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A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space. Like a circle, its two-dimensional analog, a sphere is defined mathematically as the set of points that are all the same distance r from a given point in three-dimensional space. This distance r is the radius of the sphere, and the given point is the center of the sphere. The maximum straight distance through the sphere passes through the center and is thus twice the radius; it is the diameter.
In mathematics, a distinction is made between the sphere (a two-dimensional closed surface embedded in three-dimensional Euclidean space) and the ball (a three-dimensional shape that includes the interior of a sphere).
The surface area of a sphere is:
Archimedes first derived this formula from the fact that the projection to the lateral surface of a circumscribed cylinder (i.e. the Lambert cylindrical equal-area projection) is area-preserving; it equals the derivative of the formula for the volume with respect to r because the total volume inside a sphere of radius r 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 r. At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius r is simply the product of the surface area at radius r and the infinitesimal thickness.
At any given radius r, the incremental volume (δV) equals the product of the surface area at radius r (A(r)) and the thickness of a shell (δr):
The total volume is the summation of all shell volumes:
In the limit as δr approaches zero this equation becomes:
Differentiating both sides of this equation with respect to r yields A as a function of r:
Which is generally abbreviated as:
The total area can thus be obtained by integration:
where r is the radius of the sphere and π is the constant pi. Archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. (This assertion follows from Cavalieri's principle.) In modern mathematics, this formula can be derived using integral calculus, i.e. disk integration to sum the volumes of an infinite number of circular disks of infinitesimally small thickness stacked centered side by side along the x axis from x = 0 where the disk has radius r (i.e. y = r) to x = r where the disk has radius 0 (i.e. y = 0).
At any given x, the incremental volume (δV) equals the product of the cross-sectional area of the disk at x and its thickness (δx):
The total volume is the summation of all incremental volumes:
In the limit as δx approaches zero this equation becomes:
At any given x, a right-angled triangle connects x, y and r to the origin; hence, applying the Pythagorean theorem yields:
Thus, substituting y with a function of x gives:
Which can now be evaluated as follows:
Therefore the volume of a sphere is:
Alternatively this formula is found using spherical coordinates, with volume element
For most practical purposes, the volume inside a sphere inscribed in a cube can be approximated as 52.4% of the volume of the cube, since . For example, a sphere with diameter 1m has 52.4% the volume of a cube with edge length 1m, or about 0.524m3.
The points on the sphere with radius r can be parameterized via
A sphere of any radius centered at zero is an integral surface of the following differential form:
This equation reflects that position and velocity vectors of a point traveling on the sphere are always orthogonal to each other.
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. 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 sphere is called the specific surface area and can be expressed from the above stated equations as
where is the ratio of mass to volume.
A sphere can also be defined as the surface formed by rotating a circle about any diameter. 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.
Pairs of points on a sphere that lie on a straight line through the sphere's 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 along the surface between two distinct non-antipodal points on the surface is on the unique great circle that includes the two points. Equipped with the great-circle distance, a great circle becomes the Riemannian circle.
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 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 such approximately spheroidal astronomical bodies as the planet Earth (see geoid).
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 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, which can also be thought of as the northern hemisphere with antipodal points of the equator identified.
The circles of intersection of any plane not intersecting the sphere's center and the sphere's surface are called spheric sections.
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 that 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 centered 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.
Another expression for the surface area is
and the volume is the surface area times or
More generally, in a metric space (E,d), the sphere of center x and radius r > 0 is the set of points y such that d(x,y) = r.
If the center is a distinguished point that is considered to be the 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 to equal one, as in the case of a unit 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.
Smale's paradox shows that it is possible to turn an ordinary sphere inside out in a three-dimensional space with possible self-intersections but without creating any crease, a process more commonly and historically called sphere eversion.
The basic elements of Euclidean 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. Measuring by arc length yields that the shortest path between two points that entirely lie in the sphere is a segment of the great circle the includes the points; see geodesic. Many, but not all (see parallel postulate) theorems from classical geometry hold true for this spherical geometry as well. In spherical trigonometry, angles are defined between great circles. Thus spherical trigonometry differs 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.
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:
|This section requires expansion. (April 2012)|
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