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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:
.
Its parametric equation is:
![{\displaystyle x(u)=a\operatorname {sn} (u,k)+(a/k){\big (}(1-k^{2})u-E(u,k){\big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b668ce28e4beff0778162e6ccd5b1d7bf1304d3)
![{\displaystyle y(u)=-a\operatorname {cn} (u,k)+(a/k)\operatorname {dn} (u,k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b479aa814fa036c15291f0e9c61c46bfdb06743)
where
is the elliptic modulus and
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
- ^ a b John Oprea, Differential Geometry and its Applications, MAA 2007. pp. 147–148