physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...

and

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the nodary is the curve that is traced by the focus of a

hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, c ...

as it rolls without slipping along the axis, a

roulette curve In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, epitrochoids, hypotrochoids, and involutes. Definition Informal definition Roughly speaking, a roulette is th ...

. John Oprea, Differential Geometry and its Applications, MAA 2007. pp. 147–148 The differential equation of the curve is:

y^2 + \frac=b^2

. Its parametric equation is: :

x(u)=a\operatorname(u,k)+(a/k)\big((1-k^2)u - E(u,k)\big)

y(u)=-a\operatorname(u,k)+(a/k)\operatorname(u,k)

where

k= \cos(\tan^(b/a))

is the elliptic modulus and

E(u,k)

is the

incomplete elliptic integral of the second kind In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...

and sn, cn and dn are

Jacobi's elliptic functions In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While tr ...

. The surface of revolution is the

nodoid In differential geometry, a nodoid is a surface of revolution with constant nonzero mean curvature obtained by rolling a hyperbola along a fixed line, tracing the focus Focus, or its plural form foci may refer to: Arts * Focus or Focus Fest ...

constant mean curvature surface In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature.Carl Johan Lejdfors, Surfaces of Constant Mean Curvature. Master’s thesis Lund University, Centre for Mathematical Sciences Mathematics ...

References

Plane curves {{geometry-stub