Nodary Explained

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]

The differential equation of the curve is:

y ^{2} + (2 ay)/(\sqrt{ 1+ y' ^{2}) }=b^2.

Its parametric equation is:

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

Notes and References

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