Unduloid

In geometry, an unduloid, or onduloid, is a surface with constant nonzero mean curvature obtained as a surface of revolution of an elliptic catenary: that is, by rolling an ellipse along a fixed line, tracing the focus, and revolving the resulting curve around the line. In 1841 Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid.

Let ${\displaystyle \operatorname {sn} (u,k)}$ represent the normal Jacobi sine function and ${\displaystyle \operatorname {dn} (u,k)}$ be the normal Jacobi elliptic function and let ${\displaystyle \operatorname {F} (z,k)}$ represent the normal elliptic integral of the first kind and ${\displaystyle \operatorname {E} (z,k)}$ represent the normal elliptic integral of the second kind. Let a be the length of the ellipse's major axis, and e be the eccentricity of the ellipse. Let k be a fixed value between 0 and 1 called the modulus.

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