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.

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.