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Cassini Surface

\begin{figure}\begin{center}\BoxedEPSF{CassiniSurface.epsf scaled 500}\quad\BoxedEPSF{CassiniSurfacePOV.epsf scaled 500}\end{center}\end{figure}

The Quartic Surface obtained by replacing the constant $c$ in the equation of the Cassini Ovals

\begin{displaymath}[(x-a)^2+y^2][(x+a)^2+y^2]=c^2
\end{displaymath} (1)

by $c=z^2$, obtaining
\begin{displaymath}[(x-a)^2+y^2][(x+a)^2+y^2]=z^4.
\end{displaymath} (2)

As can be seen by letting $y=0$ to obtain
\begin{displaymath}
(x^2-a^2)^2=z^4
\end{displaymath} (3)


\begin{displaymath}
x^2+z^2=a^2,
\end{displaymath} (4)

the intersection of the surface with the $y=0$ Plane is a Circle of Radius $a$.


References

Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, p. 20, 1986.

Fischer, G. (Ed.). Plate 51 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 51, 1986.




© 1996-9 Eric W. Weisstein
1999-05-26