Catalan's Surface

$\begin{figure}\begin{center}\BoxedEPSF{CatalansSurface.epsf}\end{center}\end{figure}$

A Minimal Surface given by the parametric equations

$\displaystyle x(u,v)$	$\textstyle =$	$\displaystyle u-\sin u\cosh v$	(1)
$\displaystyle y(u,v)$	$\textstyle =$	$\displaystyle 1-\cos u\cosh v$	(2)
$\displaystyle z(u,v)$	$\textstyle =$	$\displaystyle 4\sin({\textstyle{1\over 2}}u)\sinh({\textstyle{1\over 2}}v)$	(3)

(Gray 1993), or

$\displaystyle x(r,\phi)$	$\textstyle =$	$\displaystyle a\sin(2\phi)-2a\phi+{\textstyle{1\over 2}}av^2\cos(2\phi)$	(4)
$\displaystyle y(r,\phi)$	$\textstyle =$	$\displaystyle -a\cos(2\phi)-{\textstyle{1\over 2}}av^2\cos(2\phi)$	(5)
$\displaystyle z(r,\phi)$	$\textstyle =$	$\displaystyle 2av\sin\phi,$	(6)

where

$\begin{displaymath} v=-r+{1\over r} \end{displaymath}$

(7)

(do Carmo 1986).

References

Catalan, E. ``Mémoire sur les surfaces dont les rayons de courbures en chaque point, sont égaux et les signes contraires.'' C. R. Acad. Sci. Paris 41, 1019-1023, 1855.

do Carmo, M. P. ``Catalan's Surface'' §3.5D in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 45-46, 1986.

Fischer, G. (Ed.). Plates 94-95 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 90-91, 1986.

Gray, A. Modern Differential Geometry of Curves and Surfaces.Boca Raton, FL: CRC Press, pp. 448-449, 1993.