Catalan's minimal surface

This article is about the minimal surface. For the ruled surfaces, see Catalan surface.

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

Catalan's minimal surface.

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]

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

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