Trochoid

In geometry, a trochoid (from Greek trochos 'wheel') is a roulette curve formed by a circle rolling along a line.

It is the curve traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle) as it rolls along a straight line.

The word "trochoid" was coined by Gilles de Roberval, referring to the special case of a cycloid.

[2] 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.

[4] A prolate trochoid is traced by the tip of a paddle (relative to the water's surface) 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 line 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

, which axis is being translated in the x-y-plane at a constant rate in either a straight line, or a circular path (another orbit) around

(the hypotrochoid/epitrochoid case), The ratio of the rates of motion and whether the moving axis translates in a straight or circular path determines the shape of the trochoid.

In the case of a circular path for the moving axis, the locus is periodic only if the ratio of these angular motions,

The special cases of the epicycloid and hypocycloid, generated by tracing the locus of a point on the perimeter of a circle of radius

while it is rolled on the perimeter of a stationary circle of radius

The number of cusps given above also hold true for any epitrochoid and hypotrochoid, with "cusps" replaced by either "radial maxima" or "radial minima".