Astroidal Ellipsoid

The surface which is the inverse of the Ellipsoid in the sense that it ``goes in'' where the Ellipsoid ``goes out.'' It is given by the parametric equations

$\displaystyle x$	$\textstyle =$	$\displaystyle (a\cos u\cos v)^3$
$\displaystyle y$	$\textstyle =$	$\displaystyle (b\sin u\cos v)^3$
$\displaystyle z$	$\textstyle =$	$\displaystyle (c\sin v)^3$

for $u\in [-\pi/2,\pi/2]$ and $v\in[-\pi,\pi]$ . The special case

corresponds to the Hyperbolic Octahedron.

See also Ellipsoid, Hyperbolic Octahedron

References

Nordstrand, T. ``Astroidal Ellipsoid.'' http://www.uib.no/people/nfytn/asttxt.htm.