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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 $a=b=c=1$ corresponds to the Hyperbolic Octahedron.

See also Ellipsoid, Hyperbolic Octahedron


Nordstrand, T. ``Astroidal Ellipsoid.''

© 1996-9 Eric W. Weisstein