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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

$\displaystyle \kappa(s)$ $\textstyle =$ $\displaystyle -\omega\sin(\mu s)$ (1)
$\displaystyle \tau(s)$ $\textstyle =$ $\displaystyle \omega\cos(\mu s)$ (2)

$\displaystyle x(s)$ $\textstyle =$ $\displaystyle {\alpha+\mu\over 2\alpha} {\sin[(\alpha-\mu)s]\over\alpha-\mu}-{\alpha-\mu\over 2\alpha}
{\sin[(\alpha+\mu)s]\over \alpha+\mu}$ (3)
$\displaystyle y(s)$ $\textstyle =$ $\displaystyle -{\alpha+\mu\over 2\alpha} {\cos[(\alpha-\mu)s]\over\alpha-\mu}+{\alpha-\mu\over 2\alpha}{\cos[(\alpha+\mu)s]\over \alpha+\mu}$  
$\displaystyle z(s)$ $\textstyle =$ $\displaystyle {\omega\over\mu\alpha}\sin(\mu s),$ (5)

\end{displaymath} (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}.
\end{displaymath} (7)

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


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

© 1996-9 Eric W. Weisstein