Catalan's minimal surface explained

In differential geometry, Catalan's minimal surface is a minimal surface originally studied by Eugène Charles Catalan in 1855.^[1]

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.^[2]

The surface has the mathematical characteristics exemplified by the following parametric equation:^[3]

\begin{align} x(u,v) & = u - \sin(u)\cosh(v)\\ y(u,v) & = 1 - \cos(u)\cosh(v)\\ z(u,v) & = 4 \sin(u/ 2 ) \sinh(v/ 2 ) \end{align}

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. https://web.archive.org/web/20130317011222/http://www.math.hmc.edu/~gu/curves_and_surfaces/surfaces/catalan.html

Notes and References

1. Catalan, E. "Mémoire sur les surfaces dont les rayons de courbures en chaque point, sont égaux et les signes contraires." Comptes rendus de l'Académie des Sciences de Paris 41, 1019–1023, 1855.
2. Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny, Minimal Surfaces, Volume 1. Springer 2010
3. Gray, A. "Catalan's Minimal Surface." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, Florida: CRC Press, pp. 692–693, 1997