Nodary

Nodary curve.

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: $y^2+\frac{2ay}{\sqrt{1+y{\text{'}}^2}}=b^2$.

Its parametric equation is:

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

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

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.

References

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