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# Henneberg’s Minimal Surface

The semicubical parabola of the form $y = ax^{3/2}$, with constant $a$, is often called Neile’s parabola (sometimes called Neil’s parabola).

image: wolfram.mathworld.com

In 1657, the English mathematician William Neile at the age of nineteen discovered this curve and computed the arc length. Neile’s parabola was actually the first nontrivial curve to have its arc length computed.

Björling’s Problem. Given a real analytic curve $\alpha: I \rightarrow \mathbb{R}^3$ and a real analytic vector field $N$ along $\alpha$ with $N \cdot \alpha \ ' = 0$, construct a minimal surface $M$ with parametrization $\textbf{x}(u,v)$ having $u$-parameter curve $\alpha(u) = \textbf{x} (u,0)$ such that the vector field $N$ along $\alpha$ is the unit normal of $M$, $N(u) = U(u,0)$, for $u \in I$.

Henneberg’s surface can be obtained by solving Björling’s problem for the parabola $2x^3 = 9z^2$.  The Neile’s parabola $2x^3 = 9z^2$ may be parametrized,

$\alpha(u) = (cosh 2u -1, 0, -sinh u + \frac{1}{3}sinh 3u).$

Whereas Henneberg’s surface may be parametrized:

$\textbf{x}(u,v) = (-1 + cosh 2u cos 2v, sinh u sin v + \frac{1}{3}sinh 3u sin 3v, -sinh u cos v + \frac{1}{3}sinh 3u cos 3v).$

This surface is a minimal surface and can be realized as the soap film obtained by dipping a wire frame into soapy water.

I am currently coding Henneberg’s minimal surface using Python and OpenGL to be viewed and interacted with in the Beckman Institute Illinois Simulator Lab’s total immersion environment, the Cube, at the University of Illinois.

Here is screenshot of my program:

http://ocw.mit.edu/courses/mathematics/18-994-seminar-in-geometry-fall-2004/projects/main1.pdf

Oprea, John. The Mathematics of Soap Films: Explorations With Maple. AMS Student Mathematical Library, (2000), Volume 10, p96, p104 – 105.