Crystal oscillator frequencies: Difference between revisions

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[[File:Focaloid3d.jpg|thumb|right|Focaloid in 3D]]
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In [[geometry]], a '''focaloid''' is a shell bounded by two concentric, [[confocal]] [[ellipse]]s (in 2D) or [[ellipsoid]]s (in&nbsp;3D).
 
==Mathematical definition (3D)==
If one boundary surface is given by
 
: <math>
\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1
</math>
 
with semiaxes ''a'',&nbsp;''b'',&nbsp;''c'' the second surface is given by
 
: <math>
\frac{x^2}{a^2+\lambda}+\frac{y^2}{b^2+\lambda}+\frac{z^2}{c^2+\lambda}=1.
</math>
 
In the limit as&nbsp;''λ''&nbsp;→&nbsp;∞ 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 coordinates|confocal ellipsoidal coordinate system]].
 
==Confocal==
[[Confocal]] [[ellipsoid]]s share the same [[Focus (geometry)|foci]], which are given for the example above by
 
: <math>
f_1^2=a^2-b^2=(a^2+\lambda)-(b^2+\lambda), \,
</math>
 
: <math>
f_2^2=a^2-c^2=(a^2+\lambda)-(c^2+\lambda), \,
</math>
 
: <math>
f_3^2=b^2-c^2=(b^2+\lambda)-(c^2+\lambda).
</math>
 
== 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.
 
== Photo Sharing Website ==
www.focaloid.com is a photosharing social networking website where photographers can share their pictures with the world. Photographers can also give the precise camera settings with which those photos were captured.
 
== See also ==
* [[Homoeoid]]
 
== References ==
* [[Subrahmanyan Chandrasekhar]] (1969): ''Ellipsoidal Figures of Equilibrium.''  [[Yale University Press]], London, Connecticut
* [[Edward Routh|Routh, E. J.]]: ''A Treatise on Analytical Statics, Vol II'', [[Cambridge University Press]], Cambridge (1882).
 
[[Category:Surfaces]]
[[Category:Potential theory]]

Latest revision as of 20:19, 22 October 2014

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