Trochoid
In geometry, a trochoid (from the Greek word for wheel, "trochos") is a roulette 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. If the point is on the circle, the trochoid is called "common" (also known as a cycloid); if the point is inside the circle, the trochoid is "curtate"; and if the point is outside the circle, the trochoid is "prolate". The word "trochoid" was coined by  Gilles de Roberval. Basic description. 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. Curtate, common, prolate. 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". General description. A more general approach would define a trochoid as the locus of a point formula_3 orbiting at a constant rate around an axis located at formula_4, 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 formula_7 (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 straight path, one full rotation coincides with one period of a periodic (repeating) locus. In the case of a circular path for the moving axis, the locus is periodic only if the ratio of these angular motions, formula_9, is a rational  number, say formula_10, where formula_11 & formula_12 are coprime, in which case, one period consists of formula_11 orbits around the moving axis and formula_12 orbits of the moving axis around the point formula_7. The special cases of the epicycloid and hypocycloid, generated by tracing the locus of a point on the perimeter of a circle of radius formula_16 while it is rolled on the perimeter of a stationary circle of radius formula_17, have the following properties: where formula_19 is the radius of the orbit of the moving axis. The number of cusps given above also hold true for any epitrochoid and  hypotrochoid, with "cusps" replaced by either "radial maxima" or "radial minima".