# Unduloid

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Computer generated unduloid

In [geometry](/source/Geometry), an **unduloid**, or **onduloid**, is a [surface](/source/Surface_(mathematics)) with [constant nonzero mean curvature](/source/Constant-mean-curvature_surface) obtained as a [surface of revolution](/source/Surface_of_revolution) of an [elliptic catenary](/source/Elliptic_catenary): that is, by [rolling](/source/Roulette_(curve)) an [ellipse](/source/Ellipse) along a fixed line, tracing the [focus](/source/Focus_(geometry)), and revolving the resulting curve around the line. In 1841 [Delaunay](/source/Charles-Eug%C3%A8ne_Delaunay) proved that the only [surfaces of revolution](/source/Surfaces_of_revolution) with constant mean curvature were the surfaces obtained by rotating the [roulettes](/source/Roulette_(curve)) of the conics. These are the plane, cylinder, sphere, the [catenoid](/source/Catenoid), the unduloid and [nodoid](/source/Nodoid).[1]

## Formula

Let sn ⁡ ( u , k ) {\displaystyle \operatorname {sn} (u,k)} represent the normal [Jacobi sine function](/source/Jacobi_sine_function) and dn ⁡ ( u , k ) {\displaystyle \operatorname {dn} (u,k)} be the normal [Jacobi elliptic function](/source/Jacobi_elliptic_function) and let F ⁡ ( z , k ) {\displaystyle \operatorname {F} (z,k)} represent the normal [elliptic integral](/source/Elliptic_integral) of the first kind and E ⁡ ( z , k ) {\displaystyle \operatorname {E} (z,k)} represent the normal elliptic integral of the second kind. Let *a* be the length of the ellipse's [major axis](/source/Major_axis), and *e* be the [eccentricity](/source/Eccentricity_(mathematics)) of the ellipse. Let *k* be a fixed value between 0 and 1 called the modulus.

Given these variables,

- x ⁡ ( u ) = − a ( 1 − e ) ( F ⁡ ( sn ⁡ ( u , k ) , k ) + F ⁡ ( 1 , k ) ) − a ( 1 + e ) ( E ⁡ ( sn ⁡ ( u , k ) , k ) + E ⁡ ( 1 , k ) ) {\displaystyle \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))\,}

- y ⁡ ( u ) = a ( 1 + e ) dn ⁡ ( u , k ) {\displaystyle \operatorname {y} (u)=a(1+e)\operatorname {dn} (u,k)\,}

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

- X ⁡ ( u , v ) = ⟨ x ⁡ ( u ) , y ⁡ ( u ) cos ⁡ ( v ) , y ⁡ ( u ) sin ⁡ ( v ) ⟩ {\displaystyle \operatorname {X} (u,v)=\langle \operatorname {x} (u),\operatorname {y} (u)\cos(v),\operatorname {y} (u)\sin(v)\rangle \,}

## Properties

One interesting property of the unduloid is that the [mean curvature](/source/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](/source/Geodesics) on an unduloid obey the [Clairaut relation](/source/Clairaut's_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-[tempered](/source/Tempering_(metallurgy))) [silver](/source/Silver) wire will result in unduloids forming along its length.[2] This phenomenon was later discovered to also occur in [molybdenum](/source/Molybdenum) wire.[3] Unduloids have also been formed with [ferrofluids](/source/Ferrofluid).[4] By passing a current axially through a cylinder coated with a viscous magnetic fluid film, the [magnetic dipoles](/source/Magnetic_dipole) of the fluid interact with the magnetic field of the current, creating a droplet pattern along the cylinder’s length.

## References

1. **[^](#cite_ref-1)** Delaunay, Ch. (1841). ["Sur la surface de révolution dont la courbure moyenne est constante"](https://eudml.org/doc/234757). *Journal de Mathématiques Pures et Appliquées*. **6**: 309–314.

1. **[^](#cite_ref-2)** Lipski, T.; Furdal, A. (1970), ["New observations on the formation of unduloids on wires"](https://digital-library.theiet.org/content/journals/10.1049/piee.1970.0425), *Proceedings of the Institution of Electrical Engineers*, **117** (12): 2311-2314, [doi](/source/Doi_(identifier)):[10.1049/piee.1970.0425](https://doi.org/10.1049%2Fpiee.1970.0425)

1. **[^](#cite_ref-3)** [“Periodic Videos, Exploding wires”](https://www.youtube.com/watch?v=kexteGM2V2s&t=365) on [YouTube](/source/YouTube_video_(identifier))

1. **[^](#cite_ref-4)** Weidner, D.E. (2017), ["Drop formation in a magnetic fluid coating a horizontal cylinder carrying an axial electric current"](https://doi.org/10.1063/1.4982618), *[Physics of Fluids](/source/Physics_of_Fluids)*, **29** (5): 052103, [doi](/source/Doi_(identifier)):[10.1063/1.4982618](https://doi.org/10.1063%2F1.4982618)

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Adapted from the Wikipedia article [Unduloid](https://en.wikipedia.org/wiki/Unduloid) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Unduloid?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.