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[[File:Unduloid.svg|thumb|Computer generated unduloid]]

In [[geometry]], an '''unduloid''', or '''onduloid''', is a [[surface (mathematics)|surface]] with [[Constant-mean-curvature surface|constant nonzero mean curvature]] obtained as a [[surface of revolution]] of an [[elliptic catenary]]: that is, by [[Roulette (curve)|rolling]] an [[ellipse]] along a fixed line, tracing the [[Focus (geometry)|focus]], and revolving the resulting curve around the line. In 1841 [[Charles-Eugène Delaunay|Delaunay]] proved that the only [[surfaces of revolution]] with constant mean curvature were the surfaces obtained by rotating the [[Roulette (curve)|roulettes]] of the conics. These are the plane, cylinder, sphere, the [[catenoid]], the unduloid and [[nodoid]].<ref>{{cite journal |last1=Delaunay |first1=Ch. |title=Sur la surface de révolution dont la courbure moyenne est constante |journal=Journal de Mathématiques Pures et Appliquées |date=1841 |volume=6 |pages=309–314 |url=https://eudml.org/doc/234757}}</ref>

== Formula ==
Let <math>\operatorname{sn}(u,k)</math> represent the normal [[Jacobi sine function]] and <math>\operatorname{dn}(u,k)</math> be the normal [[Jacobi elliptic function]] and let <math>\operatorname{F}(z,k)</math> represent the normal [[elliptic integral]] of the first kind and <math>\operatorname{E}(z,k)</math> represent the normal elliptic integral of the second kind. Let ''a'' be the length of the ellipse's [[major axis]], and ''e'' be the [[eccentricity (mathematics)|eccentricity]] of the ellipse. Let ''k'' be a fixed value between 0 and 1 called the modulus.

Given these variables,

: <math>\operatorname{x}(u) = -a(1-e)( \operatorname{F}(\operatorname{sn}(u,k),k) + \operatorname{F}(1,k)) - a(1+e)( \operatorname{E}( \operatorname{sn}(u,k),k) + \operatorname{E}(1,k)) \, </math>

: <math>\operatorname{y}(u) = a(1+e)\operatorname{dn}(u,k) \, </math>

The formula for the surface of revolution that is the unduloid is then

: <math>\operatorname{X}(u,v) = \langle \operatorname{x}(u), \operatorname{y}(u) \cos(v), \operatorname{y}(u) \sin(v)\rangle \, </math>

== Properties ==

One interesting property of the unduloid is that the [[mean curvature]] is constant. In fact, the mean curvature across the entire surface is always the reciprocal of twice the major axis length: 1/(2''a'').

Also, [[geodesics]] on an unduloid obey the [[Clairaut's relation|Clairaut relation]], and their behavior is therefore predictable.

== Occurrence in material science ==
Unduloids are not a common pattern in nature. However, there are a few circumstances in which they form. First documented in 1970, passing a strong electric current through a thin (0.16—1.0mm), horizontally mounted, hard-drawn (non-[[Tempering_(metallurgy)|tempered]]) [[silver]] wire will result in unduloids forming along its length.<ref>{{citation |first1=T. |last1=Lipski |first2= A. | last2=Furdal |title=New observations on the formation of unduloids on wires |journal= Proceedings of the Institution of Electrical Engineers|volume=117 |issue=12 |year=1970 |page=2311-2314 |doi=10.1049/piee.1970.0425 |url=https://digital-library.theiet.org/content/journals/10.1049/piee.1970.0425}}</ref>
This phenomenon was later discovered to also occur in [[molybdenum]] wire.<ref>{{YouTube|id=kexteGM2V2s|title=“Periodic Videos, Exploding wires”|time=365}}</ref>
Unduloids have also been formed with [[Ferrofluid|ferrofluids]].<ref>{{citation |first1=D.E. |last1=Weidner |title= Drop formation in a magnetic fluid coating a horizontal cylinder carrying an axial electric current |journal=[[Physics of Fluids]] |volume=29 |issue=5 |year=2017 |page=052103 |doi=10.1063/1.4982618 |url=https://doi.org/10.1063/1.4982618}}</ref> By passing a current axially through a cylinder coated with a viscous magnetic fluid film, the [[Magnetic dipole|magnetic dipoles]] of the fluid interact with the magnetic field of the current, creating a droplet pattern along the cylinder’s length.

==References==
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[[Category:Surfaces]]