Focaloid

Focaloid in 3D

Focaloid in 2D

A focaloid is a geometric figure that is bounded by confocal ellipses (2D) or by confocal ellipsoids (3D).

Mathematical definition (3D)

Becomes a boundary through an implicitly given ellipsoid

${\ displaystyle {\ frac {x ^ {2}} {a ^ {2}}} + {\ frac {y ^ {2}} {b ^ {2}}} + {\ frac {z ^ {2} } {c ^ {2}}} = 1}$

with the semiaxes described, the second boundary is through ${\ displaystyle a, b, c}$

${\ displaystyle {\ frac {x ^ {2}} {a ^ {2} + \ lambda}} + {\ frac {y ^ {2}} {b ^ {2} + \ lambda}} + {\ frac {z ^ {2}} {c ^ {2} + \ lambda}} = 1}$ given.

In the borderline case one speaks of thin, otherwise thick focaloid. ${\ displaystyle \ lambda \ to 0}$

Confocality

The above confocal ellipsoids have the same focal points . The following applies to their distances from the center O: ${\ displaystyle a \ geq b \ geq c}$ ${\ displaystyle f_ {1}, \; f_ {2}, \; f_ {3}}$

{\ displaystyle f_ {1} ^ {2} = a ^ {2} -b ^ {2} = (a ^ {2} + \ lambda) - (b ^ {2} + \ lambda)}

{\ displaystyle f_ {2} ^ {2} = a ^ {2} -c ^ {2} = (a ^ {2} + \ lambda) - (c ^ {2} + \ lambda)}

{\ displaystyle f_ {3} ^ {2} = b ^ {2} -c ^ {2} = (b ^ {2} + \ lambda) - (c ^ {2} + \ lambda)}

Definition of focal distribution

A confocal or focaloid distribution is e.g. For example, if the layers of constant density of a mass distribution or the layers of the same charge density are given by confocal ellipsoids or ellipses (see picture).

Lines of constant density of a confocal distribution

Physical meaning

Focaloids also play a role in physical potential theory . It lies in the fact that two confocal ellipsoids, which are homogeneously filled with mass or charge, cause forces in a test specimen located outside which point in the same direction and are proportional to the respective masses or charges of the respective individual ellipsoids.

From this one can conclude that different confocal, homogeneously filled with mass or charge, focalaloids of the same mass or charge outside of their extension, regardless of their geometry, produce the same effect.

This also means that the external effect of a focaloid distribution can be described by the external effect of a confocal ellipsoid that is homogeneously filled with the same mass.

Furthermore, the external field of a segment (thin rod) with homogeneously distributed mass or constant potential along the rod can be described as a focal distribution, whereby the ends of the segment (of the rod) are the focal points (foci) of the focal field. The field vectors are perpendicular to the ellipsoids of the same field amount.

literature

S. Chandrasekhar : Ellipsoidal Figures of Equilibrium. Yale Univ. Press, London 1969.
EJ Routh : A Treatise on Analytical Statics. Volume II. Cambridge University Press, Cambridge 1882.