Nodary

In physics and geometry, the nodary is the curve that is traced by the focus of a hyperbola as it rolls without slipping along the axis, a roulette curve.

Nodary curve.

The differential equation of the curve is: $y^{2}+{\frac {2ay}{\sqrt {1+y'^{2}}}}=b^{2}$.

Its parametric equation is:

$x(u)=a\operatorname {sn} (u,k)+(a/k){\big (}(1-k^{2})u-E(u,k){\big )}$

$y(u)=-a\operatorname {cn} (u,k)+(a/k)\operatorname {dn} (u,k)$

where $k=\cos(\tan ^{-1}(b/a))$ is the elliptic modulus and $E(u,k)$ is the incomplete elliptic integral of the second kind and sn, cn and dn are Jacobi's elliptic functions.

The surface of revolution is the nodoid constant mean curvature surface.