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ASTROIDAL ELLIPSOID

Other names : hyperbolic octahedron, astroidoctahedron. |

Cartesian equation: .
Algebraic surface of degree 18. Special case of Lame surface. Cartesian parametrization:. |

The astroidal ellipsoid is the surface with the above equation. Its name comes from the fact that its sections by planes parallel to the axes are astroids.

It is the envelope
of the planes intersecting the three axes at the three vertices of a triangle
for which the distance between the center of gravity and *O* is constant,
equal to *a* /3.

It has the same vertices and same symmetries as the surface
of the regular octahedron:.

Figure made by Alain Esculier, obtained by intersection of tangent planes, each of them being represented in a different color. |

Similarly, the envelope of the planes intersecting the
three axes at three points *A,B,C* such that the tetrahedron *OABC*
has, now, a *constant volume* is the cubic
surface with equation
*xyz = a* ^{3}.

Another variation: let a disk rest on the three faces of a trihedron with right angles (think of a bike wheel resting on a corner of a room); the center of the disk describes a portion of a sphere, and the plane of the disk envelopes the surface below:

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© Robert FERRÉOL 2017