The name of this surface comes from the property that its sections with planes parallel to the axes are astroids.[br]A parameterization of its equation is: [br][math]x=a\ cos^3\left(u\right)\cos^3\left(v\right)[/math], [math]y=b\sin^3\left(u\right)\cos^3\left(v\right)[/math], [math]z=c\sin^3\left(v\right)[/math][br]and the Cartesian equation is [math]\sqrt[3]{\frac{x^2}{a^2}}+\sqrt[3]{\frac{y^2}{b^2}}+\sqrt[3]{\frac{z^2}{c^2}}=1[/math][br][br]For the special case [math]a=b=c=1[/math] this surface is named [i]hyperbolic octahedron[/i], and has the following properties:[br][list][*]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 [math]\frac{a}{3}[/math].[/*][/list][br]Explore this surface in the app below. Use the mouse wheel to zoom, and drag the 3D [i]View[/i] 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