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{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}

External links

References

  1. ↑ 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.
  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, FL: CRC Press, pp. 692–693, 1997