Sphere

A sphere is a surface with three dimensions of which all the points are located at the same distance from a point called center . The value of this distance common to the center is called the ray of the sphere. It thus does not include the points located at a distance lower than the ray, contrary to the swell. Concretely, one can see a sphere like an infinitely thin empty shell.

An approximate sphere is called géosphère in reference to the Ground whose surface is not a perfect sphere. This term is frequently used in Astrophysique and sometimes in Architecture.

In a Euclidean Espace, it acts of the balloon that everyone associates at the end of sphere. In an not-Euclidean space or the case of the definition of a nonEuclidean Distance, the form can be more complex.

A sphere can also be defined as the surface formed by the Rotation of a Cercle around its Diamètre. If the circle is replaced by a ellipse, one obtains a Sphéroïde.

Representation

In Cartesian Geometry, a sphere of center ( X 0, there 0, Z 0) and of ray R is the whole of the points ( X , there , Z ) such as:

\ displaystyle (x-x_0) ^2+ (there there _0) ^2+ (z-z_0) ^2 = r^2.

The points of the sphere of radius R and center the origin of the reference mark can be parameterized by:

\ left \ { \begin{matrix} X & = & R \ cos \ theta \; \ cos \ phi \ \ there & = & R \ cos \ theta \; \ sin \ phi \ \ Z & = & R \ sin \ theta \end{matrix} \ right. \ qquad (\ frac {- \ pi} {2} \ the \ theta \ the \ frac {\ pi} {2} \ mbox {and} - \ pi \ the \ phi \ \ pi) One can see \ theta \, like the latitude and \ phi \, like longitude. (See goniometrical functions and Coordinated spherical.)

Formulas

The surface of a sphere of ray r is:

A=4 \ pi r^ {2} \,

The Volume which it contains is:

V= \ frac {4 \ pi r^ {3}} {3}

Its compactness is of:

C= \ frac {has} {V} = \ frac {3} {R}

The Moment of inertia of a homogeneous sphere full with ray R, of density \ rho, mass M compared to an axis passing by its center is:

I= \ frac {2 M R^2} {5} = \ frac {8 \ pi \ rho R^5} {15}

The Moment of inertia of a homogeneous sphere vacuum of ray R, of mass M compared to an axis passing by its center is:

I= \ frac {2 M R^2} {3}

The element of surface of the sphere of ray r \, in the coordinates latitude-longitude is d \ sigma=r^2 \ cos \ theta D \ theta D \ phi \, . One from of deduced that the surface from a spindle (portion limited by two half-circles uniting the poles and forming an angle \ alpha \, expressed in radians is 2 \ alpha r^2 \, .

That also makes it possible to calculate the surface of a segment of a sphere (one says also segment of sphere ), i.e. of a portion of sphere limited by two parallel plans of distance h \, one being able to be tangent with the sphere. One finds 2 \ pi Rh \, : the surface is the same one as that of a circular cylinder of the same height tangent with the sphere (circumscribed cylinder). This remarkable result was known of Archimedes, which would have required that it be mentioned on its tomb.

The Cylindre circumscribes with a given sphere has a volume equal to 3/2 times the volume of the sphere.

The sphere with the smallest surface among surfaces containing a given volume and contains the highest volume among surfaces of a given surface. For this reason, the sphere appears in nature, for example the bubbles and drops of water (in the absence of Gravité) are spheres because the surface Tension tries to minimize the surface.

Owner

One can show that there does not exist owner of the sphere.

See too

  • D. Hilbert and S. Cohn-Vossen, Geometry and imagination , Chelsea 1952

  • B. Shepherd, Geometry (CH. 18), Nathan 1990, ISBN 209.191.730-3

Related articles

External bonds

  • A. Javary, Treated descriptive geometry , 1881, (on Gallica): Cones and cylinders, sphere and surfaces of the second degree

  • the shortest way on the sphere

Simple: Sphere Zh-yue: 波

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