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{{About|the minimal surface|the ruled surfaces|Catalan surface}}

[[File:Catalan's Minimal Surface.png|thumb|Catalan's minimal surface.]]

In [[differential geometry]], '''Catalan's minimal surface''' is a [[minimal surface]] originally studied by [[Eugène Charles Catalan]] in 1855.<ref>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.</ref>

It has the special property of being the minimal surface that contains a [[cycloid]] as a [[geodesic]]. It is also swept out by a family of parabolae.<ref>Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny, Minimal Surfaces, Volume 1. Springer 2010</ref>

The surface has parametric equation:<ref>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</ref>
:<math>\begin{align}
x(u,v) &= u - \cosh(v)\sin(u)\\
y(u,v) &= 1 - \cos(u)\cosh(v)\\
z(u,v) &= 4 \sin(u/2) \sinh(v/2)
\end{align}</math>

== External links ==
*  Weisstein, Eric W. "Catalan's Surface." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/CatalansSurface.html
*  Weiqing Gu, The Library of Surfaces. http://www.math.hmc.edu/~gu/curves_and_surfaces/surfaces/catalan.html

==References==
{{reflist}}

{{DEFAULTSORT:Catalans minimal surface}}
[[Category:Minimal surfaces]]
[[Category:Differential geometry]]