9758

The Unduloid

First studied by Delaunay [1], the unduloid (or onduloid) is a surface of revolution of an elliptic catenary with constant nonzero mean curvature (in this case, the reciprocal of twice the major axis length, ). It can be obtained by tracing the focus of a rolling ellipse along a fixed line. This surface appears in fields like string theory [2] and carbon nanotubes [3].

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

The unduloid is the solution of the differential equation
,
where and are constants.
A parametrization of an unduloid can be represented by
,
where
.
References
[1] C. Delaunay, "Sur la surface de révolution dont la courbure moyenne est constante," Journal de Mathématiques Pures et Appliquées, 6, 1841 pp. 309–320. portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_ 1841_ 1_ 6_A24 _ 0.
[2] K. Maeda and U. Miyamoto, "Black Hole-Black String Phase Transitions from Hydrodynamics." arxiv:0811.2305.
[3] I. M. Mladenov, M. Ts. Hadzhilazova, V. M. Vassilev, and P. A. Djondjorov, "Unduloid-like Equilibrium Shapes of Carbon Nanotubes Subjected to Hydrostatic Pressure," Geometric Methods in Physics, Trends in Mathematics (P. Kielanowski, S. T. Ali, A. Odesskii, A. Odzijewicz, M. Schlichenmaier, and T. Voronov, eds.), Basel: Birkhauser, 2013 pp. 195–202. doi:10.1007/978-3-0348-0645-9_ 18.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+