Cycloid

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.

References

[a1]	J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)