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

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

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

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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