In geometry, a focaloid is a shell bounded by two concentric, confocal ellipses (in 2D) or ellipsoids (in 3D). When the thickness of the shell becomes negligible, it is called a thin focaloid.
If one boundary surface is given by
x2 | + | |
a2 |
y2 | + | |
b2 |
z2 | |
c2 |
=1
with semiaxes a, b, c the second surface is given by
x2 | + | |
a2+λ |
y2 | + | |
b2+λ |
z2 | |
c2+λ |
=1.
The thin focaloid is then given by the limit
λ\to0
In general, a focaloid could be understood as a shell consisting out of two closed coordinate surfaces of a confocal ellipsoidal coordinate system.
Confocal ellipsoids share the same foci, which are given for the example above by
2=a | |
f | |
1 |
2-b2=(a2+λ)-(b2+λ),
2=a | |
f | |
2 |
2-c2=(a2+λ)-(c2+λ),
2=b | |
f | |
3 |
2-c2=(b2+λ)-(c2+λ).
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.
A Treatise on Analytical Statics, Vol II, Cambridge University Press, Cambridge (1882).