Catalan's minimal surface

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

Catalan's minimal surface.

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 parametric equation:^[3]

{\begin{aligned}x(u,v)&=u-\cos(v)\sin(u)\\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. http://www.math.hmc.edu/~gu/curves_and_surfaces/surfaces/catalan.html

↑ 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.
↑ Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny, Minimal Surfaces, Volume 1. Springer 2010
↑ 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

Minimal surfaces

Associate family Bour's Catalan's Catenoid Chen–Gackstatter Costa's Enneper Gyroid Helicoid Henneberg K-noid Lidinoid Neovius Richmond Riemann's Saddle tower Scherk Schwarz Triply periodic

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