Curve of Constant Precession

A curve whose Centrode revolves about a fixed axis with constant Angle and Speed when the curve is traversed with unit Speed. The Tangent Indicatrix of a curve of constant precession is a Spherical Helix. An Arc Length parameterization of a curve of constant precession with Natural Equations

$$\kappa(s) = -\omega\sin(\mu s)$$ (1)

$$\tau(s) = \omega\cos(\mu s)$$ (2)

is

$$x(s) = {\alpha+\mu\over 2\alpha} {\sin[(\alpha-\mu)s]\over\alpha-\mu}-{\alpha-\mu\over 2\alpha} {\sin[(\alpha+\mu)s]\over \alpha+\mu}$$ (3)

$$y(s) = -{\alpha+\mu\over 2\alpha} {\cos[(\alpha-\mu)s]\over\alpha-\mu}+{\alpha-\mu\over 2\alpha}{\cos[(\alpha+\mu)s]\over \alpha+\mu}$$ (4)

$$z(s) = {\omega\over\mu\alpha}\sin(\mu s),$$ (5)

where

$$\alpha\equiv\sqrt{\omega^2+\mu^2}$$ (6)

and $\omega$, and $\mu$ are constant. This curve lies on a circular one-sheeted Hyperboloid

$$x^2+y^2-{\mu^2\over\omega^2} z^2={4\mu^2\over\omega^4}.$$ (7)

The curve is closed Iff $\mu/\alpha$ is Rational.

References

Scofield, P. D. ``Curves of Constant Precession.'' Amer. Math. Monthly 102, 531-537, 1995.