Henneberg surface

Henneberg surface.

In differential geometry, the Henneberg surface is a non-orientable minimal surface[1] named after Lebrecht Henneberg.[2]

It has parametric equation

\begin{align} x(u,v) &= 2\cos(v)\sinh(u) - (2/3)\cos(3v)\sinh(3u)\\ y(u,v) &= 2\sin(v)\sinh(u) + (2/3)\sin(3v)\sinh(3u)\\ z(u,v) &= 2\cos(2v)\cosh(2u) \end{align}

and can be expressed as an order-15 algebraic surface.[3] It can be viewed as an immersion of a punctured projective plane.[4] Up until 1981 it was the only known non-orientable minimal surface.[5]

The surface contains a semicubical parabola ("Neile's parabola") and can be derived from solving the corresponding Björling problem.[6][7]

References

  1. ↑ L. Henneberg, Über salche minimalfläche, welche eine vorgeschriebene ebene curve sur geodätishen line haben, Doctoral Dissertation, Eidgenössisches Polythechikum, Zürich, 1875
  2. ↑ Lebrecht Henneberg from the German-language Wikipedia. Retrieved on September 25, 2012.
  3. ↑ Weisstein, Eric W. "Henneberg's Minimal Surface." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/HennebergsMinimalSurface.html
  4. ↑ Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny, Minimal Surfaces, Volume 1. Springer 2010
  5. ↑ M. Elisa G. G. de Oliveira, Some New Examples of Nonorientable Minimal Surfaces, Proceedings of the American Mathematical Society, Vol. 98, No. 4, Dec., 1986
  6. ↑ L. Henneberg, Über diejenige minimalfläche, welche die Neil'sche Paralee zur ebenen geodätischen line hat, Vierteljschr Natuforsch, Ges. Zürich 21 (1876), 66–70.
  7. ↑ Kai-Wing Fung, Minimal Surfaces as Isotropic Curves in C3: Associated minimal surfaces and the Björling's problem. MIT BA Thesis. 2004 http://ocw.mit.edu/courses/mathematics/18-994-seminar-in-geometry-fall-2004/projects/main1.pdf