A minimal surface given by the parametric equations

			(1)
			(2)
			(3)

(Gray 1997), or

			(4)
			(5)
			(6)

where

(7)

(do Carmo 1986).

The first fundamental form has coefficients

			(8)
			(9)
			(10)

and the second fundamental form has coefficients

			(11)
			(12)
			(13)

The principal curvatures are

			(14)
			(15)

the mean curvature is

(16)

and the Gaussian curvature is

(17)

Catalan, E. "Mémoire sur les surfaces dont les rayons de courbures en chaque point, sont égaux et les signes contraires." Comptes Rendus 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 aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp. 90-91, 1986.

Gray, A. "Catalan's Minimal Surface." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 692-693, 1997.

JavaView. "Classic Surfaces from Differential Geometry: Catalan Surface." http://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_Catalan.html.

CITE THIS AS:

Weisstein, Eric W. "Catalan's Surface." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CatalansSurface.html