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Hyperbolic Octahedron

\begin{figure}\begin{center}\BoxedEPSF{octahedron_hyperbolic.epsf scaled 1000}\end{center}\end{figure}

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

$\displaystyle x$ $\textstyle =$ $\displaystyle (\cos u\cos v)^3$  
$\displaystyle y$ $\textstyle =$ $\displaystyle (\sin u\cos v)^3$  
$\displaystyle z$ $\textstyle =$ $\displaystyle \sin^3v$  

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

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


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

Nordstrand, T. ``Astroidal Ellipsoid.''

mathematica.gif Rivin, I. ``Hyperbolic Polyhedron Graphics.''

© 1996-9 Eric W. Weisstein