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. [1]

Nodary curve.

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

Its parametric equation is:

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

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

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

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

References

1. John Oprea, Differential Geometry and its Applications, MAA 2007. pp. 147–148