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ASTROIDAL ELLIPSOID

See this page by Alain Esculier.

 
Cartesian equation: .
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