Monoid factorisation

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 $y^2+\frac{2ay}{\sqrt{1+y{\text{'}}^2}}=b^2$. It has parametric equation

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

$y(u)=-\mathrm{c}\mathrm{n}\mathrm{(}\mathrm{u},\mathrm{k}\mathrm{)}+(a/k)\mathrm{d}\mathrm{n}\mathrm{(}\mathrm{u},\mathrm{k}\mathrm{)}$

where $k=\cos ({\tan }^{}(b/a))$ 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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