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

for
and .

The special case
corresponds to the hyperbolic octahedron .
Like the hyperbolic octahedron , the astroidal
ellipse is an algebraic surface of degree 18
with very complicated terms.

The astroidal ellipsoid has first fundamental
form coefficients

while the coefficients of the second fundamental
form are more complicated.

The Gaussian curvature is

(7)

while the mean curvature has a complicated expression.

See also Astroid ,

Ellipsoid ,

Hyperbolic Octahedron ,

Superegg ,

Superellipsoid
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References Nordstrand, T. "Astroidal Ellipsoid." http://jalape.no/math/asttxt .
Cite this as:
Weisstein, Eric W. "Astroidal Ellipsoid."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/AstroidalEllipsoid.html

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