Catalan's minimal 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 the mathematical characteristics exemplified by the following parametric equation:[3]

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 ) {\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}}}
  • 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

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
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