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. [1]

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

Its parametric equation is:

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

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

References

  1. 1 2 John Oprea, Differential Geometry and its Applications, MAA 2007. pp. 147–148
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