Cassini Surface

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

$$[(x-a)^2+y^2][(x+a)^2+y^2]=c^2$$ (1)

by $c=z^2$, obtaining

$$[(x-a)^2+y^2][(x+a)^2+y^2]=z^4.$$ (2)

As can be seen by letting $y=0$ to obtain

$$(x^2-a^2)^2=z^4$$ (3)

$$x^2+z^2=a^2,$$ (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.