Focaloid
Encyclopedia
In geometry
, a focaloid is a shell bounded by two concentric, confocal
ellipse
s (2D) or ellipsoids (3D).
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a focaloid is a shell bounded by two concentric, confocal
Confocal
In geometry, confocal means having the same foci.* For an optical cavity consisting of two mirrors, confocal means that they share their foci...
ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
s (2D) or ellipsoids (3D).
Mathematical definition (3D)
If one boundary surface is given by-
with semiaxes a, b, c the second surface is given by
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In the limit as λ → ∞ one speaks of thin focaloids.
In general, a focaloid could be understood as a shell consisting out of two closed coordinate surfaces of a confocal ellipsoidal coordinate system.
Confocal
ConfocalConfocalIn geometry, confocal means having the same foci.* For an optical cavity consisting of two mirrors, confocal means that they share their foci...
ellipsoids share the same fociFocus (geometry)In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola...
, which are given for the example above by
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Physical meaning
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.
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