  A popular classroom demonstration is to draw a cycloid on a blackboard with a
piece of chalk inserted through a hole at a point P with radius r = R from the
center of a wood disk of radius R that is rolling without slipping along the
chalk tray of the blackboard. Here the parametric equations versus time are
derived for the path of P from the superposition of the translational motion of
the center of mass (cm) of the disk and the rotational motion of P about this
cm for r = R (cycloid), r < R (curtate cycloid) and r > R (prolate cycloid). It
is further shown that the path of P is still a cycloidal function for rolling
with frictionless slipping, but where the time dependence of the sinusoidal
Cartesian coordinates of the position of P is modified. In a similar way the
parametric equations versus time for the orbit with respect to a star of a moon
in a circular orbit about a planet that is in a circular orbit about a star are
derived, where the orbits are coplanar. Finally, the general parametric
equations versus time for the path of the magnetization vector during undamped
electron-spin resonance are found, which show that cycloidal paths can occur
under certain conditions.
