Focaloid
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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a focaloid is a shell bounded by two
concentric In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center point ...
, confocal ellipses (in 2D) or
ellipsoids An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as the ...
(in 3D). When the thickness of the shell becomes negligible, it is called a thin focaloid.


Mathematical definition (3D)

If one boundary surface is given by : \frac+\frac+\frac=1 with semiaxes ''a'', ''b'', ''c'' the second surface is given by : \frac+\frac+\frac=1. The thin focaloid is then given by the limit \lambda \to 0. In general, a focaloid could be understood as a shell consisting out of two closed coordinate surfaces of a confocal ellipsoidal coordinate system.


Confocal

Confocal In geometry, confocal means having the same foci: confocal conic sections. * For an optical cavity consisting of two mirrors, confocal means that they share their foci. If they are identical mirrors, their radius of curvature, ''R''mirror, equals ' ...
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as the ...
s share the same
foci Focus, or its plural form foci may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film *''Focus'', a 1962 TV film starring James Whitmore * ''Focus'' (2001 film), a 2001 film based ...
, which are given for the example above by : f_1^2=a^2-b^2=(a^2+\lambda)-(b^2+\lambda), \, : f_2^2=a^2-c^2=(a^2+\lambda)-(c^2+\lambda), \, : f_3^2=b^2-c^2=(b^2+\lambda)-(c^2+\lambda).


Physical significance

A focaloid can be used as a construction element of a matter or charge distribution. The particular importance of focaloids lies in the fact that two different but confocal focaloids of the same mass or charge produce the same action on a test mass or charge in the exterior region.


See also

*
Homoeoid A homoeoid is a shell (a bounded region) bounded by two concentric, similar ellipses (in 2D) or ellipsoids (in 3D). When the thickness of the shell becomes negligible, it is called a thin homoeoid. The name homoeoid was coined by Lord Kelvin and ...
*
Confocal conic sections In geometry, two conic sections are called confocal, if they have the same foci. Because ellipses and hyperbolas possess two foci, there are confocal ellipses, confocal hyperbolas and confocal mixtures of ellipses and hyperbolas. In the mixture of ...


References

*
Subrahmanyan Chandrasekhar Subrahmanyan Chandrasekhar (; ) (19 October 1910 – 21 August 1995) was an Indian-American theoretical physicist who spent his professional life in the United States. He shared the 1983 Nobel Prize for Physics with William A. Fowler for "... ...
(1969): ''Ellipsoidal Figures of Equilibrium.''
Yale University Press Yale University Press is the university press of Yale University. It was founded in 1908 by George Parmly Day, and became an official department of Yale University in 1961, but it remains financially and operationally autonomous. , Yale Universi ...
, London, Connecticut * Routh, E. J.: ''A Treatise on Analytical Statics, Vol II'',
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...
, Cambridge (1882).


External links

*{{Commonscatinline Surfaces Potential theory