Hyperbolic Octahedron

A hyperbolic version of the Euclidean Octahedron, which is a special case of the Astroidal Ellipsoid with $a=b=c=1$. It is given by the parametric equations

$$x = (\cos u\cos v)^3$$

$$y = (\sin u\cos v)^3$$

$$z = \sin^3v$$

for $u\in[-\pi/2,\pi/2]$ and $v\in[-\pi,\pi]$.

See also Astroidal Ellipsoid, Hyperbolic Cube, Hyperbolic Dodecahedron, Hyperbolic Tetrahedron

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 305-306, 1993.

Nordstrand, T. ``Astroidal Ellipsoid.'' http://www.uib.no/people/nfytn/asttxt.htm.

Rivin, I. ``Hyperbolic Polyhedron Graphics.'' http://www.mathsource.com/cgi-bin/MathSource/Applications/Graphics/3D/0201-788.