# Cycloid

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The plane transcendental curve that is the trajectory of a point of a circle rolling along a straight line (Fig. a). Figure: c027540a

The parametric equations are:  where is the radius of the circle and the angle of rotation of the circle. In Cartesian coordinates the equation is: A cycloid is a periodic curve: the period (basis) is . The points , are cusps. The points and are the so-called vertices. The area is , the radius of curvature is .

If the curve is described by a point lying outside (inside) a circle rolling along a line, then it is called an extended, (or elongated, Fig. b), a contracted, (or shortened, Fig. c) cycloid or sometimes a trochoid. Figure: c027540b Figure: c027540c

The parametric equations are  where is the distance of the point from the centre of the rolling circle.

The cycloid is a tautochronic (or isochronic) curve, that is, a curve for which the time of descent of a material point along this curve from a certain height under the action of gravity does not depend on the original position of the point on the curve.