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 + 2 a y 1 + y ′ 2 = b 2 {\displaystyle y^{2}+{\frac {2ay}{\sqrt {1+y'^{2}}}}=b^{2}} . Its parametric equation is:
x ( u ) = a sn ( u , k ) + ( a / k ) ( ( 1 − k 2 ) u − E ( u , k ) ) {\displaystyle x(u)=a\operatorname {sn} (u,k)+(a/k){\big (}(1-k^{2})u-E(u,k){\big )}}
y ( u ) = − a cn ( u , k ) + ( a / k ) dn ( u , k ) {\displaystyle y(u)=-a\operatorname {cn} (u,k)+(a/k)\operatorname {dn} (u,k)}
where k = cos ( tan − 1 ( b / a ) ) {\displaystyle k=\cos(\tan ^{-1}(b/a))} is the elliptic modulus and E ( u , k ) {\displaystyle 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.
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