Equations
The cycloid through the origin, generated by a circle of radius r, consists of the points (x, y), with
where t is a real parameter, corresponding to the angle through which the rolling circle has rotated, measured in radians. For given t, the circle's centre lies at x = rt, y = r.
Solving for t and replacing, the Cartesian equation would be
The first arch of the cycloid consists of points such that
When y is viewed as a function of x, the cycloid is differentiable everywhere except at the cusps where it hits the x-axis, with the derivative tending toward or as one approaches a cusp. The map from t to (x, y) is a differentiable curve or parametric curve of class C∞ and the singularity where the derivative is 0 is an ordinary cusp.
The cycloid satisfies the differential equation:
Read more about this topic: Cycloid