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The semicubical parabola of the form , with constant , is often called Neile’s parabola (sometimes called Neil’s parabola).
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 and a real analytic vector field along with , construct a minimal surface with parametrization having -parameter curve such that the vector field along is the unit normal of , , for .
Henneberg’s surface can be obtained by solving Björling’s problem for the parabola . The Neile’s parabola may be parametrized,
Whereas Henneberg’s surface may be parametrized:
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:
Oprea, John. The Mathematics of Soap Films: Explorations With Maple. AMS Student Mathematical Library, (2000), Volume 10, p96, p104 – 105.