Use this tag for questions about the curve traced by a point on a circle as it rolls along a straight line.
A cycloid is the curve traced by a point on a circle as it rolls along a straight line. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve.
The cycloid, with the cusps pointing upward, is the curve of fastest descent under constant gravity and is also the form of a curve for which the period of an object in descent on the curve does not depend on the object's starting position.
The cycloid through the origin generated by a circle of radius r rolling on top of the x-axis consists of points (x, y) such that
x = r (t − sin t)
y = r (1 − cos t)
where t is a real parameter corresponding to the angle through which the rolling circle has rotated. For given t, the circle's center is at x = rt, y = r.
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 smooth, and the singularity where the derivative is zero is an ordinary cusp.
A cycloid segment from one cusp to the next is called an arch of the cycloid. The first arch of the cycloid consists of points such that 0 ≤ t ≤ 2 π.
The cycloid satisfies the differential equation (dy/dx)² = 2 r/y − 1.
