Trochoid

In geometry, a trochoid () 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. 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. 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 = 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 (relative to the ground) when a normally geared bicycle is pedaled along a straight line. 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 straight path, one full rotation coincides with one period of a periodic (repeating) locus.