  
a^{2} 
.
f(x,y,z)=a(x^{2}+y^{2}+z^{2)}+2(bx+cy+dz)+e=0
\rho<0
\rho=0
f(x,y,z)=0
P_{0}=(x_{0,y}_{0,z}_{0)}
\rho>0
f(x,y,z)=0
P_{0}
\sqrt\rho
If in the above equation is zero then is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius whose center is a point at infinity.^{[3]}
The points on the sphere with radius
r>0
(x_{0,y}_{0,z}_{0)}
\begin{align}x&=x_{0}+r\sin\theta \cos\varphi\\ y&=y_{0}+r\sin\theta \sin\varphi (0\leq\theta\leq\pi, 0\leq\varphi<2\pi)\\ z&=z_{0}+r\cos\theta\end{align}
\theta
\varphi
A sphere of any radius centered at zero is an integral surface of the following differential form:
xdx+ydy+zdz=0.
This equation reflects that position and velocity vectors of a point, and, traveling on the sphere are always orthogonal to each other.
A sphere can also be constructed as the surface formed by rotating a circle about any of its diameters. Since a circle is a special type of ellipse, a sphere is a special type of ellipsoid of revolution. Replacing the circle with an ellipse rotated about its major axis, the shape becomes a prolate spheroid; rotated about the minor axis, an oblate spheroid.^{[5]}
In three dimensions, the volume inside a sphere (that is, the volume of a ball, but classically referred to as the volume of a sphere) is
V=
4  
3 
\pir^{3}=
\pi  
6 
d^{3} ≈ 0.5236 ⋅ d^{3}
At any given, the incremental volume equals the product of the crosssectional area of the disk at and its thickness :
\deltaV ≈ \piy^{2} ⋅ \deltax.
The total volume is the summation of all incremental volumes:
V ≈ \sum\piy^{2} ⋅ \deltax.
In the limit as approaches zero, this equation becomes:
V=
r  
\int  
r 
\piy^{2}dx.
At any given, a rightangled triangle connects, and to the origin; hence, applying the Pythagorean theorem yields:
y^{2}=r^{2}x^{2.}
Using this substitution gives
V=
r  
\int  
r 
\pi\left(r^{2}x^{2\right)dx,}
which can be evaluated to give the result
V=\pi\left[r^{2x}
x^{3}  
3 
r  
\right]  
r 
=\pi\left(r^{3}
r^{3}  
3 
\right)\pi\left(r^{3}+
r^{3}  
3 
\right)=
43\pi  
r 
^{3.}
An alternative formula is found using spherical coordinates, with volume element
dV=r^{2\sin\theta}drd\thetad\varphi
2\pi  
V=\int  
0 
\pi  
\int  
0 
r  
\int  
0 
r'^{2\sin\theta}dr'd\thetad\varphi=2\pi
\pi  
\int  
0 
r  
\int  
0 
r'^{2\sin\theta}dr'd\theta =4\pi
r  
\int  
0 
r'^{2}dr' =
43\pi  
r 
^{3.}
The surface area of a sphere of radius is:
A=4\pir^{2.}
Archimedes first derived this formula from the fact that the projection to the lateral surface of a circumscribed cylinder is areapreserving.^{[8]} Another approach to obtaining the formula comes from the fact that it equals the derivative of the formula for the volume with respect to because the total volume inside a sphere of radius can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius . At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius is simply the product of the surface area at radius and the infinitesimal thickness.
At any given radius, the incremental volume equals the product of the surface area at radius and the thickness of a shell :
\deltaV ≈ A(r) ⋅ \deltar.
The total volume is the summation of all shell volumes:
V ≈ \sumA(r) ⋅ \deltar.
In the limit as approaches zero^{[9]} this equation becomes:
V=
r  
\int  
0 
A(r)dr.
Substitute :
43\pi  
r 
^{3}=
r  
\int  
0 
A(r)dr.
Differentiating both sides of this equation with respect to yields as a function of :
4\pir^{2}=A(r).
This is generally abbreviated as:
A=4\pir^{2,}
Alternatively, the area element on the sphere is given in spherical coordinates by . In Cartesian coordinates, the area element is
dS=  r  

The total area can thus be obtained by integration:
A=
2\pi  
\int  
0 
\pi  
\int  
0 
r^{2}\sin\thetad\thetad\varphi=4\pir^{2.}
The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area.^{[10]} The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the surface tension locally minimizes surface area.
The surface area relative to the mass of a ball is called the specific surface area and can be expressed from the above stated equations as
SSA=
A  
V\rho 
=
3  
r\rho 
,
A sphere is uniquely determined by four points that are not coplanar. More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc.^{[11]} This property is analogous to the property that three noncollinear points determine a unique circle in a plane.
Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle.
By examining the common solutions of the equations of two spheres, it can be seen that two spheres intersect in a circle and the plane containing that circle is called the radical plane of the intersecting spheres.^{[12]} Although the radical plane is a real plane, the circle may be imaginary (the spheres have no real point in common) or consist of a single point (the spheres are tangent at that point).^{[13]}
The angle between two spheres at a real point of intersection is the dihedral angle determined by the tangent planes to the spheres at that point. Two spheres intersect at the same angle at all points of their circle of intersection.^{[14]} They intersect at right angles (are orthogonal) if and only if the square of the distance between their centers is equal to the sum of the squares of their radii.
If and are the equations of two distinct spheres then
sf(x,y,z)+tg(x,y,z)=0
A great circle on the sphere has the same center and radius as the sphere—consequently dividing it into two equal parts. The plane sections of a sphere are called spheric sections—which are either great circles for planes through the sphere's center or small circles for all others.
Any plane that includes the center of a sphere divides it into two equal hemispheres. Any two intersecting planes that include the center of a sphere subdivide the sphere into four lunes or biangles, the vertices of which coincide with the antipodal points lying on the line of intersection of the planes.
Any pair of points on a sphere that lie on a straight line through the sphere's center (i.e. the diameter) are called antipodal points—on the sphere, the distance between them is exactly half the length of the circumference. Any other (i.e. not antipodal) pair of distinct points on a sphere
Spherical geometry shares many analogous properties to Euclidean once equipped with this "greatcircle distance".
And a much more abstract generalization of geometry also uses the same distance concept in the Riemannian circle.
The hemisphere is conjectured to be the optimal (least area) isometric filling of the Riemannian circle.
The antipodal quotient of the sphere is the surface called the real projective plane, which can also be thought of as the Northern Hemisphere with antipodal points of the equator identified.
Terms borrowed directly from geography of the Earth, despite its spheroidal shape having greater or lesser departures from a perfect sphere (see geoid), are widely wellunderstood. In geometry unrelated to astronomical bodies, geocentric terminology should be used only for illustration and noted as such, unless there is no chance of misunderstanding.
If a particular point on a sphere is (arbitrarily) designated as its north pole, its antipodal point is called the south pole. The great circle equidistant to each is then the equator. Great circles through the poles are called lines of longitude (or meridians). A line not on the sphere but through its center connecting the two poles may be called the axis of rotation. Circles on the sphere that are parallel (i.e. not great circles) to the equator are lines of latitude.
In topology, an sphere is defined as a space homeomorphic to the boundary of an ball; thus, it is homeomorphic to the Euclidean sphere, but perhaps lacking its metric.
The sphere is denoted . 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 (an exotic sphere).
The Heine–Borel theorem implies that a Euclidean sphere is compact. The sphere is the inverse image of a onepoint set under the continuous function . Therefore, the sphere is closed. is also bounded; therefore it is compact.
Remarkably, it is possible to turn an ordinary sphere inside out in a threedimensional space with possible selfintersections but without creating any crease, in a process called sphere eversion.
See main article: Spherical geometry. The basic elements of Euclidean plane geometry are points and lines. On the sphere, points are defined in the usual sense. The analogue of the "line" is the geodesic, which is a great circle; the defining characteristic of a great circle is that the plane containing all its points also passes through the center of the sphere. Measuring by arc length shows that the shortest path between two points lying on the sphere is the shorter segment of the great circle that includes the points.
Many theorems from classical geometry hold true for spherical geometry as well, but not all do because the sphere fails to satisfy some of classical geometry's postulates, including the parallel postulate. In spherical trigonometry, angles are defined between great circles. Spherical trigonometry differs from ordinary trigonometry in many respects. For example, the sum of the interior angles of a spherical triangle always exceeds 180 degrees. Also, any two similar spherical triangles are congruent.
See main article: Circle of a sphere. Circles on the sphere are, like circles in the plane, made up of all points a certain distance from a fixed point on the sphere. The intersection of a sphere and a plane is a circle, a point, or empty.
More complicated surfaces may intersect a sphere in circles, too: the intersection of a sphere with a surface of revolution, whose axis contains the center of the sphere (are coaxial), consists of circles and/or points if not empty. For example, the diagram to the right shows the intersection of a sphere and a cylinder, which consists of two circles. If the cylinder radius were that of the sphere, the intersection would be a single circle. If the cylinder radius were larger than that of the sphere, the intersection would be empty.
See main article: Rhumb line. In navigation, a rhumb line or loxodrome is an arc crossing all meridians of longitude at the same angle. Loxodromes are the same as straight lines in the Mercator projection. A rhumb line is not a spherical spiral. Except for some simple cases, the formula of a rhumb line is complicated.
\varphi
\theta
\varphi=c \theta, c>0
c=1
c>2
See main article: Spherical conic. The analog of a conic section on the sphere is a spherical conic, a quartic curve which can be defined in several equivalent ways, including:
Many theorems relating to planar conic sections also extend to spherical conics.
If a sphere is intersected by another surface, there may be more complicated spherical curves.
See main article: Sphere–cylinder intersection.
The intersection of the sphere with equation
x^{2+y}^{2+z}^{2=r}^{2 }
2+z  
(yy  
0) 
^{2=a}^{2,} y_{0\ne}0
x^{2+y}^{2+z}^{2r}^{2=0}
2+z  
(yy  
0) 
^{2a}^{2=0 .}
In their book Geometry and the Imagination,^{[15]} David Hilbert and Stephan CohnVossen 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 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.
This property defines the sphere uniquely.
The width of a surface is the distance between pairs of parallel tangent planes. Numerous other closed convex surfaces have constant width, for example the Meissner body. The girth of a surface is the circumference of the boundary of its orthogonal projection on to a plane. Each of these properties implies the other.
At any point on a surface a normal direction is at right angles to the surface because the sphere these are the lines radiating out from the center of the sphere. The intersection of a plane that contains the normal with the surface will form a curve that is called a normal section, and the curvature of this curve is the normal curvature. For most points on most surfaces, different sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures. 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 curvatures 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.
For a given normal section exists a circle of curvature that equals the sectional curvature, is tangent to the surface, and the center lines of which lie along on the normal line. For example, the two centers corresponding to the maximum and minimum sectional curvatures 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 that are each a surface and meet at umbilical points. Several cases are special:
* For channel surfaces one sheet forms a curve and the other sheet is a surface
* For cones, cylinders, tori 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 property is unique to the sphere.
Geodesics are curves on a surface that give the shortest distance between two points. They are a generalization of the concept of a straight line in the plane. For the sphere the geodesics are great circles. Many other surfaces share this property.
It follows from isoperimetric inequality. These properties define the sphere uniquely and can be seen in soap bubbles: a soap bubble will enclose a fixed volume, and surface tension minimizes its surface area for that volume. A freely floating soap bubble therefore approximates a sphere (though such external forces as gravity will slightly distort the bubble's shape). It can also be seen in planets and stars where gravity minimizes surface area for large celestial bodies.
The mean curvature is the average of the two principal curvatures, which is constant because the two principal curvatures are constant at all points of the sphere.
The sphere is the only imbedded surface that lacks boundary or singularities with constant positive mean curvature. Other such immersed surfaces as minimal surfaces have constant mean curvature.
Gaussian curvature is the product of the two principal curvatures. It is an intrinsic property that can be determined by measuring length and angles and is independent of how 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 that lacks a boundary with constant, positive Gaussian curvature. The pseudosphere is an example of a surface with constant negative Gaussian curvature.
Rotating around any axis a unit sphere at the origin will map the sphere onto itself. Any rotation about a line through the origin can be expressed as a combination of rotations around the threecoordinate axis (see Euler angles). Therefore, a threeparameter family of rotations exists such that each rotation transforms the sphere onto itself; this family is the rotation group SO(3). The plane is the only other surface with a threeparameter family of transformations (translations along the  and axes and rotations around the origin). Circular cylinders are the only surfaces with twoparameter families of rigid motions and the surfaces of revolution and helicoids are the only surfaces with a oneparameter family.
The locus of points in the space such that the sum of the
2m
d_{i}
T_{n}
R
n  
\sum  
i=1 
2m  
d  
i 
>nR^{2m}
whose center is at the centroid of the
T_{n}
The values of the
m
n
•
m
•
m
•
m
See main article: nsphere and Metric space.
Spheres can be generalized to spaces of any number of dimensions. For any natural number, an "sphere," often written as, is the set of points in dimensional Euclidean space that are at a fixed distance from a central point of that space, where is, as before, a positive real number. In particular:
Spheres for are sometimes called hyperspheres.
The sphere of unit radius centered at the origin is denoted and is often referred to as "the" sphere. Note that the ordinary sphere is a 2sphere, because it is a 2dimensional surface (which is embedded in 3dimensional space).
The surface area of the unit sphere is
 

where is Euler's gamma function.
Another expression for the surface area is
\begin{cases} \displaystyle
(2\pi)^{n/2}r^{n1}  
2 ⋅ 4 … (n2) 
,&ifniseven;\ \\ \displaystyle
2(2\pi)^{(n1)/2}r^{n1}  
1 ⋅ 3 … (n2) 
,&ifnisodd. \end{cases}
and the volume is the surface area times or
\begin{cases} \displaystyle
(2\pi)^{n/2}r^{n}  
2 ⋅ 4 … n 
,&ifniseven;\ \\ \displaystyle
2(2\pi)^{(n1)/2}r^{n}  
1 ⋅ 3 … n 
,&ifnisodd. \end{cases}
General recursive formulas also exist for the volume of an ball.
More generally, in a metric space, the sphere of center and radius is the set of points such that .
If the center is a distinguished point that is considered to be the origin of, as in a normed space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken to equal one, as in the case of a unit sphere.
Unlike a ball, even a large sphere may be an empty set. For example, in with Euclidean metric, a sphere of radius is nonempty only if can be written as sum of squares of integers.