Astroidal Ellipsoid

The name of this surface comes from the property that its sections with planes parallel to the axes are astroids. A parameterization of its equation is: , , and the Cartesian equation is For the special case this surface is named hyperbolic octahedron, and has the following properties:
  • it has the same vertices and symmetries of the regular octahedron
  • it is the envelope of the planes that intersect the axes at the vertices of a triangle whose distance between the barycenter and the origin is constant, and equal to .
Explore this surface in the app below. Use the mouse wheel to zoom, and drag the 3D View to change the point of view of the surface (or use the predefined gestures on mobile devices).
Explore the intersections of the hyperbolic octahedron with planes parallel to the axes