Trochoid
A trochoid (from the Greek word for wheel, "trochos") is the curve described by a fixed point on a circle as it rolls along a straight line. The cycloid is a notable member of the trochoid family. The word "trochoid" was coined by Gilles de Roberval.
As a circle of radius a rolls without slipping along a line L, the center C moves parallel to L, and every other point P in the rotating plane rigidly attached to the circle traces the curve called the trochoid. Let CP = b. Parametric equations of the trochoid for which L is the x-axis are
where θ is the variable angle through which the circle rolls.
If P lies inside the circle (b < a), on its circumference (b = a), or outside (b > a), the trochoid is described as being curtate ("contracted"), common, or prolate ("extended"), respectively. A curtate trochoid is traced by a pedal when a normally geared bicycle is pedaled along a straight line. A prolate trochoid is traced by the tip of a paddle when a boat is driven with constant velocity by paddle wheels; this curve contains loops. A common trochoid, also called a cycloid, has cusps at the points where P touches the L.
A more general approach would define a trochoid as the locus of a point orbiting at a constant rate around an axis located at ,
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